Differential Geometry Seminar Schedule for Spring 2007

Fridays 3:00 - 4:00pm, SH 6617

4/20 Xiaodong Cao, MSRI and Cornell University ``Cross Curvature Flow on Locally Homogenous Three-manifolds"

Abstract: Recently, Chow and Hamilton introduced the cross curvature flow on three-manifolds, which is a weakly parabolic partial differential equation system when the sectional curvatures have a definite sign. They also conjectured the long time existence and convergence of cross curvature flow on closed three-manifolds with negative sectional curvature. In this talk, we will study the cross curvature flow on locally homogenous three-manifolds. We will describe the long time behavior of the cross curvature flow for each case. This is a joint work with Yilong Ni and Laurent Saloff-Coste.

5/4 Will Wylie, UCLA ``Comparison Geometry for the Bakry-Emery Tensor"

Abstract: One of the primary tools in geometric analysis is comparison theorems. We discuss extensions of some comparsion theorems for Ricci curvature, namely the volume comparison theorem and splitting theorem, to a natural generalization of the Ricci tensor, the Bakry-Emery tensor. As an application we obtain extensions of topological results involving Ricci curvature to the Bakry Emery tensor. This is joint work with Guofang Wei.

5/11 Sophie Chen, MSRI `` Conformal invariants, fully nonlinear equations and complete Einstein metrics"

5/21 (special day) John Lott, MSRI, University of Michigan ``Dimensional reduction and long-time behavior of Ricci flow"

5/25 Pengzi Miao, Monash University ``A Localized Riemannian Penrose Inequality -- An Application of A Generalized Shi-Tam Monotonicity"

Abstract: In 2002 Shi and Tam proved the positivity of Brown-York mass. In this talk we first review the key ingredients of their proof. Then we discuss a generalziation of their monotonicity formula. As an application, we derive a Riemannian Penrose type inequality for a class of compact manifolds with boundary which physcially represent bodies enclosing horizons.

6/1 Julie Rowlett, UCSB ``Spectral Geometry and Asymptotically Conic Convergence"

Abstract: In this talk I will define asymptotically conic (AC) convergence in which a family $\{g_{\epsilon}\}$ of smooth Riemannian metrics on a fixed compact manifold $M$ degenerate to a singular metric $g_0$ on a compact manifold with boundary $M_0,$ where $g_0$ has an isolated conic singularity at the boundary. This convergence is related to the analytic surgery metric degeneration of Mazzeo-Melrose and is the model problem for ongoing work of Degeratu-Mazzeo on QALE/QAC spaces. After motivating the definition of AC convergence, I will present two spectral convergence results. \begin{enumerate} \item Convergence of the spectrum of geometric Laplacians for $g_{\epsilon}$ to the spectrum of the Friedrich's extension of geometric Laplacian for $g_0.$ \item Asymptotic expansion in $\epsilon$ of the corresponding heat kernels as $\epsilon \to 0,$ with uniform convergence in $t.$ \end{enumerate} I will briefly describe the techniques of the proofs which include rescaling arguments, parametrix construction on manifolds with corners, maximum principle, and a new resolution blowup and parameter ($\epsilon$) dependent heat operator calculus developed for this work.

6/8 Nelia Charalambous, UCI ``The Yang-Mills heat equation on compact manifolds with boundary"

Abstract: Gauge theory is the study of differential equations for fields over a principal bundle. The case of a principal bundle with a nonabelian group was first introduced by R.L. Mills and C.N. Yang in order to give a model of the weak and strong interactions in the nucleus of a particle. They wanted to mirror the invariance of physics under an infinite dimensional gauge group, also known as the principle of local invariance. In this talk we will consider a gauge-theoretic heat equation, the Yang-Mills heat equation. The underlying manifold will be smooth, three-dimensional, with a nonempty boundary. We will prove the existence and uniqueness of solutions to this equation, and consider questions about its convergence at infinite time.

6/14 Yu Ding, CSU, Long Beach `` Degenerated singularities in Ricci flow"

Abstract: We discuss the asymptotic behavior of degenerated singularities in Ricci flow, on three dimensional compact manifolds. Differential Geometry Seminar Friday 2:00-3:00pm SH 6617

2/23/07 Zair Ibragimov, Univerity of Michigan ``On a theorem of R. Bott"

Abstarct: We discuss the notion of symmetric products of topological spaces introduced by Borsuk and Ulam in 1930. For $n\geq 1$, the $n$-th symmetric product of a topological space $X$ is the space of all subsets of $X$ of cordinality less than or equal to $n$ equipped with the quotient topology coming from the product space $\prod_{1}^{n}X$. Our primary focus will be on Bott's Theorem, which says that the third symmetric product $M$ of a circle $S^1$ is homeomorphic to a $3$-sphere $S^3$. I start by discussing a new proof of Bott's Theorem based on the Poincare Conjecture. Then I will discuss metric properties of $(M, d)$, where $d$ is the Hausdorff metric induced by the Euclidean metric on $S^1$. If time permits, I will discuss metric properties of the third symmetric product of the real line $R^1$.

Differential Geometry Seminar Schedule for Fall 2006

Fridays 2:00 - 3:00pm, SH 6617

9/29 organizational meeting

10/6 Rugang Ye, UCSB ``Some convergence theorems of the Ricci flow"

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