Geometry & Analysis Seminar Winter 2018

Thursdays - 4:00pm
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

The Ends of the Hitchin Moduli Space

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.''

Thursday, February 1, 2018

Yiran Wang, University of Washington

Inverse Problems for Nonlinear Hyperbolic Equations and Applications

In this talk, we start with the inverse problem for semilinear wave equations on a 4-dimensional Lorentzian manifold. We present the results of determining the conformal and isometry class of the Lorentzian metric, and some information about the nonlinear term. A key component of the method is the (microlocal) analysis of the nonlinear interaction of (distorted) plane waves, and to show that new waves are generated from such interactions.  Next, we discuss the inverse problems for the Einstein equations in general relativity, that is the determination of space-time structures using gravitational or electromagnetic waves. 

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

formalism to describe knot invariants, specifically Khovanov homology.  I will describe the analytic 
foundations of this theory, developed with Witten, as well as the remaining challenges in carrying out
this program (mostly to do with compactness). I will then discuss a dimensional reduction of
this problem and recent work with S. He which is near to resolving this case. 

***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

We consider a class of magnetic fields defined over the interior of a manifold M which diverge to infinity at its boundary and are asymptotically tangent to $\partial M$. We then show that we may find a closed 1-form $\sigma$ on $\partial M$ such that trajectories of the magnetic flow of such fields can only escape the manifold through its zero locus. In the case where σ is nowhere vanishing we conclude that charged particles inside of M become confined to its interior for all time. We then provide a topological characterization of such manifolds and discuss various examples such as solid tori, tubular neighborhoods of loops, principal circle bundles over manifolds with boundary and log-symplectic manifolds. 

NEW DAY-Friday March 9th, 2018

Charles Hadfield, University of California Berkeley

Resonances on Asymptotically Hyperbolic Manifolds; the Ambient Metric Approach

On an asymptotically hyperbolic manifold, the Laplacian has essential spectrum. Since work of Mazzeo and Melrose, this essential spectrum has been studied via the theory of resonances; poles of the meromorphic continuation of the resolvent of the Laplacian (with modified spectral parameter). A recent technique of Vasy provides an alternative construction of this meromorphic continuation which dovetails the ambient metric approach to conformal geometry initiated by Fefferman and Graham. I will discuss the ambient geometry present in this construction, use it to define quantum resonances for the Laplacian acting on natural tensor bundles (forms, symmetric tensors) and (although time won’t permit) mention one application to a classical/quantum correspondence on convex cocompact hyperbolic manifolds.

Thursday, March 15, 2018

Mihajlo Cekić, Max-Planck Institute, Bonn