Geometry & Analysis Seminar Fall 2014

One of the most active field in geometric analysis is the rigidity phenomena involving scalar curvature, which seems to be the weakest curvature quantity. However, the study showed that scalar curvature can in fact have a great impact on the geometry and topology of the manifold. It is more rigid than one expected.
In this talk, I will introduce the so-called static spaces in general relativity and discuss both rigidity and non-rigidity results involving scalar curvature in vacuum static spaces based on some of my recent research work joint with Professor Jie Qing.

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Friday October 24, 2014 - Room 4130

"Star-shaped mean curvature flow"

Longzhi Lin, University of California, Santa Cruz

A one-parameter family of hypersurfaces in Euclidean space evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. It can be viewed as a geometric heat equation, i.e., it is locally moving in the direction of steepest descent for the volume element, deforming surfaces towards optimal ones (minimal surfaces). This equation has been used and studied in material science to model things like cell, grain, and bubble growth back in the 1920’s. In this talk we will discuss some recent work on the local curvature estimate and convexity estimate for the star-shaped mean curvature flow and the consequences. This is joint work with Robert Haslhofer.

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Friday October 31, 2014 - Room 4130

"Delay Differential Equations, Brain Dynamics and Synchronization Theory"

Yusuf Goren, University of California, Santa Cruz

I will talk about our joint work with Maria Schonbek, Gustavo Deco, and Morten Kringelbach about a delay differential model of brain dynamics. I will explain the fixed point approach in proving the existence and uniqueness. I will also talk about other possible approaches to model brain dynamics and how synchronization theory might help us understand this dynamics better.

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Friday November 7, 2014 - Room 4130

"Blow up analysis and compactness of solutions to the Yamabe problem"

Yi Fang, University of Science and Technology of China

In the talk, I will report the research on the compactness of all positive solutions to the Yamabe problem.  This program was proposed by Richard Schoen for manifolds that are not conformally diffeomorphic to the sphere. Using blow up analysis and the Positive Mass Theorem, Schoen and other mathematicians had proved that the compactness holds for dimension less than 25. When the dimension is bigger than 24, Simon Brendle and Fernando Marques constructed counterexamples.

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3:45PM - 4:45PM Wednesday November 12, 2014 - Room 4130 - DATE AND TIME CHANGE

"Wave maps from the hyperbolic plane"

Sung-Jin Oh,  UC Berkeley

In this talk, we consider equivariant wave maps from the hyperbolic plane into two model rotationally symmetric targets, namely the two sphere ($\mathbb{S}^{2}$) and the hyperbolic plane itself ($\mathbb{H}^{2}$). Due to the non-Euclidean geometry of the domain, this problem exhibits markedly different phenomena compared to its Euclidean counterpart. For instance, there exist numerous stationary solutions to not only $\mathbb{S}^{2}$ but also $\mathbb{H}^{2}$, which has a negative constant curvature. Moreover, when the target is $\mathbb{S}^{2}$, the spectrum of the linearized operator about certain stationary solutions possesses a \emph{gap eigenvalue}, i.e., a simple eigenvalue in the gap $(0, 1/4)$ between $0$ and the essential spectrum. (joint with A. Lawrie and S. Shahshahani)

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Thursday November 20, 2014 - Room 4130 - DATE CHANGE

Minimal Surfaces with Arbitrary Topology in H^2xR

Baris Coskunuzer, Koc University and MIT

In this talk, we show that any open orientable surface can be embedded in H^2xR as a complete area minimizing surface. Furthermore, we will discuss the asymptotic Plateau problem in H^2xR, and give a fairly complete solution.

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Friday November 28, 2014 NO Seminar - Thanksgiving Holiday

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Friday December 5, 2014 - Room 4130

Conley Conjecture for Lagrangian Correspondences

Yusuf Goren, University of California, Santa Cruz

The well-known Conley Conjecture states that any Hamiltonian diffeomorphism of a symplectic manifold (with some extra conditions on the manifold) has infinitely many periodic orbits. There are many proofs in the literature which proves the conjecture for symplectically aspherical manifolds, negative monotone manifolds, etc. As a generalization, motivated by the fact that the graph of a Hamiltonian diffeomorphism \(\varphi: M \rightarrow M\) is a Lagrangian isotopic to the diagonal in \(M\times M\), the question now becomes if a similar Conley Conjecture type result hold for Lagrangians \(L \subset M\times M\). I will start by introducing Lagrangian correspondences, pose the dynamics problem and prove that, under some restrictions, there exist infinitely many periodic orbits for Lagrangian correspondences.

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Friday December 12, 2014 - Room 4130

"Prescribed mean curvature equation in Riemannian manifolds"

Jorge Lira, Universidade Federal do Ceará (Brazil)

We survey some existence and non-existence results for graphs with prescribed mean curvature into Riemannian manifolds endowed with Killing or conformal vector fields. We discuss in detail gradient estimates for both elliptic and parabolic versions of the existence problem.

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