# A priori estimates for stable solutions to nonlinear elliptic equations

Kelei Wang

Wuhan University

## Time:

Tuesday, June 1, 2021 - 4:00pm to 5:00pm

## Location:

Zoom

In many elliptic variational problems, one usually needs the information on the stability or Morse index to get some good a priori estimates. In this talk I will review a common phenomenon about this estimate and some difficulties for semilinear elliptic equations. I will also discuss my joint work with J. Wei on stable solutions of Allen-Cahn equation, where we get a uniform second order estimate on clustering interfaces.

# Gravitational instantons and K3 surfaces

Jeff Viaclovsky

UCI

## Time:

Tuesday, April 6, 2021 - 4:00pm to 5:00pm

## Location:

https://uci.zoom.us/j/94657362218

There are many interesting examples of complete non-compact Ricci-flat metrics in dimension 4, which are referred to as ALE, ALF, ALG, ALH gravitational instantons. In this talk, I will describe some examples of these geometries, and other types called ALG^* and ALH^*. All of the above types of gravitational instantons arise as bubbles for sequences of Ricci-flat metrics on K3 surfaces, and are therefore important for understanding the behavior of Calabi-Yau metrics near the boundary of the moduli space.  I will describe some general aspects of this type of degeneration, and some recent work on degenerations of Ricci-flat metrics on elliptic K3 surfaces in which case ALG and ALG^* bubbles can arise. This is based on joint work with Hein-Song-Zhang (to appear in JAMS) and with Chen-Zhang (to appear in CAG).

# The construction of the splitting maps

Guoyi Xu

## Institution:

Tsinghua University

## Time:

Tuesday, March 9, 2021 - 4:00pm

## Location:

Zoom

For a geodesic ball with non-negative Ricci curvature and almost
maximal volume, we give the existence proof of splitting map without
compactness argument. There are two technical new points, the first one is
the way of finding n-directional points by induction and stratified Gou-Gu
Theorem, the second one is the error estimates of projections. The content
of the talk is technical, but we will explain the basic geometric intuition
behind the technical proof. This is a joint work with Jie Zhou.

# Ancient mean curvature flows and their applications to topology

Kyeongsu Choi

## Institution:

Korea Institute for Advanced Study (KIAS)

## Time:

Tuesday, February 9, 2021 - 4:00pm to 5:00pm

## Location:

Zoom

The mean curvature flow is an evolution of hypersurfaces satisfying a geometric heat equation. The mean curvature flow in general develops singularities, and the topology of the hypersurface is changed through singularities. To study the topological change, we consider ancient flows which can be obtained by the blow-up of the flow at singularities. In this talk, we will discuss how to use the classification result of ancient flows for singularities analysis and topology.

# Unitary Representations of 3-manifold Groups and the Atiyah-Floer Conjecture

Aliakbar Daemi

## Institution:

Washington University in St. Louis

## Time:

Tuesday, February 23, 2021 - 4:00pm

## Location:

Zoom

A useful tool to study a 3-manifold is the space of the
representations of its fundamental group, a.k.a. the 3-manifold group, into
a Lie group. Any 3-manifold can be decomposed as the union of two
handlebodies. Thus, representations of the 3-manifold group into a Lie group
can be obtained by intersecting representation varieties of the two
handlebodies. Casson utilized this observation to define his celebrated
invariant. Later Taubes introduced an alternative approach to define Casson
invariant using more geometric objects. By building on Taubes' work, Floer
refined Casson invariant into a graded vector space whose Euler
characteristic is twice the Casson invariant.  The Atiyah-Floer conjecture
states that Casson's original approach can be also used to define a graded
vector space and the resulting invariant of 3-manifolds is isomorphic to
Floer's theory. In this talk, after giving some background, I will give an
exposition of what is known about the Atiyah-Floer conjecture and discuss
some recent progress, which is based on a joint work with Kenji Fukaya and
Maksim Lipyanskyi. I will only assume a basic background in algebraic
topology and geometry.

# Weak solutions for general complex Hessian equations

Slawomir Dinew

## Institution:

Jagiellonian University

## Time:

Tuesday, January 12, 2021 - 3:00pm to 4:00pm

## Location:

Zoom

Joint with Analysis seminar.

Pluripotential solutions to the complex Monge-Ampere equations are very useful  in modern complex geometry. It is thus very natural to seek analogues for other Hessian type complex equations that appear in geometric analysis. In the talk I will discuss several approaches towards such a theory and their limitations. Then I will focus on a newly coined notion of a $L^p$-viscosity solutions and discuss uniform estimates for a large class of Hessian equations. This is a joint work with S. Abja and G. Olive.

# Hermitian geometry and curvature positivity

Freid Tong

## Institution:

Columbia University

## Time:

Tuesday, December 8, 2020 - 4:00pm to 5:00pm

## Location:

Zoom

A general theme in differential/complex geometry is that curvature positivity conditions imposes strong geometric and topological constraint on the underlying manifold. In this talk, I will discuss a new curvature positivity condition in Hermitian geometry and prove a liouville type theorem for (1, 1)-forms for manifolds satisfying the positivity condition. I will discuss various interactions between this curvature condition and other notions in Hermitian geometry. Lastly, I will discuss some examples and potential applications.

# PDEs on Non-Kahler Calabi-Yau Manifolds

Sebastien Picard

## Institution:

University of British Columbia (UBC)

## Time:

Tuesday, December 1, 2020 - 4:00pm to 5:00pm

## Location:

ZOOM

We will discuss the non-Kahler Calabi-Yau geometry introduced by string theorists C. Hull and A. Strominger. We propose to study these spaces via a parabolic PDE which is a nonlinear flow of non-Kahler metrics. This talk will survey works with T. Collins, T. Fei, D.H. Phong, S.-T. Yau, and X.-W. Zhang.

# On categorical dynamical systems

Yu-Wei Fan

UC Berkeley

## Time:

Tuesday, October 13, 2020 - 4:00pm

## Location:

Zoom

One can regard a self-map on a space as a dynamical system, and study its
long-term behavior under larger iterations. In this talk, we will introduce
the categorical version of such dynamical systems, and establish invariants
that one can associate to endofunctors of categories from the dynamical
perspective. We will also explain some results on categorical dynamical
systems that are related to holomorphic dynamics, symplectic dynamics,
Teichmuller theory, and Poincare rotation numbers. No background in
dynamical systems is assumed for this talk.

# Generalized soap bubbles and the topology of manifolds with positive scalar curvature

Chao Li

## Institution:

Princeton University

## Time:

Tuesday, October 27, 2020 - 4:00pm

## Location:

Zoom

Abstract: It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. I will present some recent progress on this question, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov), as well as complete metrics of positive scalar curvature on an arbitrary manifold connect sum with a torus. Applications include a Schoen-Yau Liouville theorem for all locally conformally flat manifolds. The proofs of these results are based on analyzing generalized soap bubbles - surfaces that are stable solutions to the prescribed mean curvature problem. This talk is based on joint work with O. Chodosh.