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** Prospective Graduate Students: ** The applied mathematics group welcomes you to apply to the graduate program. Applied mathematics offers many opportunities for exciting graduate research making connections between core areas of mathematics and important problems arising in applications. At UCSB there are also many exciting research opportunities to conduct interdisciplinary research at the interface of mathematics with fields from the sciences, engineering, and computation. To learn more about individual research areas please see the above faculty webpages.

** Prospective Graduate Students: ** The applied mathematics group welcomes you to apply to the graduate program. Applied mathematics offers many exciting opportunities for graduate research that makes connections between core areas of mathematics and important problems arising in applications. At UCSB there are also many exciting research opportunities to conduct interdisciplinary research at the interface of mathematics with fields from the sciences, engineering, and computation. To learn more about individual research areas please see the above faculty webpages.

** Apply for graduate studies in applied mathematics. ** : [more information here] \\

** Apply for graduate studies in applied mathematics. ** [more information here] \\

** Apply for graduate studies in Applied Mathematics. ** : [more information here] \\

** Apply for graduate studies in applied mathematics. ** : [more information here] \\

** Apply for graduate studies in Applied Mathematics ** : [more information here] \\

** Apply for graduate studies in Applied Mathematics. ** : [more information here] \\

** Apply for Graduate Studies in Applied Mathematics ** : [more information here] \\

** Apply for graduate studies in Applied Mathematics ** : [more information here] \\

** Apply for Graduate Studies in Applied Mathematics ** : [more information here]? \\

** Apply for Graduate Studies in Applied Mathematics ** : [more information here] \\

## Apply for Graduate Studies in Applied Mathematics : [more information here]

** Apply for Graduate Studies in Applied Mathematics ** : [more information here]?

Applied mathematics integrates the development of core areas of mathematics with the solution of specific problems arising in applications, often in the basic sciences or engineering. Faculty of the Applied Mathematics Group are active in diverse areas and participate in collaborations with many faculty on campus. Research areas include:

- Complex Fluids and Soft-Condensed Matter Physics.
- Crystalline Solids and Liquid Crystals.
- Computational Fluid Dynamics.
- Density Functional Theory.
- Analysis of Non-linear Evolutionary PDE's (existence results / finite time singularities).
- Applied Harmonic Analysis.
- Stochastic Analysis.

Here you will find information about our program in Applied Mathematics, current activities, upcoming seminar talks, and highlights from recent research.

## News

Paul J. Atzberger Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Paul J. Atzberger awarded NSF CAREER Award "Emergent Biological Mechanics of Cellular Microstructures." His proposed research aims to develop new methods combining approaches from stochastic analysis, statistical mechanics, and scientific computing to study fundamental problems related to the mechanics of biological materials. This $435K, five year grant "recognizes and supports the early career development activities of those faculty members who are most likely to become the academic leaders of the 21st century.” Significantly, this proposal will be funded by three NSF agencies: Mathematical Biology, Applied Mathematics, and the Office of Cyberinfrastructure.

Applied mathematics integrates the development of core areas of mathematics with the solution of problems arising in diverse application areas including the natural sciences, engineering, and computation. Faculty of applied mathematics are active in many areas. More information on the specific research interests of individual faculty can be found below.

## Associated Faculty:

- Paul J. Atzberger
- [webpage].

- Bjorn Birnir
- [webpage]

- Hector Ceniceros
- [webpage]

- Katy C. Craig
- [webpage].

- Carlos Garcia-Cervera
- [webpage]

- Davit Harutyunyan
- [webpage].

- Maria Isabel Bueno Cachadina
- [webpage].

- Christopher Ograin
- [webpage].

- Gustavo Ponce
- [webpage].

- Thomas Sideris
- [webpage].

- Xu Yang
- [webpage].

- Hanming Zhou
- [webpage].

** Applied Mathematics Seminar: **
A seminar is held on topics in applied mathematics and analysis. For more information on the schedule of upcoming talks, please see the calendar on the Department of Mathematics homepage. \\

** Prospective Graduate Students: ** The applied mathematics group welcomes you to apply to the graduate program. Applied mathematics offers many opportunities for exciting graduate research making connections between core areas of mathematics and important problems arising in applications. At UCSB there are also many exciting research opportunities to conduct interdisciplinary research at the interface of mathematics with fields from the sciences, engineering, and computation. To learn more about individual research areas please see the above faculty webpages.

Please feel free to reach out to the faculty and staff with any questions you may have about the graduate program. General information about the Mathematics Program at UCSB and important deadlines can be found on the Department of Mathematics homepage.

More information about Professor Atzberger's research be found on his website. \\

Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".
The Faculty Early Career Development (CAREER) Program offers the NSF’s most prestigious awards in support of early career development activities of those teacher-scholars who are most likely to become the academic leaders of the 21st century.
The awards are for a minimum of $400,000 and support his research for five years providing funding
for postdoctoral researchers and graduate students. The proposed research program has the potential
to impact fundamental computational approaches used in studying solid materials. This is the first
NSF CAREER award given to a faculty member of the department of mathematics.

Full Article

More information about Professor Garcia-Cervera's research be found on his website.

Math Circle Started at UCSB

A Math Circle has been started at UCSB for outreach to the local community. The Math Circle engages in mathematics pre-college
students from local high schools. The UCSB Math Circle offers a forum for the discussion of mathematical topics, mathematics education,
and careers involving mathematics. From the Applied Mathematics Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle Website:

UCSB Math Circle Website.

## Research Highlights

Fluid-Structure Interactions : Immersed Boundary Methods and Boundary Integral Methods

The mechanics of many physical systems depends importantly on the interaction of flexible elastic structures with a fluid flow. Examples of macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. Examples of microscopic systems include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and features hard to predict a priori. Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems both for analysis and numerics.

Faculty members working in this area include:

- Dr. Atzberger : Prof. Atzberger has done work on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method and applications can be found here [Video of IMA Talk].
- Dr. Ceniceros : Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found most likely here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

Local and Global Wellposedness of Nonlinear Evolutionary Equations

Provide conditions on initial data which ensure the existence, uniqueness, and continuous dependence of solutions to the initial value problem for nonlinear evolutionary partial differential equations. Determine whether solutions exist globally in time or develop singularities in finite time. Explore the regularity and asymptotic behavior of solutions. Applications to nonlinear dispersive equations, hydro- and elasto-dynamics.

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

## Research Gallery

## Affiliated Faculty Include:

## Applied Mathematics and PDE Seminar

For more information, please see the above faculty webpages.

Here you will find information about our program in Applied Mathematics, current activities, upcoming seminar talks, and highlights from recent research.

## News

Paul J. Atzberger Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Paul J. Atzberger awarded NSF CAREER Award "Emergent Biological Mechanics of Cellular Microstructures." His proposed research aims to develop new methods combining approaches from stochastic analysis, statistical mechanics, and scientific computing to study fundamental problems related to the mechanics of biological materials. This $435K, five year grant "recognizes and supports the early career development activities of those faculty members who are most likely to become the academic leaders of the 21st century.” Significantly, this proposal will be funded by three NSF agencies: Mathematical Biology, Applied Mathematics, and the Office of Cyberinfrastructure.

More information about Professor Atzberger's research be found on his website.

Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".
The Faculty Early Career Development (CAREER) Program offers the NSF’s most prestigious awards in support of early career development activities of those teacher-scholars who are most likely to become the academic leaders of the 21st century.
The awards are for a minimum of $400,000 and support his research for five years providing funding
for postdoctoral researchers and graduate students. The proposed research program has the potential
to impact fundamental computational approaches used in studying solid materials. This is the first
NSF CAREER award given to a faculty member of the department of mathematics.

Full Article

More information about Professor Garcia-Cervera's research be found on his website.

Math Circle Started at UCSB

A Math Circle has been started at UCSB for outreach to the local community. The Math Circle engages in mathematics pre-college
students from local high schools. The UCSB Math Circle offers a forum for the discussion of mathematical topics, mathematics education,
and careers involving mathematics. From the Applied Mathematics Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle Website:

UCSB Math Circle Website.

## Research Highlights

Fluid-Structure Interactions : Immersed Boundary Methods and Boundary Integral Methods

The mechanics of many physical systems depends importantly on the interaction of flexible elastic structures with a fluid flow. Examples of macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. Examples of microscopic systems include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and features hard to predict a priori. Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems both for analysis and numerics.

Faculty members working in this area include:

- Dr. Atzberger : Prof. Atzberger has done work on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method and applications can be found here [Video of IMA Talk].
- Dr. Ceniceros : Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found most likely here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

Local and Global Wellposedness of Nonlinear Evolutionary Equations

Provide conditions on initial data which ensure the existence, uniqueness, and continuous dependence of solutions to the initial value problem for nonlinear evolutionary partial differential equations. Determine whether solutions exist globally in time or develop singularities in finite time. Explore the regularity and asymptotic behavior of solutions. Applications to nonlinear dispersive equations, hydro- and elasto-dynamics.

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

## Research Gallery

Here you will find information about our program in Applied Mathematics, current activities, upcoming seminar talks, and highlights from recent research.

## News

Paul J. Atzberger Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Paul J. Atzberger awarded NSF CAREER Award "Emergent Biological Mechanics of Cellular Microstructures." His proposed research aims to develop new methods combining approaches from stochastic analysis, statistical mechanics, and scientific computing to study fundamental problems related to the mechanics of biological materials. This $435K, five year grant "recognizes and supports the early career development activities of those faculty members who are most likely to become the academic leaders of the 21st century.” Significantly, this proposal will be funded by three NSF agencies: Mathematical Biology, Applied Mathematics, and the Office of Cyberinfrastructure.

