# Differential Geometry Seminar Schedule for
Spring 2009

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

### 4/10 Ben Weinkove, UCSD
``The Kahler-Ricci flow on Hirzebruch surfaces"

Abstract: I will discuss the metric behavior of the Kahler-Ricci flow on Hirzebruch surfaces assuming that the initial metric is invariant under a maximal compact subgroup of the automorphism group. I will describe how, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to P^1 or contracts an exceptional divisor. This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work with Jian Song.
### 5/29 Zhiqin Lu, UCI
``Gauss-Bonnet theorem on Moduli space"

Abstract: In this talk, I will first give an overview of the
Weil-Petersson geometry on Calabi-Yau moduli space, which includes the
local theory, the semi-global theory, and the global theory of CY
moduli. Then I will present the recent joint result with M. Douglas:
the Gauss-Bonnet type theorems on moduli space, with their
applications in string theory.
# Differential Geometry Seminar Schedule for
Winter 2009

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

### 1/16 Julie Rowlett, UCSB
``Spectral Theory and Dynamics of Asymptotically Hyperbolic Manifolds"

Abstract: Manifolds with asymptotic structures at infinity are of
natural interest to mathematicians and physicists since these models
describe the geometry of the universe. Four dimensional Poincar\'e
Einstein manifolds which arise in AdS-CFT correspondence are examples
of asymptotically hyperbolic manifolds. In this talk, I will first
review the basic spectral theory of the Laplacian and dynamical theory
of the geodesic flow for compact manifolds and for infinite volume
hyperbolic manifolds. I will then present new results for (n+1)
dimension asymptotically hyperbolic manifolds with negative (but not
necessarily constant) sectional curvatures: a dynamical wave trace
formula, a prime orbit theorem for the geodesic flow based on the
dynamical zeta function, and a result which relates the pure point
spectrum of the Laplacian to the topological entropy of the geodesic
flow. Key techniques and ideas from the proofs will be summarized,
concluding with open problems. It is my aim to keep the talk widely
accessible and non-technical.
### 1/23 please attend Distinguished Lecture by Edward Frenkel

### 2/20 Guofang Wei, UCSB
``Smooth Metric Measure Spaces"

Abstract: Smooth metric measure spaces are Riemannian manifolds with a conformal change of the Riemannian measure and occur naturally as measured Gromov-Hausdorff limit of Riemannian manifolds. The important curvature quantity here is the Bakry-Emery Ricci tensor, which corresponds to the (synthetic) Ricci curvature lower bound for (nonsmooth) metric measure spaces. What geometric and topological results for Ricci curvature can be extended to Bakry-Emery Ricci tensor? Recently there are many developments. We will discuss comparison geometry and rigidity in this direction.
### 3/6 Burkhard Wilking, University of Muenster
``Ricci flow in high dimensions"

### 3/13 Owen Dearricott, UCR
``Positive curvature on 3-Sasakian 7-manifolds"

Abstract: We discuss metrics of positive curvature on 3-Sasakian 7-manifolds including one on a new diffeomorphism type.
# Differential Geometry Seminar Schedule for
Fall 2008

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

### 9/26, organization meeting

### 10/3 Yu-Jen Shu, UCSB
``Rigidity of Quasi-Einstein Metrics"

Abstract: We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics.
We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.
### 10/10 Mark Haskins, Imperial College, visiting MSRI
``Gluing constructions of special Lagrangian cones"

Abstract: Special Lagrangian submanifolds are a special type of higher-dimensional minimal submanifold that occur naturally in Calabi-Yau manifolds. They have been the focus of much attention from both mathematicians and string theorists because of their role in Mirror Symmetry. Singularities of special Lagrangians play a very important part in this story but as yet are poorly understood.
We will discuss how gluing methods can be used to construct a huge range of new special Lagrangian singularity types. In this talk we will focus on our construction of cones over compact orientable surfaces of any odd genus and sketch extensions to higher dimensions as time permits. This is joint work with Nicos Kapouleas.
### 10/17 David Trotman, University of Provence, visiting MSRI
``Tame geometry of stratified spaces"

Abstract: S. S. Chern stated in 1991 that the future of Differential
Geometry would involve extending the smooth manifold theory to stratified
spaces. As every compact smooth manifold is diffeomorphic to some real
algebraic variety (Nash-Tognoli-Akbulut-King), it is natural to study the
class A of those singular spaces which are diffeomorphic to singular real
algebraic varieties, what Grothendieck calls 'tame' stratified spaces. I
shall describe some recent results and some conjectures about the geometry
of stratified spaces in A.
### 10/24 Jeffrey Case, UCSB
``On The Nonexistence of Quasi-Einstein Metrics"

Abstract: We study complete Riemannian manifolds satisfying Ric^m_f
= 0 by
studying the associated PDE \Delta_f f+ m \mu e^{2f/m} = 0 for \mu \le 0, showing that
there are no solutions with f non-constant. We prove a generalization of the
Keller-Osserman theorem for Bakry-Emery-Ricci curvature bounded below,
analogous to the generalization of Cheng and Yau for Ricci curvature bounded
below. Together with X.-D. LiÕs generalization of the Harnack inequality for
f-harmonic functions on manifolds with Bakry-Emery-Ricci curvature lower
bounds, we are able to show nonexistence. When m < \infty, this shows that
there are no Ricci flat Einstein manifolds which can be realized as the nontrivial
warped product with base a Riemannian manifold and fiber an Einstein
manifold with nonpositive Einstein constant.
### 10/31 Lei Ni, UCSD
``Complete manifolds with pinched curvature"

Abstract: In this talk I shall explain how one can use Ricci flow to show that
any complete Riemannian manifold (with dimension $\ge 3$)
whose curvature operator is bounded and satisfies the pinching
condition $R\ge \delta \frac{\tr(R)}{2n(n-1)}\I>0$, for some
$\delta>0$, must be compact.
### 11/7 no meeting, please attend 11/6's colloquium by Fred Wilhelm instead

### 11/14 Rugang Ye, UCSB
``Nonlinear energy estimate and entropy functional for Ricci flow"

### 11/21 Rugang Ye, UCSB
``Bounded curvature in the Ricci flow at bounded distance and
in bounded time interval"

### 12/5 Rugang Ye, UCSB
``Bounded curvature in the Ricci flow at bounded distance and
in bounded time interval, continue"

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