Metrics on surfaces which extremize eigenvalues

Speaker: 

Richard Schoen

Institution: 

Stanford University

Time: 

Monday, February 25, 2013 - 3:00pm

Location: 

RH 306

For closed surfaces and for surfaces with boundary there are natural eigenvalue extremal problems whose solutions, when they exist, determine minimal surfaces in the sphere or the ball with a natural boundary condition. We will discuss the existence problem and describe some geometric properties of extremal metrics.

Singular hermitian metrics and the Hodge conjecture

Speaker: 

Gregory Pearlstein

Institution: 

Michigan State University

Time: 

Tuesday, March 12, 2013 - 4:00pm to 5:00pm

Host: 

Zhiqin Lu

Location: 

RH 306

A Hodge class on a smooth complex projective variety gives rise to an associated hermitian line bundle on a Zariski open subset of a complex projective space P^n.  I will discuss recent work with P. Brosnan which shows that the Hodge conjecture is equivalent to the existence of a particular kind of degenerate behavior of this metric near the boundary.

The Dirichlet problem for the prescribed Ricci curvature equation

Speaker: 

Artem Pulemotov

Institution: 

University of Queensland

Time: 

Thursday, April 4, 2013 - 3:00pm to 4:00pm

Location: 

RH 306

We will discuss the following question: is it possible to find a
Riemannian metric with given Ricci curvature on a manifold $M$? To
answer this question, one must analyze a weakly elliptic
second-order partial differential equation for tensors. In the first
part of the talk, we will review the relevant background and the
history of the subject. After that, our focus will be on new results
concerning the case where $M$ is a bounded domain in a cohomogeneity
one manifold.
 

Universal profiles for mean curvature flow neckpinches

Speaker: 

Dan Knopf

Institution: 

UT Austin

Time: 

Thursday, May 2, 2013 - 4:00pm

Host: 

Jeffrey D. Streets

Location: 

RH 306

We report on recent and ongoing work with Zhou Gang and I.M.
Sigal in which we prove that all MCF neckpinches are asymptotically
rotationally symmetric. Combined with recent work of other authors, this
represents strong evidence in favor of the conjecture that MCF solutions
originating from generic initial data are constrained to one of exactly
two asymptotic singularity profiles.

Long time existence of minimizing movement solutions of Calabi flow

Speaker: 

Jeff Streets

Institution: 

UCI

Time: 

Tuesday, February 12, 2013 - 4:00pm

Location: 

RH 306

In 1982 Calabi proposed studying gradient flow of the L^2 norm
of the scalar curvature (now called Calabi flow) as a tool for finding
canonical metrics within a given Kahler class. The main motivating
conjecture behind this flow (due to Calabi-Chen) asserts the smooth long
time existence of this flow with arbitrary initial data. By exploiting
aspects of the Mabuchi-Semmes-Donaldson metric on the space of Kahler
metrics I will construct a kind of weak solution to this flow, known as a
minimizing movement, which exists for all time.

HF=ECH via open book decompositions

Speaker: 

Ko Honda

Institution: 

University of Southern California

Time: 

Tuesday, January 22, 2013 - 4:00pm

Location: 

RH 306

Floer homology theories have had an enormous impact on low-dimensional topology over the last 2-3 decades.  The goal of this talk is to introduce two Floer homology theories -- Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings) -- and to sketch a proof of the equivalence of the two.  This is joint work with Vincent Colin and Paolo Ghiggini.

The talk will be accessible to beginning graduate students.

Holomorphic functions on certain Kahler manifolds

Speaker: 

Ovidiu Munteanu

Institution: 

University of Connecticut

Time: 

Tuesday, February 26, 2013 - 4:00pm

Location: 

RH 306

We first survey some development regarding the study of holomorphic functions on manifolds. We insist mostly on Liouville theorems or, more generally, dimension estimates for the space of polynomially growing holomorphic functions. Then we present some recent joint work with Jiaping Wang on this topic. Our work is motivated by the study of Ricci solitons in the theory of Ricci flow. However, the most general results we have do not require any knowledge of curvature.

Positive curvature in Sasaki geometry

Speaker: 

Weiyong He

Institution: 

University of Oregon

Time: 

Tuesday, January 15, 2013 - 4:00pm to 5:00pm

Location: 

RH 306

This is based on joint work with Song Sun.
As an analogue of Frankel conjecture (Mori, Siu-Yau theorem) in Kahler geometry, we
classify compact Sasaki manifolds with positive curvature by deforming metrics.
Roughly speaking, such Sasaki structure is a standard Sasaki structure on (odd
dimensional) spheres. Our theorem gives a new proof of Frankel conjecture as a
special case. We have also similar results as in Kahler setting for nonnegative
curvature.

An open mirror theorem for toric varieties

Speaker: 

Siu-Cheong Lau

Institution: 

Harvard University

Time: 

Tuesday, January 8, 2013 - 4:00pm

Location: 

RH 306

Mirror map is a central object in the study of mirror symmetry. They are obtained in hypergeometric series by solving Picard-Fuchs equations. In this talk, I will explain a geometric meaning of mirror maps for toric varieties in terms of counting of holomorphic discs bounded by Lagrangian submanifolds. It is motivated by the study of SYZ mirror symmetry. This is a joint work with K. Chan, N.-C. Leung and H.-H. Tseng.

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