More information about Professor Atzberger's research be found on his website.

Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".
The Faculty Early Career Development (CAREER) Program offers the NSF’s most prestigious awards in support of early career development activities of those teacher-scholars who are most likely to become the academic leaders of the 21st century.
The awards are for a minimum of $400,000 and support his research for five years providing funding
for postdoctoral researchers and graduate students. The proposed research program has the potential
to impact fundamental computational approaches used in studying solid materials. This is the first
NSF CAREER award given to a faculty member of the department of mathematics.

Full Article

More information about Professor Garcia-Cervera's research be found on his website.

Math Circle Started at UCSB

A Math Circle has been started at UCSB for outreach to the local community. The Math Circle engages in mathematics pre-college
students from local high schools. The UCSB Math Circle offers a forum for the discussion of mathematical topics, mathematics education,
and careers involving mathematics. From the Applied Mathematics Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle Website:

UCSB Math Circle Website.

## Research Highlights

Fluid-Structure Interactions : Immersed Boundary Methods and Boundary Integral Methods

The mechanics of many physical systems depends importantly on the interaction of flexible elastic structures with a fluid flow. Examples of macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. Examples of microscopic systems include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and features hard to predict a priori. Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems both for analysis and numerics.

Faculty members working in this area include:

- Dr. Atzberger : Prof. Atzberger has done work on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales and Stochastic Eulerian-Langrain Methods for Fluid-Structure Interactions subject to Thermal Fluctuations..
- Dr. Ceniceros : Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found most likely here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

Local and Global Wellposedness of Nonlinear Evolutionary Equations

Provide conditions on initial data which ensure the existence, uniqueness, and continuous dependence of solutions to the initial value problem for nonlinear evolutionary partial differential equations. Determine whether solutions exist globally in time or develop singularities in finite time. Explore the regularity and asymptotic behavior of solutions. Applications to nonlinear dispersive equations, hydro- and elasto-dynamics.

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

## Research Gallery

## Affiliated Faculty Include:

## Applied Mathematics and PDE Seminar

For more information, please see the above faculty webpages.

- Dr. Atzberger : Prof. Atzberger has done work on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method and applications can be found here [Video of IMA Talk].

- Dr. Atzberger : Prof. Atzberger has done work on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales and Stochastic Eulerian-Langrain Methods for Fluid-Structure Interactions subject to Thermal Fluctuations..

Kozato Postdoctoral Fellowship in Quantitative Biology

The University of California, Santa Barbara invites applications for the Kozato Postdoctoral Fellowship in Quantitative Biology. The fellowship will support research at the interface of the biological sciences with mathematics, computation, physics, or engineering. The postdoctoral appointment will be within the Department of Mathematics starting in September 2011, and is renewable for up to three years. UCSB offers a strong interdisciplinary environment for research with many opportunities for close interaction with both theoreticians and experimentalists on campus.

For more information see: Kozato Fellowship in Quantitative Biology.

\\

Kozato Graduate Fellowship in Quantitative Biology

Kozato Postdoctoral Fellowship in Quantitative Biology

For more information:

Kozato Fellowship in Quantitative Biology\\

For more information see: Kozato Fellowship in Quantitative Biology.\\

The Kozato Fellowship in Quantitative Biology will offer competitive multi-year support comparable to the NSF Graduate Fellowship. The position is to begin in the Fall of 2011. The fellowship will support a student who has an interest in working on an interdisciplinary thesis project investigating a biological system using a combination of mathematical analysis and computational methods. It is envisioned the supported graduate student would have a primary adviser in mathematics, but would also interact closely with theoreticians and experimental biologists on campus at UCSB. The fellowship is funded by a generous donation from Hiro Kozato, a distinguished alumnus of the Department of Mathematics.

For those students interested in mathematics and biology, please see the following page for more information:

The University of California, Santa Barbara invites applications for the Kozato Postdoctoral Fellowship in Quantitative Biology. The fellowship will support research at the interface of the biological sciences with mathematics, computation, physics, or engineering. The postdoctoral appointment will be within the Department of Mathematics starting in September 2011, and is renewable for up to three years. UCSB offers a strong interdisciplinary environment for research with many opportunities for close interaction with both theoreticians and experimentalists on campus.

For more information:

Kozato Graduate Fellowship in Quantitative Biology

The Kozato Fellowship in Quantitative Biology will offer competitive multi-year support comparable to the NSF Graduate Fellowship. The position is to begin in the Fall of 2011. The fellowship will support a student who has an interest in working on an interdisciplinary thesis project investigating a biological system using a combination of mathematical analysis and computational methods. It is envisioned the supported graduate student would have a primary adviser in mathematics, but would also interact closely with theoreticians and experimental biologists on campus at UCSB. The fellowship is funded by a generous donation from Hiro Kozato, a distinguished alumnus of the Department of Mathematics.

For those students interested in mathematics and biology, please see the following page for more information:

Kozato Fellowship in Quantitative Biology

Kozato Graduate Fellowship in Quantitative Biology

The Kozato Fellowship in Quantitative Biology will offer competitive multi-year support comparable to the NSF Graduate Fellowship. The position is to begin in the Fall of 2011. The fellowship will support a student who has an interest in working on an interdisciplinary thesis project investigating a biological system using a combination of mathematical analysis and computational methods. It is envisioned the supported graduate student would have a primary adviser in mathematics, but would also interact closely with theoreticians and experimental biologists on campus at UCSB. The fellowship is funded by a generous donation from Hiro Kozato, a distinguished alumnus of the Department of Mathematics.

For those students interested in mathematics and biology, please see the following page for more information:

Kozato Fellowship in Quantitative Biology

\\

Kozato Graduate Fellowship in Quantitative Biology

For those students interested in mathematics and biology, please see the following page for more information:

Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".
The Faculty Early Career Development (CAREER) Program offers the NSF’s most prestigious awards in support of early career development activities of those teacher-scholars who are most likely to become the academic leaders of the 21st century.
The awards are for a minimum of $400,000 and support his research for five years providing funding
for postdoctoral researchers and graduate students. The proposed research program has the potential
to impact fundamental computational approaches used in studying solid materials. This is the first
NSF CAREER award given to a faculty member of the department of mathematics.

Full Article

Kozato Fellowship in Quantitative Biology\\

More information about Professor Garcia-Cervera's research be found on his website.

Kozato Graduate Fellowship in Quantitative Biology

For those students interested in mathematics and biology, please see the following page for more information:

Kozato Fellowship in Quantitative Biology

Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

Full Article

More information about Professor Garcia-Cervera's research be found on his website.

For those students interested in mathematics and biology and this fellowship, please see the following webpage for more information:

For those students interested in mathematics and biology, please see the following page for more information:

Advisory Panel:

Paul Atzberger, Department of Mathematics.

Frank Brown, Department of Chemistry and Biochemistry.

Hector Ceniceros, Department of Mathematics.

Mustafa Kummash, Department of Mechanical Engineering.

Everett Lipman, Department of Physics.

Omar Saleh, Department of Materials.

Megan Valentine, Department of Mechanical Engineering.

Kozato Graduate Fellowship in Quantitative Biology

Advisory Panel:

Paul Atzberger, Department of Mathematics. Frank Brown, Department of Chemistry and Biochemistry. Hector Ceniceros, Department of Mathematics. Mustafa Kummash, Department of Mechanical Engineering. Everett Lipman, Department of Physics. Omar Saleh, Department of Materials. Megan Valentine, Department of Mechanical Engineering.

Advisory Panel:

Paul Atzberger, Department of Mathematics.

Frank Brown, Department of Chemistry and Biochemistry.

Hector Ceniceros, Department of Mathematics.

Mustafa Kummash, Department of Mechanical Engineering.

Everett Lipman, Department of Physics.

Omar Saleh, Department of Materials.

Megan Valentine, Department of Mechanical Engineering.\\

For those students interested in mathematics and biology and this fellowship, please see the following webpage for more information:

Kozato Fellowship in Quantitative Biology

Advisory Panel:

Paul Atzberger, Department of Mathematics. Frank Brown, Department of Chemistry and Biochemistry. Hector Ceniceros, Department of Mathematics. Mustafa Kummash, Department of Mechanical Engineering. Everett Lipman, Department of Physics. Omar Saleh, Department of Materials. Megan Valentine, Department of Mechanical Engineering.

