# Geometric Analysis Seminar

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

### 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