# Geometric Analysis Seminar

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

### Winter Quarter, 2018

- January 9, Organizational Meeting
- January 16,
**Peter Gilkey** (UO)

Constructing neutral signature 4-dimensional Bach flat manifolds using the Riemannian extension
- January 23,
**Gao Chen** (IAS)

Classification of gravitational instantons
**Abstract**: A gravitational instanton is a noncompact complete Calabi-Yau surface with faster than quadratic curvature decay. In this talk, I will discuss the classification of gravitational instantons. This is a joint work with Xiuxiong Chen.

- January 30,
**Christine Escher** (Oregon State)

Non-negatively curved 6-manifolds with almost maximal symmetry rank.
**Abstract**: In joint work with Catherine Searle we classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism. I will describe the classification result and give an outline of the proof.

- February 13,
**Weiyong He** (UO)

On existence of Kahler metrics of constant scalar curvature (csck)
**Abstract**: Recently Chen and Cheng have made surprising breakthrough on existence of Kahler metrics of constant scalar curvature, where they are able to solve the long standing problem of Calabi-Donaldson program. Technically they solved a fourth order nonlinear elliptic equation of scalar curvature type, assuming necessary properness condition; the key is to derive a priori estimates of scalar curvature type equations (fourth order on Kahler manifolds). We will report their achievements and extend their results to Calabi’s extremal metric.

- February 20,
**Adam Layne** (UO)
- February 27,
**Jiuru Zhou**
- March 6,
**Yongjia Zhang** (UCSD)
- March 13,
**Arunima Bhattacharya** (UO)

### Fall Quarter, 2017

- October 10,
**Nikolai Saveliev** (University of Miami)

Index theory on manifolds with periodic ends
**Abstract**: We extend the Atiyah, Patodi, and Singer index theorem for Dirac operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. The index is expressed in terms of a new periodic eta-invariant that equals the classical eta-invariant in the cylindrical setting. We use this theorem to study Riemannian metrics of positive scalar curvature on some even-dimensional manifolfds, as well as the Seiberg-Witten invariants of 4-manifolds with integral homology of S^1×S^3. This is a joint work with Jianfeng Lin, Tom Mrowka, and Daniel Ruberman.

- October 17,
**Micah Warren** (UO)

Continuum Nash Bargaining Solutions
**Abstract**: Nash`s classical bargaining solution suggests that n players should maximize a product of utility functions. We consider a special case: Suppose that the players are chosen from a continuum, distribution μμ and suppose they are to divide up a resource νν that is also on a continuum, whose utility to each player is the inverse of exponential of a cost function. The maximization problem becomes an optimal type transport problem, where the target density is the minimizer to the functional F(β)=Hν(β)+W2(μ,β) where H is the entropy and W is the Wasserstein distance. One may recognize this problem from a single time step in the Jordan-Kinderlehrer-Otto scheme. Thanks to optimal transport theory, the solution may be described as by a potential that solves a fourth order nonlinear elliptic PDE.

- October 24,
**Peng Lu** (UO)

New proofs of Perelman`s theorem on shrinking Breathers in Ricci flow
- October 31,
**Shaosai Huang** (Stony Brook)

ϵ -Regularity for 4-dimensional shrinking Ricci solitons
**Abstract**: A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local

-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as ϵ -regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an ϵ -regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant.

- November 14,
**Jim Isenberg** (UO)

Non-Kaehler Ricci Flows that Converge to Kaehler-Ricci Solitons
- November 17,
**Richard Bamler**

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