## Apply for Graduate Studies in Applied Mathematics [more information here] \\

## Apply for Graduate Studies in Applied Mathematics : [more information here] \\

## Apply for Graduate Studies in Applied Mathematics [more info. here] \\

## Apply for Graduate Studies in Applied Mathematics [more information here] \\

Professor Paul J. Atzberger awarded NSF CAREER Award "Emergent Biological Mechanics of Cellular Microstructures." His proposed research aims to develop new methods combining approaches from stochastic analysis, statistical mechanics, and scientific computing to study fundamental problems related to the mechanics of biological materials. This $435K, five year grant "recognizes and supports the early career development activities of those faculty members who are most likely to become the academic leaders of the 21st century.” Significantly, this proposal will be funded by three NSF agencies: Mathematical Biology, Applied Mathematics, and the Office of Cyberinfrastructure.\\

Professor Paul J. Atzberger awarded NSF CAREER Award "Emergent Biological Mechanics of Cellular Microstructures." His proposed research aims to develop new methods combining approaches from stochastic analysis, statistical mechanics, and scientific computing to study fundamental problems related to the mechanics of biological materials. This $435K, five year grant "recognizes and supports the early career development activities of those faculty members who are most likely to become the academic leaders of the 21st century.” Significantly, this proposal will be funded by three NSF agencies: Mathematical Biology, Applied Mathematics, and the Office of Cyberinfrastructure. \\

More information about Professor Atzberger's research be found on his homepage. \\

More information about Professor Atzberger's research be found on his website. \\

More information about Professor Garcia-Cervera's research be found on his homepage. \\

More information about Professor Garcia-Cervera's research be found on his website. \\

More information about Professor Atzberger's research be found at http://www.math.ucsb.edu/~atzberg/index.html. \\

More information about Professor Atzberger's research be found on his homepage. \\

More information about Professor Garcia-Cervera's research be found at http://www.math.ucsb.edu/~cgarcia/index.html. \\

More information about Professor Garcia-Cervera's research be found on his homepage. \\

More information about Professor Atzberger's research be found at http://www.math.ucsb.edu/~atzberg/index.html.

More information about Professor Garcia-Cervera's research be found at http://www.math.ucsb.edu/~cgarcia/index.html.

Paul J. Atzberger Wins NSF Faculty Early Career Development Award (NSF CAREER)

Professor Paul J. Atzberger awarded NSF CAREER Award "Emergent Biological Mechanics of Cellular Microstructures." His proposed research aims to develop new methods combining approaches from stochastic analysis, statistical mechanics, and scientific computing to study fundamental problems related to the mechanics of biological materials. This $435K, five year grant "recognizes and supports the early career development activities of those faculty members who are most likely to become the academic leaders of the 21st century.” Significantly, this proposal will be funded by three NSF agencies: Mathematical Biology, Applied Mathematics, and the Office of Cyberinfrastructure.

Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".

Professor Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".

## Apply for Graduate Studies in Applied Mathematics at UCSB [more info. here] \\

## Apply for Graduate Studies in Applied Mathematics [more info. here] \\

## Apply to the UCSB Applied Mathematics Program [more info. here] \\

## Apply for Graduate Studies in Applied Mathematics at UCSB [more info. here] \\

## Apply to the UCSB Applied Mathematics Program [more info. here] \\

and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle Website:\\

and careers involving mathematics. From the Applied Mathematics Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle Website:\\

- complex fluids and soft-condensed matter
- crystalline solids and liquid crystals
- density functional theory
- analysis of non-linear evolutionary PDE's (existence results / finite time singularities)
- applied harmonic analysis
- stochastic analysis.

- Complex Fluids and Soft-Condensed Matter Physics.
- Crystalline Solids and Liquid Crystals.
- Density Functional Theory.
- Analysis of Non-linear Evolutionary PDE's (existence results / finite time singularities).
- Applied Harmonic Analysis.
- Stochastic Analysis.

and participate in collaborations with a diverse collection of faculty on campus. Research areas include: complex fluids, soft-condensed matter, crystalline solids and liquid crystals, density functional theory, analysis of non-linear evolutionary PDE's (existence results / finite time singularities), applied harmonic analysis, and stochastic analysis.

and participate in collaborations with many faculty on campus. Research areas include:

- complex fluids and soft-condensed matter
- crystalline solids and liquid crystals
- density functional theory
- analysis of non-linear evolutionary PDE's (existence results / finite time singularities)
- applied harmonic analysis
- stochastic analysis.

and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle website:\\

and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle Website:\\

students from local high schools. The Math Circle offers a forum in which the discuss mathematical topics, mathematics education,

students from local high schools. The UCSB Math Circle offers a forum for the discussion of mathematical topics, mathematics education,

UCSB Math Circle Website.

UCSB Math Circle Website.

Math Circle Started at UCSB

A Math Circle has been started at UCSB for outreach to the local community. The Math Circle engages in mathematics pre-college
students from local high schools. The Math Circle offers a forum in which the discuss mathematical topics, mathematics education,
and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle website:

UCSB Math Circle Website.

UCSB Math Circle

A Math Circle has been started at UCSB as outreach to the local community. The Math Circle engages in mathematics pre-college
students from local high schools. The Math Circle offers a forum in which the discuss mathematical topics, mathematics education,
and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle website:

UCSB Math Circle Website.

and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in
organizing the UCSB Math Circle. For more information see the Math Circle website:

.

and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in organizing the UCSB Math Circle. For more information see the Math Circle website:

UCSB Math Circle Website.

and careers involving mathematics. Maribel ... from the Applied Mathematics Group is play a leading role in organizing the UCSB Math Circle.
For more information see the Math Circle website:

and careers involving mathematics. From the Applied Math Group, Maribel Bueno Cachadina is playing a leading role in
organizing the UCSB Math Circle. For more information see the Math Circle website:

.

UCSB Math Circle

A Math Circle has been started at UCSB as outreach to the local community. The Math Circle engages in mathematics pre-college
students from local high schools. The Math Circle offers a forum in which the discuss mathematical topics, mathematics education,
and careers involving mathematics. Maribel ... from the Applied Mathematics Group is play a leading role in organizing the UCSB Math Circle.
For more information see the Math Circle website:

and participate in collaborations with many departments on campus.

and participate in collaborations with a diverse collection of faculty on campus.

Applied mathematics strives to integrate the development of core areas of mathematics with

Applied mathematics integrates the development of core areas of mathematics with

Applied mathematics refers to the branch of mathematics which strives to integrate the development of core areas of mathematics with

Applied mathematics strives to integrate the development of core areas of mathematics with

and participate in many collaborations with other departments on campus.

and participate in collaborations with many departments on campus.

Another basic test of notifications....

Another basic test of notifications....

Test of the notification system to make sure it works. Will send e-mail summarizing changes made for the day, if any.

Test of the notification system to make sure it works. Will send e-mail summarizing changes made for the day, if any.

and participate in many collaborations with faculty from other departments on campus.

and participate in many collaborations with other departments on campus.

or engineering. Faculty of the Applied Mathematics Group are active in many areas including: complex fluids, soft-condensed matter, crystalline solids and

or engineering. Faculty of the Applied Mathematics Group are active in diverse areas and participate in many collaborations with faculty from other departments on campus. Research areas include: complex fluids, soft-condensed matter, crystalline solids and

applied harmonic analysis, and stochastic analysis. Here you will find information about our program in Applied Mathematics, current activities,

applied harmonic analysis, and stochastic analysis.

Here you will find information about our program in Applied Mathematics, current activities,

## Research Highlights (select subset of recent activities)

## Research Highlights

or engineering. Faculyt of the Applied Mathematics Group are active in many areas

or engineering. Faculty of the Applied Mathematics Group are active in many areas

Welcome to the applied mathematics group's homepage. Applied mathematics refers to the branch of mathematics which strives

Applied mathematics refers to the branch of mathematics which strives

or engineering. Specific areas in which faculty are involved include the study of complex fluids / soft-condensed matter, computational fluid dynamics, boundary integral methods, immersed boundary methods, crystalline solids and liquid crystals, density functional theory, analysis of non-linear evolutionary PDE's, study of existence or development of singularities in finite time, applied harmonic analysis, and stochastic differential equations. Here you will find research highlights, upcoming seminar talks, and information concerning recent news or current activities of the applied mathematics group.

or engineering. Faculyt of the Applied Mathematics Group are active in many areas including: complex fluids, soft-condensed matter, crystalline solids and liquid crystals, density functional theory, analysis of non-linear evolutionary PDE's (existence results / finite time singularities), applied harmonic analysis, and stochastic analysis. Here you will find information about our program in Applied Mathematics, current activities, upcoming seminar talks, and highlights from recent research.

## Research Gallery

Local and Global Wellposedness of Nonlinear Evolutionary Equations

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

Local and Global Wellposedness of Nonlinear Evolutionary Equations

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

## Research Gallery

- Dr. Atzberger : Prof. Atzberger has done work on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk].

- Dr. Ceniceros : Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk].

- Dr. Atzberger : Prof. Atzberger has done work on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk].

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Additional papers can be found on his research website.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk].

