# 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.

# 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.

### 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.

### 3/13 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.

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

### 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.