# Geometry & Analysis Seminar Winter 2018

McHenry Library Room 4130

For further information please contact Professor Francois Monard or call 831-459-2400

**Thursday, January 11, 2018**

**Rohit Kumar Mishra, University of California, Santa Cruz**

**Support Theorem for Integral Moments of a Symmetric m-Tensor Field - II**This talk is a follow up of the talk I gave on October 12, 2017 titled “ Support Theorem for Integral Moments of a Symmetric m-Tensor Field” where I explained the main results without going too much in to the details of the proofs. In this talk, we will see some main ideas involved in the proofs. More importantly, we will discuss how analytic microlocal techniques are helpful in proving such results.

**Thursday, January 18, 2018**

**Maxwell Stolarski, University of Texas at Austin**

**Ricci Flow of Doubly-Warped Product Metrics****The Ricci flow applied to metrics of the form $dx^2 + u(x)^2 g_{S^p}$ has been a source of interesting dynamics of the Ricci flow that include the formation of slow blow-up degenerate neckpinch singularities and the forward continuation of the flow through neckpinch singularities. A natural next source of examples is then the Ricci flow of doubly-warped product metrics $dx^2 + u(x)^2 g_{S^p} + v(x)^2 g_{S^q}$. In this case, the Ricci flow is equivalent to a degenerate parabolic system or, by fixing a certain choice of gauge, a parabolic system with a nonlocal term. The doubly-warped product structure also allows for a larger collection of singularity models to appear compared to the singly-warped case. Indeed, formal matched asymptotic expansions suggest that some non-generic set of initial metrics on a compact manifold form finite-time Type II singularities modeled on a Ricci-flat cone at an appropriate scale. I will outline the formal matched asymptotics of this singularity formation and discuss the topological argument used to make such a formal construction rigorous. Along the way, we’ll draw comparisons to similar constructions for the mean curvature flow and a class of semilinear heat equations.**

**Thursday, January 25, 2018**

**Laura Fredrickson, Stanford University**

**Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. In this talk, I'll describe what solutions of $SL(n,\mathbb{C})$-Hitchin's equations ``near the ends'' of the moduli space look like, and the resulting compactification of the Hitchin moduli space. Wild Hitchin moduli spaces are an important ingredient in this construction. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's construction of $SL(2,\mathbb{C})$-solutions of Hitchin's equations where the Higgs field is "simple.''**

*The Ends of the Hitchin Moduli Space*

**Thursday, February 1, 2018**

**Yiran Wang, University of Washington**

**Inverse Problems for Nonlinear Hyperbolic Equations and Applications****Thursday, February 8, 2018**

**Semyon Dyatlov, University of California, Berkeley**

**Lower Bounds on Eigenfunctions on Hyperbolic Surfaces**

I show that on a compact hyperbolic surface, the mass of an $L^2$-normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassical version, has many applications including control for the Schr\"odinger equation and the full support property for semiclassical defect measures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.

**Geometry and Analysis Seminar/Mathematics Colloquium****Thursday, February 15, 2018**

**Rafe Mazzeo, Stanford University**

**The Extended Bogomolny and Kapustin-Witten Equations**

An intriguing proposal was made by Witten a decade ago involving a new gauge-theoretic

*****Special Day*****

**Friday, February 23, 2018**

**Chao Li, Stanford University**

**A Polyhedra Comparison Theorem for 3-manifolds with Positive Scalar Curvature**We establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron.

**Thursday, March 1, 2018**

**Gabriel Martins, University of California Santa Cruz**

**Topological Aspects of Magnetic Confinement****NEW DAY-Friday March 9th, 2018**

**Charles Hadfield, University of California Berkeley**

**Resonances on Asymptotically Hyperbolic Manifolds; the Ambient Metric Approach****Thursday, March 15, 2018**

**Mihajlo Cekić, Max-Planck Institute, Bonn**