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. Application areas include polymeric fluids, gels, and lipid bilayer membranes. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here :

A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Additional papers can be found on his research website.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here : A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Additional papers can be found on his research website.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. Application areas include polymeric fluids, gels, and lipid bilayer membranes. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. Application areas include polymeric fluids, gels, and lipid bilayer membranes. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A paper on the methodology can be found here :

A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Additional papers can be found on his research website.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. Applications of the new methodology include the study of polymeric fluids, gels, and lipid bilayer membranes. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. Application areas include polymeric fluids, gels, and lipid bilayer membranes. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger uses immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. His work is also concerned with applications of the methodology, in particular, to the study of polymeric fluids, gels, and lipid bilayer membranes. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger has been working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the IBM formalism to incorporate stochastic driving fields consistent with statistical mechanics. Applications of the new methodology include the study of polymeric fluids, gels, and lipid bilayer membranes. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. His work is also concerned with applications of the methodology, in particular, to the study of polymeric fluids, gels, and lipid bilayer membranes. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger uses immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. His work is also concerned with applications of the methodology, in particular, to the study of polymeric fluids, gels, and lipid bilayer membranes. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and

thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. His work is also concerned with applications of the methodology, in particular, to the study of polymeric fluids, gels, and lipid bilayer membranes. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. His work is also concerned with applications of the methodology, in particular, to the study of polymeric fluids, gels, and lipid bilayer membranes. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary methods to study properties of soft matter materials, which contain hydrodynamically coupled microstructures subject to thermal fluctuations. To account for thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields which are consistent with statistical mechanics. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary methods to study properties of soft condensed matter. To account consistently for microstructure mechanics, hydrodynamic coupling, and

thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields consistent with statistical mechanics. The resulting stochastic partial differential equations (SPDEs) present many interesting challenges both for analysis and the development of numerical methods. His work is also concerned with applications of the methodology, in particular, to the study of polymeric fluids, gels, and lipid bilayer membranes. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

the solution of specific problems arising in applications often in the basic sciences

the solution of specific problems arising in applications, often in the basic sciences

about news and current activities of the applied mathematics group.

concerning recent news or current activities of the applied mathematics group.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary

methods to study properties of soft matter materials, which contain hydrodynamically coupled microstructures subject to thermal fluctuations. To account for thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields which are consistent with statistical mechanics. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary methods to study properties of soft matter materials, which contain hydrodynamically coupled microstructures subject to thermal fluctuations. To account for thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields which are consistent with statistical mechanics. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

methods to study properties of soft matter materials, which contain hydrodynamically coupled microstructures subject to thermal fluctuations. To account for thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields which are consistent with statistical mechanics. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

methods to study properties of soft matter materials, which contain hydrodynamically coupled microstructures subject to thermal fluctuations. To account for thermal fluctuations he has extended the immersed boundary method to incorporate stochastic driving fields which are consistent with statistical mechanics. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Atzberger : Prof. Atzberger is working on utilizing immersed boundary

- Dr. Atzberger : Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his website publications.

- Dr. Atzberger : Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his Research Website.

- Dr. Ceniceros) : Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

- Dr. Ceniceros : Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

- Dr. Atzberger (Research Website) Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his website publications.
- Dr. Ceniceros (Research Website) Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

- Dr. Atzberger : Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his website publications.
- Dr. Ceniceros) : Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his website publications.
- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

- Dr. Atzberger (Research Website) Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his website publications.
- Dr. Ceniceros (Research Website) Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods. Related papers can be found here.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here [Video of IMA Talk]. Related papers can be found on his website publications.

## Research Highlights (select subset of activities)

## Research Highlights (select subset of recent activities)

## Research Highlights (select subset of activities)

## Research Highlights (select subset of activities)

## Local and Global Wellposedness of Nonlinear Evolutionary Equations

Local and Global Wellposedness of Nonlinear Evolutionary Equations

## Fluid-Structure Interactions : Immersed Boundary Methods and Boundary Integral Methods

Fluid-Structure Interactions : Immersed Boundary Methods and Boundary Integral Methods

# Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

# Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Research Gallery

## Research Gallery

## Research Highlights (select subset of activities)

## Research Highlights (select subset of activities)

## %color$555555%Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## %color$555555%Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

News

## News

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Research Highlights

News

## Research Areas (select subset of activities)

## Research Highlights (select subset of activities)

## Research Areas (select subset of recent activities)

## Research Areas (select subset of activities)

## Research Areas (selected subset of activities)

## Research Areas (select subset of recent activities)

## Fluid-Structure Dynamics : Immersed Boundary Methods and Boundary Integral Methods

## Fluid-Structure Interactions : Immersed Boundary Methods and Boundary Integral Methods

- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method

## Liquid Crystals / Lipid Bilayer Membranes / Complex Polymeric Fluids

Brief article outlining basic area of research, interesting math. issues...

Faculty members working in this area include:

- Dr. Garcia-Cervera : Research Website.
- Dr. Ceniceros : Research Website.
- Dr. Atzbeger : Research Website.

- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method.

## Research Areas (Selected subset of activities)

## Research Areas (selected subset of activities)

## Research Areas (Selected Subset of Activities)

## Research Areas (Selected subset of activities)

## Discussion of Above Items

Above is an experiment and rough draft / brainstorm. Idea is to give undergraduate / graduate students and general public and scientific audience sense of what types of work our group does. Above is meant to give an example of some themes and what we might post to highlight our specific research programs and on-going work both at the level of light-reading and in more detail. I wrote mostly about my own work, since I know the most about this activity presumably. :) I propose up-top we have short gallery style presentation of results for the light-of-heart. We then write in more detail a few of these short articles which discuss general thematic areas of research of the group (which of course can be modified over time to freshen things up and as our collective interests develop). This means people encounter what's new and light first, and then depending on their level of interest can scroll down for a more detailed desciption of the group.

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site which is quite nicely done http://amath.unc.edu/]. In my opinion our group runs the risk of not presenting a well-defined core and anchored research program communicated to people when they visit if we only include a collection of images with captions. Maybe there is some happy medium, please write-up some example or express your ideas.

Please take a shot at posting/editing the welcome message and some materials for discussion.

test [All pages are under-construction] Please feel free to edit and taking a shot at writing these materials. We can collectively edit over time until we get something we are all happy with.

(will work on embedding some floating boxes on the right to list in an accessible format "news items and current events", see http://www.me.ucsb.edu/ for stylistic sense of what I have in mind)

Welcome to the applied mathematics group's homepage. Applied mathematics refers to the branch of mathematics which strives to integrate the development of core areas of mathematics with the solution of

Welcome to the applied mathematics group's homepage. Applied mathematics refers to the branch of mathematics which strives to integrate the development of core areas of mathematics with the solution of

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Research Gallery \\

## Research Gallery \\

## Research Areas (Selected Subset of Activities)

## Research Areas (Selected Subset of Activities)

## Discussion of Above Items

## Discussion of Above Items

## Research Highlights

## Research Highlights

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

## Carlos Garcia-Cervera Wins NSF Faculty Early Career Development Award (NSF CAREER)

dynamics, boundary integral methods, immersed boundary methods),

dynamics, boundary integral methods, immersed boundary methods,

Department of Mathematics : South Hall [Image Credit:Statistics Homepage]

South Hall

(Image from UCSB Statistics Homepage)

Department of Mathematics : South Hall [Image Credit:Statistics Homepage]

(Image from UCSB Statistics Homepage)

(Carlos: Please finish writing this synposis.)

\\

dynamics for systems with moving boundaries (boundary integral methods / immersed boundary methods), crystalline solids and liquid crystals, density functional theory, analysis of non-linear evolutionary PDE's, study of existence or development of singularities in finite time, applied harmonic analysis, and stochastic differential equations. Here you will find research highlights, upcoming seminar talks, and information

dynamics, boundary integral methods, immersed boundary methods), crystalline solids and liquid crystals, density functional theory, analysis of non-linear evolutionary PDE's, study of existence or development of singularities in finite time, applied harmonic analysis, and stochastic differential equations. Here you will find research highlights, upcoming seminar talks, and information

dynamics (boundary integral methods / immersed boundary methods), crystalline solids and liquid crystals, density functional theory, analysis of non-linear hyperbolic PDE's (?), applied harmonic analysis, stochastic differential equations. Here you will find research highlights, upcoming seminar talks, and information about news and current activities of the applied mathematics group.

test [All pages are under-construction] Please feel free to edit and taking a shot at writing these materials. We can collectively edit over time until we get something we are all happy with.

(will work on embedding some floating boxes on the right to list in an accessible format "news items and current events", see http://www.me.ucsb.edu/ for stylistic sense of what I have in mind)

dynamics for systems with moving boundaries (boundary integral methods / immersed boundary methods), crystalline solids and liquid crystals, density functional theory, analysis of non-linear evolutionary PDE's, study of existence or development of singularities in finite time, applied harmonic analysis, and stochastic differential equations. Here you will find research highlights, upcoming seminar talks, and information about news and current activities of the applied mathematics group.

test [All pages are under-construction] Please feel free to edit and taking a shot at writing these materials. We can collectively edit over time until we get something we are all happy with.

(will work on embedding some floating boxes on the right to list in an accessible format "news items and current events", see http://www.me.ucsb.edu/ for stylistic sense of what I have in mind)

The award will support his research for five years providing funding for postdoctoral researchers and graduate students. The proposed research program has the potential to impact fundamental computational approaches used in studying solid materials taking into account important quantum effects. This is the first NSF CAREER award given to a faculty member of the department of mathematics.\\

The awards are for a minimum of $400,000 and support his research for five years providing funding for postdoctoral researchers and graduate students. The proposed research program has the potential to impact fundamental computational approaches used in studying solid materials. This is the first NSF CAREER award given to a faculty member of the department of mathematics.\\

The Faculty Early Career Development (CAREER) Program offers the NSF’s most prestigious awards in support of early career development activities of those teacher-scholars who are most likely to become the academic leaders of the 21st century.

computational approaches used in studying quantum effects in a wide range of systems from semiconductors to biological molecules. This is the first NSF CAREER award given to a faculty

computational approaches used in studying solid materials taking into account important quantum effects. This is the first NSF CAREER award given to a faculty

The award will .... and ... The proposed research program has the potential to impact... This is the first NSF CAREER award given to a faculty member of the department of mathematics.\\

The award will support his research for five years providing funding for postdoctoral researchers and graduate students. The proposed research program has the potential to impact fundamental computational approaches used in studying quantum effects in a wide range of systems from semiconductors to biological molecules. This is the first NSF CAREER award given to a faculty member of the department of mathematics.\\

Not sure if this is necessary (formal rules for editing the Wiki website):**Applied math. group wiki by-laws:**

To handle the issue of any members putting inappropriate things on
the public part of the website all content editing will be password protected (admin or
core members). To avoid abuse by any one group member.
I propose we shall have the official convention that all core members "vote"
on the "hidden" page draft content before anything is put on the
homepage (password protected by admin). This way if requested we can allow
in fairness anyone in the group to post material, but everything on-line
reflects the group consensus. Anything failing the vote and revisions process
will not be copied to the homepage [kind of like a journal peer-review process].
Basically, we will post on the hidden page content "approved PJA", "revise xyz PJA",
"do-not-approve PJA", after all members post the admin will copy... We could
also agree to authorize certain individuals (by majority vote) to have the
right to post content on-line without prior approval with provision they
remove anything to which someone objects and the majority does not support.
I am not sure this is necessary, but
these paragraphs could be copied to a formal manual on by-laws for the pages
if this is required by university to avoid any formal objections being made
to how the pages are collectively managed.

