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.

 

 

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