Higher dimensional Schrodinger flow into Grassmannians

Speaker: 

Professor Bo Dai

Institution: 

Peking University

Time: 

Tuesday, January 22, 2008 - 4:00pm

Location: 

MSTB 254

Schrodinger flow is the Hamiltonian flow for energy functional on the space of maps from a Riemannian manifold into a Kahler manifold. I'll talk about some background on this flow, then focus on the special case of maps from a Euclidean space into the complex Grassmannian Gr(k,C^n). Terng and Uhlenbeck proved that Schrodinger flow of maps from R^1 into complex Grassmannian is gauge equivalent to the matrix nonlinear Schrodinger equation. Using this gauge equivalence and the result of Beals and Coifman, they obtained the global existence of Schrodinger flow with rapidly decay initial data. Applying the method of Terng and Uhlenbeck, we will see that Schrodinger flow of radial maps from R^m into the complex Grassmannian is gauge equivalent to a generalized matrix nonlinear Schrodinger equation. When the target is the 2-sphere, the gauge equivalence was studied by Lakshmanan and his colleagues by different method. They also observed that if the domain is R^2, then the corresponding matrix nonlinear Schrodinger equation is an integrable system.

On the $\sigma_2$-scalar curvature and its application

Speaker: 

Professor Yuxin Ge

Institution: 

University Paris 12 & U. Washington

Time: 

Tuesday, March 11, 2008 - 4:00pm

Location: 

MSTB 254

In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry. As application, we prove that, if a compact 3-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int
\sigma_2\ge 0$, then topologically $M$ is a quotient of sphere.

A regularity theory for solutions to the prescribed mean curvature equation with unattained Dirichlet data

Speaker: 

Dr. Theodora Bourni

Institution: 

Stanford

Time: 

Tuesday, December 4, 2007 - 4:00pm

Location: 

MSTB 254

We present some new results concerning the Dirichlet problem for the prescribed mean curvature equation over a bounded domain in R^n. In the case when the mean curvature is zero this can be posed variationally as the problem of finding a least area representative among functions of bounded variation with prescribed boundary values. We show that there is always a minimizer which is represented by a compact C^{1,alpha} manifold with boundary, with boundary given by the prescribed Dirichlet data, provided this data is C^{1,alpha} and it is of class C^{1,1} if the prescribed data is C^3.

Simply connected surfaces of general type with p_g = 0 and K^2 = 3

Speaker: 

Professor Jongil Park

Institution: 

Seoul National University and MSRI

Time: 

Tuesday, January 15, 2008 - 4:00pm

Location: 

MSTB 254

One of the fundamental problems in the classification of complex surfaces is to find a new family of simply connected surfaces with p_g = 0 and K^2 > 0. In this
talk, I will sketch how to construct a new family of simply connected symplectic 4- manifolds using a rational blow-down surgery and how to show that such 4-manifolds
admit a complex structure using a Q-Gorenstein smoothing theory. In particular, I will show explicitly how to construct a simply connected minimal surface of general
type with p_g = 0 and K^2 = 3.

If time allows, I will also sketch how to construct a simply
connected, minimal, symplectic 4-manifold with b_+2 = 1 (equivalently, p_g = 0) and K^2 = 4 using a rational blow-down surgery.

Metric Degeneration and Spectral Convergence

Speaker: 

Professor Julie Rowlett

Institution: 

UCSB

Time: 

Tuesday, March 4, 2008 - 4:00pm

Location: 

MSTB 254

Consider a family of smooth compact connected $n$ dimensional Riemannian manifolds. What can one say about the spectral geometry of a limit of these?

This question has interested many spectral geometers; my talk focuses on conical metric degeneration in which the family converges "asymptotically conically'' to an open manifold with conical singularity. I will present spectral convergence results and discuss techniques including microlocal analysis on manifolds with corners and geometric blowup constructions. I will also summarize spectral convergence results for other geometric contexts and discuss applications and open questions.

Tight contact structures on Seifert fibered 3-manifolds

Speaker: 

Professor Andras Stipsciz

Institution: 

Renyi Mathematics Institute, Budapest and Columbia

Time: 

Tuesday, November 6, 2007 - 4:00pm

Location: 

MSTB 254

After discussing the basics of contact surgery in dimension 3, and introducing contact Ozsvath-Szabo invarinats, we show that a Seifert fibered 3-manifold does admit a positive tight contact structure unless it is orientation preserving diffeomorphic to the result of (2n-1)-surgery along the T(2,2n+1) torus knot (for some positive integer n).

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