## Welcome

## Welcome

[All pages are under-construction]

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods. Related papers can be found here.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations for applications involving microscopic mechanical systems. The formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) obtained by introducing an appropriate stochastic driving field in the fluid equations. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations consistent with statistical mechanics for applications involving microscopic mechanical systems. The resulting formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) which present interesting challenges both for analysis and the development of numerical methods. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations for applications involving microscopic

mechanical systems. The formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) obtained by introducing an appropriate stochastic driving field in the fluid equations. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations for applications involving microscopic mechanical systems. The formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) obtained by introducing an appropriate stochastic driving field in the fluid equations. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

mechanical systems. The formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) obtained by introducing an appropriate stochastic driving field in the fluid equations. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

mechanical systems. The formalism is cast in terms of a system of stochastic partial differential equations (SPDEs) obtained by introducing an appropriate stochastic driving field in the fluid equations. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

- Dr. Atzberger : Research Website. Prof. Atzberger has worked on extending the immersed boundary method to incorporate thermal fluctuations for applications involving microscopic

mechanical systems. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

- Dr. Atzberger : Research Website. Prof. Atzberger has extended the immersed boundary method to incorporate thermal fluctuations for applications involving microscopic

- Dr. Atzberger : Research Website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.
- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma(USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method

- Dr. Atzberger : Research Website. Prof. Atzberger has worked on extending the immersed boundary method to incorporate thermal fluctuations for applications involving microscopic

mechanical systems. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.

- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method

## Research Areas (Selected Subset for Highlights)

## Research Areas (Selected Subset of Activities)

## Local and Global Wellposedness of Nonlinear Evolutionary Equations

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

## Local and Global Wellposedness of Nonlinear Evolutionary Equations

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

## Local and Global Well-posedness of Nonlinear Evolutionary Equations

(Tom and Gustavo, please write brief outline of research areas and edit URLs below)

Brief article outlining basic area of research, interesting math. issues. Dr. Sideris's work and Dr. Ponce's work highlighted here...

## Local and Global Wellposedness of Nonlinear Evolutionary Equations

## Analysis of XYZ... Non-linear Elasticity.... Schrodinger Equ... (please edit)

## Local and Global Well-posedness of Nonlinear Evolutionary Equations

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here:

Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method

- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma(USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here: Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method

- Dr. Cinceros : Research Website. Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures, papers can be found here [link]... (Hector please edit profile and above description as you see fit...)

- Dr. Ceniceros : Research Website. Prof. Ceniceros in joint work with UCSB grad student Jordan Fisher and Prof. Alexandre Roma (USP) has recently designed efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures. A soon to appear manuscript of the work can be found here:

Efficient solutions to robust, semi-implicit discretizations of the Immersed Boundary Method

- Dr. Cinceros : Research Website.

- Dr. Ceniceros : Research Website.

The mechanics of many physical systems depends crucially on the interaction of flexible elastic

The mechanics of many physical systems depends importantly on the interaction of flexible elastic

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. In microscopic systems examples include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and features hard to predict a priori.

The mechanics of many physical systems depends crucially on the interaction of flexible elastic structures with a fluid flow. Examples of macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. Examples of microscopic systems include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and features hard to predict a priori. Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid.

if this is required by university to avoid any issues of "free speech" and to guard against any formal objections being made to fairness.

if this is required by university to avoid any formal objections being made to how the pages are collectively managed.

**Applied math. wiki-edit by-laws:**\\

**Applied math. group wiki by-laws:**\\

**Wiki-edit By-Laws**\\

**Applied math. wiki-edit by-laws:**\\

- Dr. Cinceros : Research Website. Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures, papers can be

found here [link]... (Hector please edit profile and above description as you see fit...)

- Dr. Cinceros : Research Website. Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures, papers can be found here [link]... (Hector please edit profile and above description as you see fit...)

- Dr. Cinceros's : Research Website. Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures... (Hector please edit profile and above description as you see fit...)

- Dr. Cinceros : Research Website. Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures, papers can be

found here [link]... (Hector please edit profile and above description as you see fit...)

## Fluid-Structure Dynamics: Immersed Boundary Methods and Boundary Integral Methods

## Fluid-Structure Dynamics : Immersed Boundary Methods and Boundary Integral Methods

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. In microscopic systems examples include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and a priori hard to predict features.

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. In microscopic systems examples include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and features hard to predict a priori.

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea. In microscopic systems examples include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and a priori hard to predict features.

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea of the inner ear. In microscopic systems examples include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and a priori hard to predict features.

- Dr. Atzberger : Research Website.

A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.

- Dr. Cinceros's : Research Website.

Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures... (Hector please edit profile and above description as you see fit...)

- Dr. Atzberger : Research Website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here : IMA Workshop on Multiscale Methods.
- Dr. Cinceros's : Research Website. Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures... (Hector please edit profile and above description as you see fit...)

- Dr. Atzberger : Research Website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.
- Dr. Cinceros's : Research Website. Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures... (Hector please edit profile and above description as you see fit...)

- Dr. Atzberger : Research Website.

A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.

- Dr. Cinceros's : Research Website.

Dr. Cinceros has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures... (Hector please edit profile and above description as you see fit...)

## Research Areas (Selected)

## Research Areas (Selected Subset for Highlights)

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems both for analysis and numerics. Several faculty members of the applied group work in this area both on analysis of fluid-structure systems and the development of efficient numerical methods. These include:

- Dr. Atzberger who has extended the IB methodology to include thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.
- Dr. Cinceros's who has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures...

Faculty members working in this area include:

- Dr. Atzberger : Research Website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.

(Tom and Gustavo, please write brief outline of research areas and edit URLs below)

Faculty members working in this area include:

- Dr. Sideris : Research Website.
- Dr. Ponce : Research Website.

Brief article outlining basic area of research, interesting math. issues. Mention Dr. Garcia-Cervera's, Dr. Cinceros's, and Dr. Atzbeger's specific work in this area...

Brief article outlining basic area of research, interesting math. issues...

Faculty members working in this area include:

- Dr. Garcia-Cervera : Research Website.
- Dr. Cinceros : Research Website.
- Dr. Atzbeger : Research Website.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems both for analysis and numerics. A common theme often arising in practice is that the resulting system of equations stiff as a consequence of fast time-scales introduced into the fluid-structure dynamics either by the elastic structures or in microscopic systems by thermal fluctuations. Several faculty members of the applied group work in this area both on analysis of fluid-structure systems and the development of efficient numerical methods.

Faculty members working on fluid-structure problems include:

- Dr. Atzberger who has extended the IB methodology to include thermal fluctuations, by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. Mathematically, the formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Uses results from stochastic calculus he has developed efficient stochastic numerical methods for the formalism. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes using the SIB formalism to account for molecular level interactions (lipid-lipid and lipid-solvent) along with hydrodynamic coupling and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems both for analysis and numerics. Several faculty members of the applied group work in this area both on analysis of fluid-structure systems and the development of efficient numerical methods. These include:

- Dr. Atzberger who has extended the IB methodology to include thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.

## Research Areas (Select Subset)

## Research Areas (Selected)

## Select Research Activities

## Research Areas (Select Subset)

## Research Activities (Subset)

## Select Research Activities

## Research Activities

## Research Activities (Subset)

## Highlighted Research Areas

## Research Activities

## Research Gallery (images / movies, here, etc...) \\

## Research Gallery \\

(images / movies, here, etc...)

(will work on embedding some floating boxes that list news items and current events, see http://www.me.ucsb.edu/ for some stylistic ideas)

(will work on embedding some floating boxes that list news items and current events, see http://www.me.ucsb.edu/ for some stylistic ideas)

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site which is quite nicely done]. In my opinion our group runs the risk of not presenting a well-defined core and anchored research program communicated to people when they visit if we only include a collection of images with captions. Maybe there is some happy medium, please write-up some example or express your ideas.

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site which is quite nicely done http://amath.unc.edu/]. In my opinion our group runs the risk of not presenting a well-defined core and anchored research program communicated to people when they visit if we only include a collection of images with captions. Maybe there is some happy medium, please write-up some example or express your ideas.

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site which is quite nicely done]. In my opinion our group runs the risk of not presenting a well-defined core and anchored research program communicated on the site to people when they visit if we only include a collection of images with captions.

