Algebraic Gluing of Holomorphic Discs in K3 Surfaces and Tropical Geometry

Speaker: 

Yu-Shen Lin

Institution: 

Harvard CMSA

Time: 

Monday, October 30, 2017 - 4:00pm

Location: 

RH 340P

We will start from the motivation of the tropical geometry. Then
we will explain how to use Lagrangian Floer theory to establish the
correspondence between the weighted counting of tropical curves to the
counting of holomorphic discs in K3 surfaces. In particular, the result
provides the existence of new holomorphic discs which do not come easily
from direct gluing argument.

 

 

Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness

Speaker: 

Qi Zhang

Institution: 

UC Riverside

Time: 

Tuesday, November 21, 2017 - 4:00pm

Host: 

Jeffrey D. Streets

Location: 

RH 306

Consider the scaling invariant, sharp log entropy (functional)
introduced by Weissler on noncompact manifolds with nonnegative Ricci
curvature. It can also be regarded as a sharpened version of
Perelman's W entropy  in the stationary case. We prove that it has a
minimizer if and only if the manifold is isometric to $\R^n$.
Using this result, it is proven that a class of noncompact manifolds
with nonnegative Ricci curvature is isometric to $\R^n$. Comparing
with some well known flatness results in on asymptotically flat
manifolds and asymptotically locally Euclidean (ALE) manifolds, their
decay or integral condition on the curvature tensor is replaced by the
condition that the metric converges to the Euclidean one in C1 sense
at infinity. No second order condition on the metric is needed.

Tangent cones of Yang-Mills connections with applications to G2 instantons

Speaker: 

Adam Jacob

Institution: 

UC Davis

Time: 

Tuesday, February 27, 2018 - 4:00pm to 5:00pm

Host: 

Zhiqin Lu

Location: 

RH 306

The study of tangent cones in geometric analysis is an important tool in understanding the structure at a singular point of a geometric equation. In this talk I will discuss how to uniquely identify the tangent cone of a Yang-Mills connection with isolated singularity in the complex setting, given an initial assumption on the complex structure of the bundle. I will then discuss applications to a project with the goal of constructing examples of singular G2 instantons, using the twisted connected sum construction. This is joint work with H. Sa Earp and T. Walpuski.

Four-dimensional shrinking Ricci solitons with nonnegative isotropic curvature

Speaker: 

Xiaolong Li

Institution: 

UC Irvine

Time: 

Tuesday, October 3, 2017 - 4:00pm

Location: 

RH 306

We show that a four-dimensional complete gradient shrinking Ricci
soliton with positive isotropic curvature is either a quotient of S^4 or
a quotient of S^3 x R. We also give a classification result on
four-dimensional gradient shrinking Ricci solitons with non-negative
isotropic curvature. This is joint work with Lei Ni and Kui Wang.

Non-Kahler Ricci flow singularities that converge to Kahler-Ricci solitons

Speaker: 

Dan Knopf

Institution: 

UT Austin

Time: 

Tuesday, February 13, 2018 - 4:00pm to 5:00pm

Host: 

Jeffrey D. Streets

Location: 

RH 306

We describe Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kahler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered in 2003 by Feldman, Ilmanen, and the speaker. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kahler solutions of Ricci flow that become asymptotically Kahler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kahler metrics under Ricci flow.

Localizing the Fukaya category of a Stein manifold

Speaker: 

Sheel Ganatra

Institution: 

USC

Time: 

Tuesday, November 28, 2017 - 4:00pm

Location: 

RH 306

We introduce a new class of non-compact symplectic manifolds called
Liouville sectors and show they have well-behaved, covariantly functorial
Fukaya categories.  Stein manifolds frequently admit coverings by Liouville
sectors, which can be used to understand the Fukaya category of the total
space (we will study this geometry in examples). Our first main result in
this setup is a local-to-global criterion for generating Fukaya categories.
Our eventual goal is to obtain a combinatorial presentation of the Fukaya
category of any Stein manifold. This is joint work (in progress) with John
Pardon and Vivek Shende.

Holomorphic isometries from the Poincare disc into bounded symmetric domains

Speaker: 

Yuan Yuan

Institution: 

Syracuse University

Time: 

Tuesday, September 26, 2017 - 4:00pm to 5:00pm

Location: 

RH 306

I will first overview the classical holomorphic isometry problem between complex manifolds, in particular between bounded symmetric domains. When the source is the unit ball, in general the characterization of holomorphic isometries to bounded symmetric domains is not quite clear. With Shan Tai Chan, we recently characterized the holomorphic isometries from the Poincare disc to the product of the unit disc with the unit ball and it  provided new examples of holomorphic isometries from the Poincare disc into irreducible bounded symmetric domains of rank at least 2.

 

 

On closedness of ALE SFK metrics on minimal ALE Kahler surfaces

Speaker: 

Jiyuan Han

Institution: 

University of Wisconsin, Madison

Time: 

Tuesday, September 19, 2017 - 4:00pm to 5:00pm

Host: 

Jeff Viaclovsky

Location: 

RH 306

Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with Jeff Viaclovsky.

 

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