talks in S98, W98 \ \ \ \ talks in 99-00 \ \ \ \ talks in F00, W01 \ \ \ \ talks in 01-02 \ \ \ \ talks in 02-03 \ \ \ \ talks in 03-04 \ \ \ \ talks in 04-05 \ \ \ \ talks in 05-06 \ \ \ \ talks in 06-07 \ \ \ talks in 07-08 \ \ \ talks in 08-09 \ \ \ talks in 09-10 \ \ \ talks in 10-11 \ \ \ talks in 11-12 \ \ \ talks in 12-13 \ \ \ talks in 13-14 \ \ \ talks in 14-15 \ \ \ talks in Fall 2015 \ \ \ talks in 16-17

Differential Geometry Seminar Schedule for Winter and Spring 2018

Fridays 3:00 - 3:50pm, SH 6635

1/19   Hanming Zhou,   UCSB "Lens Rigidity for a Particle in a Yang-Mills Field"

Abstract:  In this talk, we consider an inverse problem related to the motion of a classical colored spinless particle under the influence of an external Yang-Mills potential $A$ on a compact manifold with boundary of dimension $\geq 3$. We show that under suitable convexity assumptions, one can recover the potential $A$, up to gauge transformations, from the lens data of the system, namely, scattering data plus travel times between boundary points. The talk is based on joint work with Gabriel Paternain and Gunther Uhlmann.

1/26 Changliang Wang,  McMaster University,"Perelman's functionals on compact manifolds with isolated conical singularities"

Abstract: We extend the theory of the Perelman's functionals on compact smooth manifolds to compact manifolds with isolated conical singularities.  For the lambda-functional, this is essentially an eigenvalue problem for a Schrodinger operator with singular potential. We obtain a certain asymptotic behavior of eigenfunctions near the singularities. This asymptotic behavior plays an important role for deriving the variation formulas of the lambda-functional and other applications. Moreover, we show that the infimum of the W-functional over a suitable weighted Sobolev space on compact manifolds with isolated conical singularities is finite, and the minimizing function exists. We also obtain a certain asymptotic behavior for the minimizing function near the singularities. This is a joint work with Professor Xianzhe Dai.

2/2  Sho Seto,   UCSB "Learning seminar on needle decomposition"

2/16  Distinguished Lecture by Prof. Gunther Uhlmann

2/23  Sho Seto,   UCSB "Learning seminar on needle decomposition"

3/2   Nicholas Brubaker, California State University, Fullerton,  "A numerical method for computing constant mean curvature surfaces with boundary"

Abstract: Constant mean curvatures (CMC) surfaces, defined as critical points of surface area subject to a volume constraint, describe mathematical idealizations of physical interfaces occurring between two immiscible fluids. Accordingly, their predicted shapes give insight into the behavior of many micro-scale systems, such as beading or stiction in microelectromechanical system devices. However, explicitly computing such shapes is often impossible, especially when the boundary of the interface is fixed and/or parameters vary. In this talk, we will propose a robust novel numerical method for computing families of discrete versions of CMC surfaces that is based on solving a partial differential equation (PDE) via arc-length continuation. The method computes both stable and unstable surfaces, unlike many direct optimization methods, and naturally identifies bifurcations. Multiple examples will be presented to highlight the efficacy and accuracy of the proposed approach, including the reconstruction of a branch of asymmetric surfaces appearing from a symmetry-breaking bifurcation.

3/9   Ruobing Zhang,  Stony Brook University "Nilpotent Structure and Examples of Collapsed Einstein Spaces"

Abstract: In this talk, we will focus on the construction of new examples of collapsed Einstein spaces. A special case is to construct a family of hyperkähler metrics on a K3 surface which are collapsing to a closed interval [0,1]. Geometrically, the regular collapsing fibers in our example are 3D-Heisenberg (nilpotent) manifolds with almost flat metrics, while the singular collapsing fibers are singular circle fibrations over a flat torus. Moreover, there is a natural constant mean curvature foliation from the regular fiber to the singular fiber. Compared with the known examples of codim-1 and codim-2 collapsed Ricci-flat hyperkähler spaces, the collapsing fiber in our example are non-abelian. We will also see how such a collapsing phenomenon is related to a general regularity theorem.

