# 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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