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: 

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: 

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.

 

From Barbilian Geometries to Gromov Hyperbolic Spaces

Speaker: 

Bogdan Suceava

Institution: 

Cal State, Fullerton

Time: 

Tuesday, October 17, 2017 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In 1934, Wilhelm Blaschke’s attention focused on a recent construction in metric geometry proposed by Dan Barbilian as a generalization of various models of hyperbolic geometry. It was the year when S.-S. Chern started his doctoral program under Blaschke’s supervision in Hamburg and when in several academic centers in Europe scholars were interested in generalizations of Riemannian geometry. Introduced originally in 1934, Barbilian’s metrization procedure induces a distance on a planar domain through a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian generalized this process to more general domains. In our discussion we plan to show that these spaces are naturally related to Gromov hyperbolic spaces. In several works written with W.G. Boskoff, we explore this connection. We conclude our talk by stating several open problems related to this content.

Isometric embeddings via heat kernel

Speaker: 

Ke Zhu

Institution: 

Minnesota State University

Time: 

Monday, June 5, 2017 - 4:00pm

Location: 

RH 340N

The Nash embedding theorem states that every Riemannian
manifold can be isometrically embedded into some Euclidean space with
dimension bound. Isometric means preserving the length of every
path. Nash's proof involves sophisticated perturbations of the
initial embedding, so not much is known about the geometry of the
resulted embedding.
     In this talk, using the eigenfunctions of the Laplacian
operator, we construct canonical isometric embeddings of compact
Riemannian manifolds into Euclidean spaces, and study the geometry of
embedded images. They turn out to have large mean curvature
(intuitively, very bumpy), but the extent of oscillation is about the
same at every point. More can be said about global quantities like
the center of mass. This is a joint work with Xiaowei Wang.

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