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site which is quite nicely done]. In my opinion our group runs the risk of not presenting a well-defined core and anchored research program communicated to people when they visit if we only include a collection of images with captions. Maybe there is some happy medium, please write-up some example or express your ideas.

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site].

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site which is quite nicely done]. In my opinion our group runs the risk of not presenting a well-defined core and anchored research program communicated on the site to people when they visit if we only include a collection of images with captions.

Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

Mention Dr. Garcia-Cervera's, Dr. Cinceros's, and Dr. Atzbeger's specific work in this area...

## Gallery of Recent Results (images / movies, here, etc...) \\

## Research Gallery (images / movies, here, etc...) \\

## Liquid Crystals / Lipid Bilayer Membranes / Complex Polymer Fluids

## Analysis of XYZ... Non-linear Elasticity.... Schrodinger Equ... (please edit)

Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

Dr. Sideris's work and Dr. Ponce's work highlighted here...

## Analysis of XYZ... Non-linear Elasticity.... Schrodinger Equ... (please edit)

## Liquid Crystals / Lipid Bilayer Membranes / Complex Polymeric Fluids

Dr. Sideris's work and Dr. Ponce's work highlighted here...

Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

## Discussion of Above Items

Above is an experiment and rough draft / brainstorm. Idea is to give newbie graduate students and general audience sense of what our group does. Above is meant to give an example of some themes and what we might post to highlight our specific research programs and on-going work. I wrote perhaps a little too much about myself, but this is since I know the most about this activity presumably. :) I propose we write several of these short articles which will be persistent for themes of the group (and modify over time as things develop). Up-top we then include some short descriptions of "fresh work" before these thematic short-articles. This means people encounter what's new first, but can scroll down for a more detailed view of our group.

Above is an experiment and rough draft / brainstorm. Idea is to give undergraduate / graduate students and general public and scientific audience sense of what types of work our group does. Above is meant to give an example of some themes and what we might post to highlight our specific research programs and on-going work both at the level of light-reading and in more detail. I wrote mostly about my own work, since I know the most about this activity presumably. :) I propose up-top we have short gallery style presentation of results for the light-of-heart. We then write in more detail a few of these short articles which discuss general thematic areas of research of the group (which of course can be modified over time to freshen things up and as our collective interests develop). This means people encounter what's new and light first, and then depending on their level of interest can scroll down for a more detailed desciption of the group.

We can discuss style of the research highlights and how we present things if you think this is too much... In any case, the idea is to post things here before we go public to discuss overall theme and criteria for content... Please also take a shot at posting some materials for discussion [present in near-final form]. The working process will eventually be to have a "hidden" page we can all edit where we write the materials and review before copying to the homepage.

Not sure if this is necessary:\\

Please take a shot at posting/editing the welcome message and some materials for discussion.

Not sure if this is necessary (formal rules for editing the Wiki website):\\

## Gallery of Recent Results (images / movies, here, etc...)

## Analysis of XYZ...

## Analysis of XYZ... Non-linear Elasticity.... Schrodinger Equ... (please edit)

Mention Dr. Sideris's work and Dr. Ponce's work...

Dr. Sideris's work and Dr. Ponce's work highlighted here...

Above is an experiment and rough draft / brainstorm. Above is meant to give an example of what we might post to highlight our research programs and on-going work. I wrote about myself since I know the most about this activity presumably. :) We may have a few short articles and either leave in place for awhile or copy these to "hidden" pages and each month rotate through the content descriptions making modifications, new articles, and updates as research develops...

Another model is to have simply an image with a short caption and links to researchers webpages with more detailed into... [see UNC site].

Above is an experiment and rough draft / brainstorm. Idea is to give newbie graduate students and general audience sense of what our group does. Above is meant to give an example of some themes and what we might post to highlight our specific research programs and on-going work. I wrote perhaps a little too much about myself, but this is since I know the most about this activity presumably. :) I propose we write several of these short articles which will be persistent for themes of the group (and modify over time as things develop). Up-top we then include some short descriptions of "fresh work" before these thematic short-articles. This means people encounter what's new first, but can scroll down for a more detailed view of our group.

Another model is not to have thematic articles, but to only have a few images up-top with a short caption and links to researchers webpages with more detailed info... [see UNC site].

Not sure if this is necessary:

\\

## Other highlights here...(images / movies, etc...)

## Immersed Boundary Methods and Boundary Integral Methods for Fluid-Structure Dynamics:

## Fluid-Structure Dynamics: Immersed Boundary Methods and Boundary Integral Methods

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems for analysis. A common theme often arising in practice os that fast time scales are introduced into the fluid-structure dynamics by the elastic structures or in microscopic system by thermal fluctuations making the resulting system of equations stiff. Several faculty members of the applied group work in this area.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems both for analysis and numerics. A common theme often arising in practice is that the resulting system of equations stiff as a consequence of fast time-scales introduced into the fluid-structure dynamics either by the elastic structures or in microscopic systems by thermal fluctuations. Several faculty members of the applied group work in this area both on analysis of fluid-structure systems and the development of efficient numerical methods.

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea. In microscopic systems examples include the rheology of complex fluids / soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy.

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea. In microscopic systems examples include the rheology of complex fluids and soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) which interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy. These microscopic processes often result macroscopically in material properties exhibiting interesting counter-intuitive phenomena and a priori hard to predict features.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of possibly non-linear integral equations. The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems for analysis. A common theme often arising in practice os that fast time scales are introduced into the fluid-structure dynamics by the elastic structures or in microscopic system by thermal fluctuations making the resulting system of equations stiff. Several faculty members of the applied group work in this area.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of integral equations (possibly non-linear). The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems for analysis. A common theme often arising in practice os that fast time scales are introduced into the fluid-structure dynamics by the elastic structures or in microscopic system by thermal fluctuations making the resulting system of equations stiff. Several faculty members of the applied group work in this area.

## Immersed Boundary Methods for Fluid-Structure Dynamics:

Immersed Boundary Methods (IBM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Immersed structures in IBM can be used to represent the mechanics of a variety of hydrodynamic systems from the macroscopic to microscopic. Examples include for macroscopic systems blood flow in the heart and interaction with valves, lift general in insect flight, and wave propagation in the cochlear. In microscopic systems examples include rheology of complex fluids / soft-matter as a results of microstructures interacting with fluid flows at small-scales where immersed structures represent solute particles, polymers, or membrane structures.

## Immersed Boundary Methods and Boundary Integral Methods for Fluid-Structure Dynamics:

Immersed Boundary Methods (IBM) and Boundary Integral Methods (BIM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Examples macroscopic systems include the pumping of the heart in which blood flow interacts with valves, lift general in insect flight, and wave propagation in the cochlea. In microscopic systems examples include the rheology of complex fluids / soft-matter which depends importantly on microstructures (such as colloids, lipids, polymers, vesicles) interact with shear and extensional fluid flows serving through small-scale deformations to elastically store or dissipate energy.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Many interesting mathematical challenges arise both in the study of specific fluid-structure systems and in obtaining in general efficient numerical methods. In practice, fast time scales are often introduced into the fluid-structure dynamics by the elastic structures or in microscopic system by thermal fluctuations making the resulting system of equations stiff.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. In the BIM formalism the hydrodynamic equations are reduced to a description on the surface of an immersed structure usually taking the form of possibly non-linear integral equations. The fluid-structure interaction problem and these underlying formulations present many interesting mathematical problems for analysis. A common theme often arising in practice os that fast time scales are introduced into the fluid-structure dynamics by the elastic structures or in microscopic system by thermal fluctuations making the resulting system of equations stiff. Several faculty members of the applied group work in this area.

## Stochastic Immersed Boundary Methods for Fluid-Structure Dynamics:

Paul J. Atzberger

Stochastic Immersed Boundary Methods (SIB) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in microscopic systems where thermal fluctuations play an important role. Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures.

## Immersed Boundary Methods for Fluid-Structure Dynamics:

Immersed Boundary Methods (IBM) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid. Immersed structures in IBM can be used to represent the mechanics of a variety of hydrodynamic systems from the macroscopic to microscopic. Examples include for macroscopic systems blood flow in the heart and interaction with valves, lift general in insect flight, and wave propagation in the cochlear. In microscopic systems examples include rheology of complex fluids / soft-matter as a results of microstructures interacting with fluid flows at small-scales where immersed structures represent solute particles, polymers, or membrane structures.

In the SIB formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. Mathematically, the formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's).

Many interesting mathematical challenges arise in obtaining efficient numerical methods as a result of fast time scales introduced into the fluid dynamics by the thermal fluctuations making the resulting system of equations stiff. Dr. Atzberger's group uses results from stochastic calculus to develop efficient stochastic numerical methods for the formalism. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes using the SIB formalism to account for molecular level interactions (lipid-lipid and lipid-solvent) along with hydrodynamic coupling and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website.

A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.

In the IBM formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Many interesting mathematical challenges arise both in the study of specific fluid-structure systems and in obtaining in general efficient numerical methods. In practice, fast time scales are often introduced into the fluid-structure dynamics by the elastic structures or in microscopic system by thermal fluctuations making the resulting system of equations stiff.

Faculty members working on fluid-structure problems include:

- Dr. Atzberger who has extended the IB methodology to include thermal fluctuations, by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. Mathematically, the formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Uses results from stochastic calculus he has developed efficient stochastic numerical methods for the formalism. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes using the SIB formalism to account for molecular level interactions (lipid-lipid and lipid-solvent) along with hydrodynamic coupling and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website. A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.
- Dr. Cinceros's who has recently formulated efficient implicit methods for the IBM formalisms significantly advancing the time-scales accessible in simulations of systems with stiff elastic structures...

