# Fridays 3:30 - 4:30pm, SH 6617

### 4/19 Guofang Wei, UCSB Universal Covers for Hausdorff Limits of Noncompact Spaces"

Abstract: Joint with Christina Sormani, we prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature then $Y$ has a universal cover, extending our earlier result on compact case.

### 4/26 Zhiqin Lu, UCI Weil-Petersson geometry and Yau-Schwartz Lemma"

Abstract: It is known that the curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau manifolds are neither positive nor negative. We give a natural metric, called Hodge metric, on the moduli space with negative holomorphic sectional curvature. Using the Yau-Schwartz Lemma, we than proved that, the Weil-Petersson volume of the Calabi-Yau moduli space is finite. We also defined what the Weil-Petersson geometry is. The work is joint with Xiaofeng Sun.

### 5/10 Xianzhe Dai, UCSB L^2 cohomology of non-isolated singularities"

Abstract: Singular spaces occur naturally in all kinds of problems. L^2 cohomology has been proven to be the right replacement for the usual cohomology. We will start with the de Rham cohomology theory, laying down all the basics here. We then will explain the L^2 cohomology theory, leading to our result on L^2 signature of non-isolated singularities. We will discuss the application to some problems of physical interests. The talk should be accessible to graduate students.

### 5/31 Michel Boileau, visiting UCSB A fibration theorem for small hyperbolic cone 3-manifolds""

Abstract: This result involves both differential geometry and toplogy, and it is an important step in the proof of Thurston's orbifold theorem. The statement is : Consider a sequence of hyperboilic cone structures with cone angles smaller than \pi and diameter \geq 1 on a small 3-orbifold O. If the sequence collapses, then the orbifold O is Seifert fibered.

### 6/7 Carlos J. Garcia-Cervera, UCSB Applications of Geometry and Topology in Applied Mathematics"

Abstract: Differential Geometry has played an important role in the development of the physical sciences. One example is the connection between Geometry and Gravitation, which is part of Einstein's General Theory of Relativity. In this talk I will describe some physical systems, in a much smaller scale, with connections to well known concepts in Geometry and Topology. In particular, I will consider ferromagnetic materials, and liquid crystals. Both systems are usually described via a unit length vector field, which makes this connection between Geometry, Topology, and Applied Mathematics more apparent. This will be an introductory talk, and students are particularly encouraged to attend.

### 6/14 11am - 12noon, Huai-Dong Cao, IPAM and Texas A&M University On translating Kaehler-Ricci solitons and dimension reduction"

Abstract: translating Kaehler-Ricci solitons arise as blow-up limit of singularities of Hamilton's Ricci flow on Kaehler manifolds, and in some sense are natural extensions of Calabi-Yau metrics on noncompact complex manifolds. In this talk we will survey the developments on both geometric and complex analytic aspects of translating Kaehler-Ricci solitons of nonnegative curvature. We will also present our recent work on dimension reduction for the Ricci flow on compact Kaehler surfaces and its applications to the study of singularities.

# Fridays 12:50 - 1:50pm, SH 6617

### 1/18 Vitali Kapovitch, UCSB On some examples of manifolds of almost nonnegative Ricci curvature"

Abstract: For every $n\ge 1$ we construct infinitely many nonhomeomorphic simply connected $4n$-dimensional manifolds which admit metrics of almost nonnegative Ricci curvature but do not admit any metrics of nonnegative Ricci curvature.

### 2/1 Xiao Zhang, UCSC The positive mass theorem for asymptotically hyperbolic 3-manifolds"

Abstract: Related to the Bondi mass, we give a definition of the total energy, the total linear momentum and the total angular momentum for asymptotically hyperbolic 3-manifolds. We prove also a positive mass theorem.

### 2/15 John Lott, University of Michigan Noncommutative geometry and the heat equation"

Abstract: The basic idea underlying Connes' noncommutative geometry is to define a noncommutative space by its ring of functions'', which is a possibly-noncommutative ring. Geometric examples in which such spaces arise include quotient spaces by group actions, and leaf spaces of foliations. In the first part of the talk I will explain how one attaches noncommutative spaces to these geometric examples. Connes proved an index theorem for families of operators which are parametrized by such noncommutative spaces. His original proof used K-theory methods. In recent joint work with Sasha Gorokhovsky we gave a more explicit proof of Connes' theorem, by means of heat equation methods. I will outline the idea of the proof and discuss its applications.

### 2/22 Zhuang-dan Guan, UCR Geodesic stability and existence"

Abstract: For many years people were looking for a proper stability which is equivalent to the existence of a K\"ahler-Einstein metric. Motivated by the Kobayashi-Hitchin correspondence for the holomorphic vector bundles and many examples, we shall describe our "geodesic stability" which is related to the existence of extremal metrics and some related results.

