# Geometric Analysis Seminar

The geometric analysis seminar is held on Tuesdays at 11am in 210 Deady Hall.

### Winter Quarter, 2020

- January 28,
**Iva Stavrov** (Lewis and Clark)

Interaction energy and effective mass of relativistic dust clouds
**Abstract**: The non-linearity of general relativity manifests itself in the form of interaction energy between different parts of a matter distribution, making it difficult to view a cloud of matter as being made up of point-source constituents. Consequently, and in contrast to Newtonian (linear) theory of gravity, it is not possible to view relativistic mass as being an integral of mass density. In fact, defining a suitable notion of quasi-local mass is still one of the most investigated problems in general relativity. The goal of this talk is to argue that in certain relativistic frameworks it is in fact possible to view a cloud of relativistic dust as being made up of point-source constituents.

- February 4,
**Christine Escher** (Oregon State University)

Odd-dimensional non-negatively curved GKM-manifolds
**Abstract**: A long-standing problem in Riemannian geometry is the topological classification of Riemannian manifolds M with positive or non-negative sectional curvature. All known results require high symmetry, in particular the existence of a “large” torus action on M. In contrast, the so-called GKMk-manifolds are (even dimensional) manifolds with arbitrary torus actions. Named after Goresky, Kottwitz and MacPherson GKMk-manifolds satisfy conditions that allow a combinatorial description of their equivariant cohomology rings. Examples of GKM2-manifolds are complex projective spaces, torus manifolds with vanishing odd degree cohomology and certain homogeneous spaces. In this talk I will give an overview of known classification results of GKMk-manifolds with positive and non-negative curvature, define a notion of an odd dimensional GKMk-manifold, and show how to generalize some of the classification results to odd dimensional GKMk-manifolds. In particular, I will outline ideas of the proof that for odd-dimensional, closed, non-negatively curved GKM3-manifolds both the equivariant and the ordinary rational cohomology split off the cohomology of an odd-dimensional sphere.

This is joint work with Oliver Goertsches and Catherine Searle.

- February 11,
**Ailana Fraser** (UBC)

Some results on higher eigenvalue optimization
**Abstract**: In this talk I will discuss some recent results concerning the optimization higher eigenvalues of manifolds with boundary, both in two and higher dimensional cases. This talk is based on joint work with P. Sargent and R. Schoen.

- February 18,
**Yu Yuan**, (University of Washington)

Regularity for convex viscosity solutions to special Lagrangian equations
**Abstract**: We present regularity for convex viscosity solutions to the special Lagrangian equation. The “gradient” graphs of all solutions are (Lagrangian) minimal surfaces in Euclidean space. In contrast, there are Pogorelov-like singular (convex) solutions to the Monge-Ampere equation, which is also the potential equation for “gradient” graphs of scalar functions being maximal surfaces in pseudo-Euclidean space.

This is joint work with Jingyi Chen and Ravi Shankar.

- March 10,
**Jeong Hyeong Park** (Sungkyunkwan University, Korea)

Harmonic manifolds
**Abstract**: A Riemannian manifold is harmonic if a volume density

function centered at a point depends only on the distance from the center. In this talk, we focus on two questions:

1. To what extent does information about the volume density function of a harmonic manifold determine its geometry?

2. To what extent the property of a manifold to be harmonic is inherited by a submanifold? We characterize harmonic manifolds via density functions and radial eigen functions and we also examine the totally geodesic submanifolds of harmonic manifolds.

### Fall Quarter, 2019

- October 15,
**Eric Bahuad** (Seattle University)

Geometrically Finite Asymptotically Hyperbolic Einstein metrics
- October 22,
**Liviu Nicolaescu** (University of Notre Dame)

A probabilistic proof of the Gauss-Bonnet-Chern theorem
**Abstract**: Suppose that E → M is an oriented real vector bundle of even rank 2r over a smooth compact oriented manifold M of dimension m ≥ 2r. To a metric h and compatible connection ∇ the Chern-Weil construction associates the Euler form e(E,h,∇)∈ Ω^{2r}(M). I will show that there is a Gaussian probability measure on the space of smooth section of E such that the expected value of the zero locus of such a random section is equal, in the sense of currents, to the Euler form e(E,h,∇).

In the process I will also describe how to recover probabilistically the geometry of the triplet (E,h,∇).

- October 29,
**Paul Allen** (Lewis and Clark)

Boundary Resolution for the Singular Yamabe Problem
- November 12,
**Matthew Gursky** (Notre Dame)

Gap and index estimates for some conformally invariant problems in dimensions 2 and 4
**Abstract**: In this talk I want to discuss sharp ‘gap’ estimates for the energy of Yang-Mills connections in 4-d, and the related question of the energy of harmonic two-spheres. I will also talk about index estimates for Yang-Mills connections and Einstein metrics in 4-d.

- November 19,
**Peng Lu** (UO)

Monotonicity of functionals for conformal Ricci flow
- December 3,
**Julie Rowlett** (Chalmers University)

When does a domain have a complete set of trigonometric eigenfunctions for the Laplace eigenvalue equation?
**Abstract**: This talk is based on joint work with my students, Max Blom, Henrik Nordell, Oliver Thim, and Jack Vahnberg. In 2008, Brian McCartin proved that the only polygonal domains in the plane which have a complete set of trigonometric eigenfunctions are: rectangles, equilateral triangles, hemi-equilateral triangles, and isosceles right triangles.

Trigonometric eigenfunctions are, as the name suggests, functions which can be expressed as a finite linear combination of sines and cosines. In 1980, Pierre Bérard proved that a certain type of polytopes in n dimensional Euclidean space, known as an alcoves associated to a root system and its Weyl group, also have a complete set of trigonometric eigenfunctions. In our work, we connect these results with the notion of `strictly tessellating polytope.’ We prove that the following are equivalent: (1) a polytope in R^n has a complete set of trigonometric eigenfunctions (2) a polytope strictly tessellates R^n (3) a polytope is an alcove associated to a root system and its Weyl group. This talk is intended for a general mathematical audience including students! No experience with any of these mathematical concepts is required, as everything will be explained so that everyone can follow along.

**Previous Schedule:** 2019, 2018, 2017, 2016, 2015 Spring, 2015 Winter, 2014 Fall, 2014 Spring, 2014 Winter, 2012 Winter, 2011 Fall, 2010 Spring, 2010 Winter