## Implicit Immersed Boundary Methods...

## Liquid Crystals / Lipid Bilayer Membranes / Complex Polymer Fluids

Mention Dr. Cinceros's group... (maybe merge SIB and IIBM approaches into one article on IB methods and boundary integral methods and then reference group members.)

Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

## Liquid Crystals and XYZ

## Analysis of XYZ...

Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

## Analysis of XYZ...

Brief article outlining basic area of research, interesting math. issues. Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

Mention Dr. Sideris's work and Dr. Ponce's work...

Dr. Atzberger's group works on Stochastic Immersed Boundary Methods (SIB) which are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in systems at small length-scales where thermal fluctuations play an important role. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, he is developing efficient stochastic numerical methods for the formalism.

Stochastic Immersed Boundary Methods (SIB) are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in microscopic systems where thermal fluctuations play an important role. Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes using the SIB formalism to account for molecular level interactions (lipid-lipid and lipid-solvent) along with hydrodynamic coupling and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website.

In the SIB formalism the hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. Mathematically, the formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's).

Many interesting mathematical challenges arise in obtaining efficient numerical methods as a result of fast time scales introduced into the fluid dynamics by the thermal fluctuations making the resulting system of equations stiff. Dr. Atzberger's group uses results from stochastic calculus to develop efficient stochastic numerical methods for the formalism. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes using the SIB formalism to account for molecular level interactions (lipid-lipid and lipid-solvent) along with hydrodynamic coupling and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website.

## Implicit Immersed Boundary Methods...

Brief article outlining basic area of research, interesting math. issues. Mention Dr. Cinceros's group... (maybe merge SIB and IIBM approaches into one article on IB methods and boundary integral methods and then reference group members.)

## Liquid Crystals and XYZ

Brief article outlining basic area of research, interesting math. issues. Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

## Analysis of XYZ...

Brief article outlining basic area of research, interesting math. issues. Mention Dr. Garcia-Cervera's group and Dr. Cinceros's group...

Above is (rough draft / brainstorm) Above is meant to give an example of what we might post about our work. We can discuss style of the research highlights and how we present things if you think this is too much... In any case, the idea is to post things here before we go public to discuss overall theme and criteria for content... Please also take a shot at posting some materials for discussion [present in near-final form]. The working process will eventually be to have a "hidden" page we can all edit where we write the materials and review before copying to the homepage.

Above is an experiment and rough draft / brainstorm. Above is meant to give an example of what we might post to highlight our research programs and on-going work. I wrote about myself since I know the most about this activity presumably. :) We may have a few short articles and either leave in place for awhile or copy these to "hidden" pages and each month rotate through the content descriptions making modifications, new articles, and updates as research develops...

Another model is to have simply an image with a short caption and links to researchers webpages with more detailed into... [see UNC site].

We can discuss style of the research highlights and how we present things if you think this is too much... In any case, the idea is to post things here before we go public to discuss overall theme and criteria for content... Please also take a shot at posting some materials for discussion [present in near-final form]. The working process will eventually be to have a "hidden" page we can all edit where we write the materials and review before copying to the homepage.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in systems at small length-scales where thermal fluctuations play an important role. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in systems at small length-scales where thermal fluctuations play an important role. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, he is developing efficient stochastic numerical methods for the formalism.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes using the SIB formalism to account for molecular level interactions (lipid-lipid and lipid-solvent) along with hydrodynamic coupling and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid at small length scales in which thermal fluctuations play an important role. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in systems at small length-scales where thermal fluctuations play an important role. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid at small length scales in which thermal fluctuations play an important role. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism.

## Stochastic Immersed Boundary Methods for Computational Fluid-Structure Dynamics:

## Stochastic Immersed Boundary Methods for Fluid-Structure Dynamics:

## Stochastic Immersed Boundary Methods for Computational Fluid Dynamics:

## Stochastic Immersed Boundary Methods for Computational Fluid-Structure Dynamics:

## Stochastic Immersed Boundary Methods for Computational Fluid Dynamics: Dr. Paul J. Atzberger

## Stochastic Immersed Boundary Methods for Computational Fluid Dynamics:

Paul J. Atzberger

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism.

The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see Dr. Atzberger's research website.

## Stochastic Immersed Boundary Methods for Computational Fluid Dynamics:

Dr. Paul J. Atzberger

## Stochastic Immersed Boundary Methods for Computational Fluid Dynamics: Dr. Paul J. Atzberger

Dr. Paul J. Atzberger

Dr. Paul J. Atzberger

The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In the figure some simulations demonstrating the methodology for a few basic physical systems are shown. Click on the images to play the associated movies. From top to bottom are: (i) polymer knot simulations showing SIB preservation of knot topology without the need for excluded volume interactions, (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling, (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics.

The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

(rough draft / brainstorm) Above is meant to give an example of what we might post about

Above is (rough draft / brainstorm) Above is meant to give an example of what we might post about

the website all content editing will be password protected (admin or

the public part of the website all content editing will be password protected (admin or

also agree to authorize certain individuals to post content with provision they remove anything to which a majority in the group objects. These paragraphs could be copied to a formal manual on by-laws for the pages

also agree to authorize certain individuals (by majority vote) to have the right to post content on-line without prior approval with provision they remove anything to which someone objects and the majority does not support. I am not sure this is necessary, but these paragraphs could be copied to a formal manual on by-laws for the pages

To handle the issue of any members putting in appropriate things on

To handle the issue of any members putting inappropriate things on

core members). To avoid abuse by any group members, mainly the usual one, we shall have the convention that all core members "vote"

core members). To avoid abuse by any one group member. I propose we shall have the official convention that all core members "vote"

"do-not-approve PJA", after all members post the admin will copy... [This paragraph will be copied to our manual of by-laws for the pages to avoid any university issues if anyone formally objects to not being able to post anything they like...]. Another work around might be to start out own "Center" or "Off-Center" around Applied and Computational Analysis and make the pages based on that group to avoid having to deal with certain members. In any case, the group should brainstorm here.

"do-not-approve PJA", after all members post the admin will copy... We could also agree to authorize certain individuals to post content with provision they remove anything to which a majority in the group objects. These paragraphs could be copied to a formal manual on by-laws for the pages if this is required by university to avoid any issues of "free speech" and to guard against any formal objections being made to fairness.

To avoid the all-too common wacky-ness from occuring by some appl. members,

To handle the issue of any members putting in appropriate things on the website all content editing will be password protected (admin or core members). To avoid abuse by any group members, mainly the usual one,

in fairness his wacki-ness to post material, but everything on-line reflects the group consensus. Anything failing the vote and revisions process will not be copied to the homepage [kind of like a journal peer-review process]. Basically, we will post on the hidden page content "approved PJA", "revise xyz PJA",

in fairness anyone in the group to post material, but everything on-line reflects the group consensus. Anything failing the vote and revisions process will not be copied to the homepage [kind of like a journal peer-review process]. Basically, we will post on the hidden page content "approved PJA", "revise xyz PJA",

to post anything they like...].

to post anything they like...]. Another work around might be to start out own "Center" or "Off-Center" around Applied and Computational Analysis and make the pages based on that group to avoid having to deal with certain members. In any case, the group should brainstorm here.

[This paragraph will be copied to our manual on by-laws for the pages to avoid any university issues if any wacko's object...].

[This paragraph will be copied to our manual of by-laws for the pages to avoid any university issues if anyone formally objects to not being able to post anything they like...].

where we write the materials and review before copying to the homepage.

where we write the materials and review before copying to the homepage.

To avoid the all-too common wacky-ness from occuring by some appl. members, we shall have the convention that all core members "vote" on the "hidden" page draft content before anything is put on the homepage (password protected by admin). This way if requested we can allow in fairness his wacki-ness to post material, but everything on-line reflects the group consensus. Anything failing the vote and revisions process will not be copied to the homepage [kind of like a journal peer-review process]. Basically, we will post on the hidden page content "approved PJA", "revise xyz PJA", "do-not-approve PJA", after all members post the admin will copy... [This paragraph will be copied to our manual on by-laws for the pages to avoid any university issues if any wacko's object...].

(Carlos: Please finish writing this synposis.)

the study of complex fluids / soft condense matter, computational fluid

the study of complex fluids / soft-condensed matter, computational fluid

analysis of non-linear hyperbolic PDE's (?), applied harmonic analysis. Here you will find research highlights, upcoming seminar talks, and information about news and current activities of the applied mathematics group.

(Please edit and take a shot at writing this materials. We can collectively edit over time until we get something we are all happy with.).

analysis of non-linear hyperbolic PDE's (?), applied harmonic analysis, stochastic differential equations. Here you will find research highlights, upcoming seminar talks, and information about news and current activities of the applied mathematics group.

Please feel free to edit and taking a shot at writing these materials. We can collectively edit over time until we get something we are all happy with.

(We can discuss style of the research highlights and how we present things if you think this is too much... In any case, the idea is to post things here before we go public to discuss overall theme and criteria for content... Please also take a shot at posting some materials for discussion [present in near-final form].)

