Friday January 11: Davit Harutyunyan (UCSB)

On geometric rigidity of the rotation group and thin domains

The rotation group $SO(n)$ is known to be rigid in the following sense: A famous theorem by Reshetnyak states that if the
gradient of a $W^{1,2}$ function belongs to the rotation group a.e. in an open and connected set $\Omega\subset R^n$, then in fact
the gradient must be a constant rotation. For a connected open subset $\Omega$ of $R^n$, and given two matrices $A, B\in M^{n\times n}$,
there exists a Lipschitz map $u\colon\Omega\to R^n$ such that $\nabla u=A$ or $B$ a.e. in $\Omega,$ if and only if $rank(A-B)\leq 1.$
Similar result is true for 3 and 4 matrices, but fails to hold for 5 matrices. This shows the speciality of the rotation group in
some sense. In this talk we will discuss stronger versions of Reshetnyak's theorem as well as some optimal forms in the
case of thin domain $\Omega$, which is applicable in nonlinear elasticity and plasticity.

Wednesday January 23: Burkhard Wilking (University of Münster)

On Ricci flow on manifolds with integral lower curvature bounds

TBA

Friday January 25: Yuchin Sun (University of Chicago)

Morse Index Bound for Min-Max Two Spheres

We prove that given a Riemannian manifold of dimension at least three, with a generic metric and nontrivial homotopy group \pi_3, there exists a collection of finitely many harmonic spheres whose sum of areas realizes the width with Morse index bound one.

Friday February 15: Guofang Wei (UCSB)

Volume entropy estimate for integral Ricci curvature

We give an optimal estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate of Aubry and give an application on the algebraic entropy of its fundamental group. We also extend the quantitative almost maximal volume entropy rigidity of Chen-Rong-Xu and almost minimal volume rigidity of Bessieres-Besson-Courtois-Gallot to integral Ricci curvature. This is a joint work with Lina Chen.

Friday February 22: Xiaolong Li (UC, Irvine)

Ancient Solutions to the Ricci Flow in Higher Dimensions

It is well-known that the Ricci flow will generally develop singularities if one flows an arbitrary initial metric. Ancient solutions arise as limits of suitable blow-ups as the time approaches the singular time and thus play a central role in understanding the formation of singularities. By the work of Hamilton, Perelman, Brendle, and many others, ancient solutions are now well-understood in two and three dimensions. In higher dimensions, only a few
classification results were obtained and many examples were constructed. In this talk, we show that for any dimension $n \geq 4$, every noncompact rotationally symmetric ancient $kappa$-solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton, extending a recent result of Brendle to higher dimensions. This is joint work with Yongjia Zhang.

Friday March 8: Fred Wilhelm (UC Riverside)

Stability, Finiteness, and Dimension 4

I will discuss the history and proof of the following result.
Theorem. For any $k\in \mathbb{R}$, $v>0$, and $D>0$, there are only finitely many diffeomorphism types of Riemannian 4-manifolds with sectional curvature $\geq k$, volume $\geq v$, and diameter $\leq D$.

Friday March 15: Steven Gindi (UC Riverside)

TBA

TBA

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).