Tuesday September 4: Jesse Ratzkin (University of Wuerzburg), SH4607

On the rate of change of the first eigenvalue of a moving domain

Extremal Sobolev functions on a bounded domain solve a nonlinear eigenvalue problem, and there is a large literature examining the relation between eigenvalues, the extremal functions, and the geometry of the underlying domain. We examine the case of an expanding domain, and in this case estimate the rate of change of the best constant in the Poincare-Sobolev inequality. Our estimates are isoperimetric, and along the way we prove an inequality which reverses the usual Holder inequality and may be of independent interest.

Friday October 5: Guofang Wei (UCSB)

Eigenvalue Estimates for Integral Curvature

Studying the eigenvalues of the Laplacian is both important in mathematics and physics. Some classical results are Lichnerowicz and Zhong-Yang estimates for the first nonzero eigenvalue of the Laplacian on closed manifolds with positive and zero Ricci curvature lower bounds. We will discuss extensions of these results to manifolds with integral Ricci curvature lower bound, which is a much weaker condition. This is a joint work with Xavier Ramos Oliver, Shoo Seto and Qi Zhang.

Friday October 12: Guangbo Xu (Stony Brook University)

Bershadsky--Cecotti--Ooguri--Vafa torsion in Landau--Ginzburg models

In the celebrated work of Bershadsky--Cecotti--Ooguri--Vafa
the genus one string amplitude in the B-model is identified with certain
analytic torsion of the Hodge Laplacian on a K\"ahler manifold. In a
joint work with Shu Shen (IMJ-PRG) and Jianqing Yu (USTC) we study the
analogous torsion in Landau--Ginzburg models. I will explain the
corresponding index theorem based on the asymptotic expansion of the
heat kernel of the Schr\"odinger operator. I will also explain the
rigorous definition of the BCOV torsion for homogeneous polynomials on
${\mathbb C}^N$. Lastly I will explain the conjecture stating that in
the Calabi--Yau case the BCOV torsion solves the holomorphic anomaly
equation for marginal deformations.

Friday October 19: Yousef Chahine (UCSB)

Volume estimates for tubes around submanifolds using integral curvature bounds

We generalize an inequality of E. Heintze and H. Karcher for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic and generalizes their result. Using similar ideas we also prove a Hessian comparison theorem for k-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications.

Friday October 26: Jiayin Pan (UCSB)

Nonnegative Ricci curvature, stability at infinity, and structure of fundamental groups

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We show that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a priorly fixed metric space, then $\pi_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth, then we can further bound the index of that abelian subgroup in terms of $n$ and the volume growth constant.

Friday November 30: Junrong Yan (UCSB)

Positive Scalar Curvature and Index Theorem

A fundamental problem in differential geometry concerns the relationship between local geometry and global topology. In particular, how curvatures control topology of manifolds. In this talk, we will review several such results with respect to scalar curvature. We are going to answer two questions: What kinds of manifolds admit metrics of positive scalar curvature? Can we say something about the fundamental group of such manifolds?
We will see that index theorem plays a fundamental role in addressing these questions.

Friday December 7: Jesús Núñez-Zimbrón (Centro de Ciencias Matemáticas UNAM)

On the Borel conjecture for Alexandrov 3-spaces

The Borel conjecture (BC) states that if two closed, aspherical
n-manifolds are homotopy equivalent then they are homeomorphic. The validity
of this conjecture for n=3 follows from Perelman's resolution of the
Geometrization Conjecture. Generalizations of the BC outside of the manifold
category have been obtained, for example, for CAT(0)-spaces and certain
classes of topological orbifolds. It is therefore natural to inquire whether
the BC holds for the class of Alexandrov 3-spaces (with curvature bounded
below). I will speak about work in progress in this direction which shows that
two aspherical, irreducible Alexandrov 3-spaces which are sufficiently
collapsed with respect to their diameters satisfy the BC. The results
presented here are joint with Noé Bárcenas.

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.

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).