(rough draft / brainstorm) Above is meant to give an example of what we might post about our work. We can discuss style of the research highlights and how we present things if you think this is too much... In any case, the idea is to post things here before we go public to discuss overall theme and criteria for content... Please also take a shot at posting some materials for discussion [present in near-final form]. The working process will eventually be to have a "hidden" page we can all edit where we write the materials and review before copying to the homepage.

development of core areas of mathematics motivated by the solution to specific problems arising in applications, such as arise in the basic sciences

development of core areas of mathematics with the solution of specific problems arising in applications often in the basic sciences

problems arising in applications, such as arise in the basic sciences

specific problems arising in applications, such as arise in the basic sciences

development of core areas of mathematics while motivated by the solution to problems arising in applications, such as arise in the sciences and engineering. Specific areas in which faculty are involved include

development of core areas of mathematics motivated by the solution to problems arising in applications, such as arise in the basic sciences or engineering. Specific areas in which faculty are involved include

Welcome to the applied mathematics group's homepage. Here you will find information about current research activities and events around campus related to applied mathematics. Applied mathematics refers to the branch of mathematics which strives to integrate the development of core areas of mathematics while motivated by the solution to

Welcome to the applied mathematics group's homepage. Applied mathematics refers to the branch of mathematics which strives to integrate the development of core areas of mathematics while motivated by the solution to

and engineering. Here you will find research highlights, upcoming seminar talks, and information about news and current activities of the applied mathematics group.

(please take a shot at writing this and we can collectively edit).

and engineering. Specific areas in which faculty are involved include the study of complex fluids / soft condense matter, computational fluid dynamics (boundary integral methods / immersed boundary methods), crystalline solids and liquid crystals, density functional theory, analysis of non-linear hyperbolic PDE's (?), applied harmonic analysis. Here you will find research highlights, upcoming seminar talks, and information about news and current activities of the applied mathematics group.

(Please edit and take a shot at writing this materials. We can collectively edit over time until we get something we are all happy with.).

Welcome to the applied mathematics group's homepage... blurb about unifying themes and mission...

Welcome to the applied mathematics group's homepage. Here you will find information about current research activities and events around campus related to applied mathematics. Applied mathematics refers to the branch of mathematics which strives to integrate the development of core areas of mathematics while motivated by the solution to problems arising in applications, such as arise in the sciences and engineering. Here you will find research highlights, upcoming seminar talks, and information about news and current activities of the applied mathematics group.

## Other highlights here...(images / movies, etc...)

## Other highlights here...(images / movies, etc...)

## Stochastic Immersed Boundary Methods / Computational Fluid Dynamics:

## Stochastic Immersed Boundary Methods for Computational Fluid Dynamics:

are a numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

are numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

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The Stochastic Immersed Boundary Method (SIB) is a numerical approach for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

Dr. Atzberger's group works on Stochastic Immersed Boundary Methods (SIB) which are a numerical approaches for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In the figure some simulations demonstrating the methodology for a few basic physical systems are shown. Click on the images to play the associated movies. From top to bottom are: (i) polymer knot simulations showing SIB preservation of knot topology without the need for excluded volume interactions, (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling, (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In the figure some simulations demonstrating the methodology for a few basic physical systems are shown. Click on the images to play the associated movies. From top to bottom are: (i) polymer knot simulations showing SIB preservation of knot topology without the need for excluded volume interactions, (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling, (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load. In collaboration with the Brown Group, Department of Chemistry, Dr. Atzberger's group is developing dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

Dr. Paul J. Atzberger:
\\

Dr. Paul J. Atzberger

(Dr. Paul J. Atzberger)

Dr. Paul J. Atzberger: \\

## Stochastic Immersed Boundary Methods / Computational Fluid Dynamics:\\

## Stochastic Immersed Boundary Methods / Computational Fluid Dynamics:

## Stochastic Immersed Boundary Methods / Computational Fluid Dynamics:

(Dr. Paul J. Atzberger)

The Stochastic Immersed Boundary Method (SIB) is a numerical approach for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see Dr. Atzberger's papers.

Attach.SIB_Schematic.png

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In the figure some simulations demonstrating the methodology for a few basic physical systems are shown. Click on the images to play the associated movies. From top to bottom are: (i) polymer knot simulations showing SIB preservation of knot topology without the need for excluded volume interactions, (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling, (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

A recent talk given by Dr. Atzberger on the Stochastic Immersed Boundary Method can be found here.

(please take a shot at writing this and we can collectively edit).

Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".

Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal "Multilevel Physics in the Study of Solids: Modeling, Analysis and Simulations".

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Carlos Garcia-Cervera receives the prestigious NSF Career Award for his proposal
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## How to use and edit wiki-pages.

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## How to use and edit wiki-pages.

How to use and edit wiki-pages.\\

## How to use and edit wiki-pages.

For practice for when the site goes

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## Applied Math and PDE Seminar

## Applied Mathematics Research at UCSB

Welcome to the applied mathematics group's homepage... blurb about unifying themes and mission... \\

*Dynamics of a Stochastically Driven Neuronal Network Model.*

Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607.

Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.*Abstract:* We study an all-to-all coupled network of identical excitatory
integrate-and-fire (I\&F) neurons driven by an external spike train
modeled as a Poisson process. Numerical simulations demonstrate that
over a broad range of parameters, the network enters a synchronized
state in which the neurons all fire together at regular intervals. We
identify mechanisms leading to this synchronization for two regimes of
the external driving current: superthreshold and subthreshold. In the
former, a probabilistic argument similar to the proof of the Central
Limit Theorem yields the oscillation period, while in the latter, this
period is analyzed via an exit time calculation utilizing a diffusion
approximation of the Kolmogorov forward equation. In both cases,
stochastic fluctuations play a central role in determining the
oscillation period. We also develop a criterion for
synchrony in the network through a probabilistic argument. This work
is in collaboration with Katherine Newhall, Gregor Kovacic, David Cai,
and Aaditya Rangan.

*Something.*

Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607.

Someone, Dept. Something, Somewhere.*Abstract:* Some description of talk.

Research Highlights here...(images / movies, etc...) \\

**Title:** Talking about something. *Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*. **Abstract:** A blurb about what to talk about...\\

*Something.*

Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607.

Someone, Dept. Something, Somewhere.*Abstract:* Some description of talk.

Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607. \\

Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607. \\

**Abstract:** We study an all-to-all coupled network of identical excitatory

*Abstract:* We study an all-to-all coupled network of identical excitatory

Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.\\

Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.\\

**Dynamics of a Stochastically Driven Neuronal Network Model.** \\

** Dynamics of a Stochastically Driven Neuronal Network Model.** \\

Dynamics of a Stochastically Driven Neuronal Network Model. \\

**Dynamics of a Stochastically Driven Neuronal Network Model.** \\

+Dynamics of a Stochastically Driven Neuronal Network Model.+ \\

Dynamics of a Stochastically Driven Neuronal Network Model. \\

*Dynamics of a Stochastically Driven Neuronal Network Model.* \\

+Dynamics of a Stochastically Driven Neuronal Network Model.+ \\

+Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.+\\

Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.\\

Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.\\

+Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.+\\

*Dynamics of a Stochastically Driven Neuronal Network Model.*' *Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.**Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607.*. \\

*Dynamics of a Stochastically Driven Neuronal Network Model.*

Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.

Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607. \\

**Title:** Dynamics of a Stochastically Driven Neuronal Network Model. \\

*Dynamics of a Stochastically Driven Neuronal Network Model.*' \\

**Title:** Talking about something. *Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*. **Abstract:** A blurb about what to talk about...\\

**Title:** Dynamics of a Stochastically Driven Neuronal Network Model. *Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.**Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607.*. **Abstract:** We study an all-to-all coupled network of identical excitatory
integrate-and-fire (I\&F) neurons driven by an external spike train
modeled as a Poisson process. Numerical simulations demonstrate that
over a broad range of parameters, the network enters a synchronized
state in which the neurons all fire together at regular intervals. We
identify mechanisms leading to this synchronization for two regimes of
the external driving current: superthreshold and subthreshold. In the
former, a probabilistic argument similar to the proof of the Central
Limit Theorem yields the oscillation period, while in the latter, this
period is analyzed via an exit time calculation utilizing a diffusion
approximation of the Kolmogorov forward equation. In both cases,
stochastic fluctuations play a central role in determining the
oscillation period. We also develop a criterion for
synchrony in the network through a probabilistic argument. This work
is in collaboration with Katherine Newhall, Gregor Kovacic, David Cai,
and Aaditya Rangan.

http://pmichaud.com/img/misc/gem.jpg

/* http://pmichaud.com/img/misc/gem.jpg */

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*. \\

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*. \\

**Title:** Talking about something. \\

**Title:** Talking about something. \\

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*. \\

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*. \\

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

**Title:**
**Abstract:**

\\

**Title:** Talking about something. **Abstract:** A blurb about what to talk about...

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

**Title:** Talking about something. **Abstract:** A blurb about what to talk about...\\

- Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777).
- Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777).
- Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777).

*Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777)*.

**Title:**
**Abstract:**

## Research Project II

Research project will be discussed here... Image and some links

## Research Project III

Research project will be discussed here... Image and some links

## Research Project I

## Applied Math and PDE Seminar

Research project will be discussed here... Image and some links

- Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777).
- Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777).
- Some Speaker, (University of Somewhere), Tuesday, April XX, 3:00pm - 4:00pm (South Hall 7777).

## Research Project I http://pmichaud.com/img/misc/gem.jpg

## Research Project I

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## Research Project I

## Research Project I http://pmichaud.com/img/misc/gem.jpg

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