# Geometric Analysis Seminar

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

### Spring Quarter, 2018

- April 10,
**Peter Gilkey** (UO)

Homogeneous affine connections on surfaces
- April 17,
**Adam Layne** (UO)
- May 1,
**Boris Bottvinick** (UO)
- May 8,
**Xuemiao Chen** (UC Berkeley)

Singularities of Hermitian-Yang-Mills connection
**Abstract**: Given a singular Hermitian-Yang-Mills connection A and assuming certain conditions, we will characterize the analytic tangent cones of A at an isolated singularity p by using the local algebraic geometric data near p given by the holomorphic bundle E determined by A. The uniqueness is a consequence. (Joint with Song Sun.)

- May 15,
**Jiuru Zhou**

On tamed almost complex four-manifolds
**Abstract**: In this talk, by adding a technical assumption, we give an affirmative answer to Donaldsons tamed to compatible question.

- May 22,
**Micah Warren** (UO)

Compactness of Hamiltonian stationary submanifolds
- May 29,
**Siqi He** (CalTech)

**Note** Joint seminar with Topology seminar at 11am.

A Kobayashi-Hitchin correspondence for the extended Bogomonly Equations
**Abstract**: We will discuss Witten’s gauge theory approaches to define the Jones polynomial for a knot over general 3-manifold by counting solutions to some gauge theory equations. We will discuss a Kobayashi-Hitchin type correspondence for the dimensional reduction of these gauge equations. This talk will base on joint works with R. Mazzeo.

### 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,
**Florian Beyer** (University of Otago)

Singular solutions of the Einstein-Euler equations
**Abstract**: Einstein’s equations can be considered as a particular geometric evolution system where the underlying PDEs are essentially of nonlinear wave equation type. Similar to other famous geometric evolution problems, there has been a lot of effort over the last decades to study the formation of singularities of solutions. In general this turns out to be a formidable (and essentially unsolved) task due to the complexity of Einstein’s equations. In this talk I will discuss a recent result by P. LeFloch and myself (based on joint work with J. Isenberg and E. Ames) regarding singular solutions of the coupled Einstein-Euler equations. I will also explain the underlying notion of a “singular initial value problem”.

- February 27,
**Jiuru Zhou**

J-anti-invariant cohomology on almost Kahler 4-manifolds
**Abstract**: In this talk, we calculate the dimension of the J-anti-invariant cohomology subgroup on torus T^4. Inspired by this concrete example, we get that: On a closed symplectic 4-dimensional manifold

,dimension of the J-anti-invariant cohomology subgroup vanishes for generic

-compatible almost complex structures.

- March 6,
**Yongjia Zhang** (UCSD)

Noncollapsed Type I ancient solutions to the Ricci flow in low dimensions
**Abstract**: Ancient solutions to the Ricci flow are very important for the understanding of singularity formation, and strongly utilized in Hamilton-Perelman`s proof of geometrization conjecture. In this talk, I will discuss my proof of the classification of three-dimensional Type I noncollapsed ancient solutions to the Ricci flow. If there were time, I will also talk about a similar rigidity result in dimension four.

- March 13,
**Arunima Bhattacharya** (UO)

Schauder estimates for the Hamiltonian stationary equation
**Abstract**: We consider the Hamiltonian stationary equation for all phases in dimension two. We show that solutions that are

will be smooth and we also derive a

estimate for it.

### 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