4/6   Qiongling Li,   Caltech,  "On cyclic Higgs bundles"

Abstract: Given a closed Riemann surface and a Lie group G, the non-abelian Hodge theory gives a correspondence between the space of representations of the surface group into G with the moduli space of G-Higgs bundles. The correspondence is through looking for an equivariant harmonic map to the symmetric space associated to G, to a given representation or a given Higgs bundle. We derive a maximum principle for a type of elliptic systems and apply it to study cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of the minimal immersion.

4/13  Shoo Seto, UCSB,  "The first eigenvalue of the p-Laplacian on Riemann and Kahler manifolds"

4/20  Katy Craig,   UCSB "Gradient flow in the Wasserstein Metric"

Abstract:   For a range of partial differential equations–including the porous medium equation, the Fokker-Planck equation, and the Keller-Segel equation—solutions of the equations can be characterized as gradient flows with respect to the Wasserstein metric on the space of probability measures. This gradient flow structure lies at the heart of many recent analytic and numerical results regarding questions of stability, uniqueness, and singular limits.

Gradient flows with respect to Hilbert space norms are a classical tool in the study of partial differential equations, but the geometry of the Wasserstein metric presents new challenges. First, even for probability measures on Euclidean space, the Wasserstein metric it is positively curved in dimensions higher than one. Second, the metric lacks a rigorous Riemannian structure, which one would normally use to make sense of the “gradient” in a “gradient flow”. In this talk, I will introduce a time discretization of the gradient flow problem, due to Jordan, Kinderlehrer, and Otto, by which these problems can be overcome and present new results which extend the convergence of the time discrete scheme to a new class of partial differential equations of applied interest.

5/4 Geometry and Analysis on Manifolds, UC Santa Barbara, May 4-6, 2018

5/11  Li Ma,  University of Science and Technology Beijing   "Results of Lichnerowicz equations on manifolds"

5/18  Jeffrey A Viaclovsky,    UCI "Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces"

ABSTRACT:  I will discuss a new construction of families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding singular fibration from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds. This is joint work with Hans-Joachim Hein, Song Sun, and Ruobing Zhang.

6/1  Luca Spolaor,   MIT & Princeton "(Log-)epiperimetric inequality and regularity at isolated singularities for almost Area-Minimizing currents"

Abstract: The uniqueness of blow-up and regularity of multiplicity-one minimal surfaces at isolated singularities has been successfully investigated by Allard-Almgren [Ann. of Math. '81], in the integrable case, and by L. Simon [Ann. of Math. '83], in its full generality.

In this talk I will present a simple and completely variational approach to this problem, achieved by proving a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity. In contrast to classical epiperimetric inequalities by Reifenberg [Ann. of Math. '64], Taylor [Invent. Math. '73, Ann. of Math. '76] and White [Duke '83], we require no a priori assumptions on the structure of the cone (e.g. integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. Epiperimetric inequalities of logarithmic type were first introduced by M.Colombo, B. Velichkov and myself in the context of the obstacle and thin-obstacle problems.

As a consequence of our analysis we give a new proof of Allard-Almgren and Simon results in the case of minimizers and we deduce a new epsilon-regularity result for almost area-minimizing currents at isolated singularities.

This is joint work with M. Engelstein (MIT) and B. Velichkov (Grenoble).

6/8  Xavier Ramos Olive, UCR  "Li-Yau Gradient Estimate under Integral Ricci Curvature Bounds"

Abstract:  Li-Yau gradient estimates are one of the key ingredients in many results in Geometric Analysis. To obtain them, one usually needs to assume a lower bound on the Ricci curvature and some convexity condition on the boundary. Following the work of Q.S.Zhang and M.Zhu, we will describe a new Li-Yau gradient estimate under integral Ricci curvature conditions for the Neumann heat kernel. We will show how we can get an estimate on domains that are not necessarily convex, but that satisfy the interior rolling $R-$ball condition introduced by J.Wang and R.Chen.

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