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

## Welcome

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.

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.