Homological rigidity of Schubert varieties in compact Hermitian symmetric spaces

Speaker: 

Professor Colleen Robles

Institution: 

Texas A&M

Time: 

Tuesday, March 1, 2011 - 3:00pm

Location: 

RH 306

The integral homology of a compact Hermitian symmetric spaces (CHSS) is generated by the homology classes of its Schubert varieties. Most Schubert varieties are singular. In 1961 Borel and Haefliger asked: when can the homology class [X] of a singular Schubert variety be represented by a smooth subvariety Y of the CHSS?

Remarkably, the subvarieties Y with [Y] = [X] are integrals of a (linear Pfaffian) differential system. I will discuss recent work with Dennis The in which we give a complete list of those Schubert varieties X for which there exists a first-order obstruction to the existence of a smooth Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong.

The sine qua non of our analysis is a new characterization of the Schubert varieties by a non-negative integer and a marked Dynkin diagram. The description generalizes the well-known characterization of the smooth Schubert varieties by subdiagrams of the Dynkin diagram associated to the CHSS.

I will illustrate the talk with examples.

How to recognize a Kahler metric?

Speaker: 

Professor Maciej Dunajski

Institution: 

University of Cambridge, United Kingdom

Time: 

Tuesday, April 12, 2011 - 4:00pm

Location: 

RH 306

Let (M, g) be Riemannian four-manifold. Does there exist a
non-zero function f:M->R such that
(*) f^2 g is flat?
(**) f^2 g satisfies Einstein equations?
Most people know the answer to (*). Nobody (really) knows the full
answer to (**). In this talk I will provide the answer to
(***) f^2 g is Kahler for some Kahler form?

Short-time existence of the Ricci flow on non-compact Riemannian manifolds

Speaker: 

Professor Guoyi Xu

Institution: 

UCI

Time: 

Tuesday, October 26, 2010 - 4:00pm

Location: 

RH 306

In this talk, using the local Ricci flow, we prove the short-time
existence of the Ricci flow on noncompact manifolds, whose Ricci curvature
has global lower bound and sectional curvature has only local average integral
bound. The short-time existence of the Ricci flow on noncompact manifolds
was studied by Wan-Xiong Shi in 1990s, who required a point-wise bound of
curvature tensors. As a corollary of our main theorem, we get the short-time existence part of Shis theorem in this more general context.

Local Curvature Bound in Ricci flow

Speaker: 

Professor Peng Lu

Institution: 

U. Oregon

Time: 

Tuesday, November 2, 2010 - 4:00pm

Location: 

RH 306

In this talk by using the idea in the proof of Perelman's pseudo locality theorem we will derive a local curvature bound in Ricci flow assuming only local sectional curvature bound and local volume lower bound for the initial metric.

This result is closely related to Theorem 10.3 in Perelman's entropy paper.

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