### 3/1 Rafael Herrera, UCR Classification of quaternion-Kahler 12-manifolds with positive scalar curvature "

Abstract: Recently there has been a renewed interest in the geometry of Riemannian manifolds which admit a certain action of the (unit) quaternions on their tangent space at every point. In this sense these manifolds are quaternionic. Hyperkahler manifolds are the most popular examples of such manifolds, although there are many others depending, among other things, on the level of compatibility with the metric that one may require of the action (eg. hypercomplex, hyperhermitian, etc). In this talk we shall discuss a classification problem on quaternion-Kahler geometry, which is a close relative of hyperkahler geometry. Namely, we classify the complete quaternion-Kahler 12-manifolds with positive scalar curvature showing that they must be homogeneous spaces. The proof involves the use of tools such as the elliptic genus.

### 3/15 10am Ozlem Imamoglu, UCSB Theta, zeta and eta"

Abstract: This talk will be a review of some classical results from analytic number theory. In particular I will derive the classical limit formula of Kronecker which leads one to the Dedekind eta function.

# Fridays 11:00 - 11:50am, SH 6635

### 9/28 Vitali Kapovitch, UCSB Restrictions on collapsing with a lower sectional curvature bound"

Abstract: We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we obtain a partial result towards a conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.

### 10/12 Xianzhe Dai, UCSB Index theorem: even vs. odd"

Abstract: The Atiyah-Singer index theorem is a vast generalization of the Gauss-Bonnet formula in differential geometry, the Riemann-Roch formula in algebraic geometry, and the Hirzebruch signature formula in topology. Needless to say, such a beautiful result has beautiful applications from diverse field of mathematics (and physics). The Atiyah-Patodi-Singer index theorem is the generaliztion of Atiyah-Singer theorem to manifolds with boundary. We will introduce these results, and explain why the Atiyah-Patodi-Singer theorem is more geometric, leading to our recent result (joint with Weiping Zhang), which is an odd dimensional analogue of the Atiyah-Patodi-Singer theorem.

### 10/19 Guofang Wei, UCSB Metrics of positive Ricci curvature on bundles"

Abstract: We construct the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature. In general, we also prove that if $E$ is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then $E\times\mathbb R^p$ admits a complete metric of positive Ricci curvature for all large $p$.

### 10/26 Yue Lei, UCSB The eta invariant on manifolds with boundary"

Abstract: In Professor Xianzhe Dai's talk, the eta invariant for a closed manifold was introduced as the boundary correction term in the Atiyah-Patodi-Singer index theorem on manifolds with boundary. We will start by introducing the heat kernel formula for the eta invariant on closed manifolds and then generalize it to manifolds with boundary. Various global boundary conditions will be discussed.

### 11/9 Siye Wu, University of Colorado Cutting and Gluing in Geometric Quantization"

Abstract: A symplectic manifold can be regarded as the phase space of a classical system. Given a symplectic manifold and a polarization, one can construct the Hilbert space of the corresponding quantum system. This process is called geometric quantization. In this talk, I will study the behavior of the quantum Hilbert space under the cutting and gluing of the symplectic manifold. When the underlying space is a symplectic orbifold or Kahler manifold, the results are relevant to number theory or complex geometry.

### 11/16 Xianzhe Dai, UCSB S^1 action and equivariant index"

Abstract: Just as Lefschetz number generalizes the Euler number, the equivariant index generalizes the usual index in the same way when there is a group action present. We will introduce these, leading to the Atiyah-Hirzebruch vanishing theorem: if a closed spin manifold admits a nontrivial S^1 action then its A-roof genus is zero.

### 11/30 John Ennis, UCSB John Ennis' Advancement talk: Hausdorff Convergence with a Ricci Curvature Bound"

Abstract: Ever since Gromov introduced the Hausdorff distance between metric spaces, a natural question has been to what extent geometric and topological information is preserved under Hausdorff convergence. Hausdorff convergence is very weak in general, but by restricting the class of spaces one can obtain positive results. For example, Perelman has shown that when a sequence of manifolds with a uniform lower bound on sectional curvature converges to a limit space, the limit space is locally contractible. In particular, the limit space has a simply connected universal cover. In this talk, I will discuss to what extent this result holds in the Ricci curvature setting. Specifically, I will consider manifolds with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, and will provide a characterization of the universal cover of the limit of such manifolds.

### 12/7 Yu Ding, UCI Asymptotic formulae for heat kernels and Green's functions"

Abstract: We study the behavior of the Laplacian on a sequence of manifolds $\{M_i^n\}$ with a lower bound in Ricci curvature, that converges to a metric-measure space $M_\infty$. We prove that the heat kernels and Green's functions on $M_i^n$ converge to some integral kernels on $M_\infty$ which is, in different cases, the heat kernel and Green's function on $M_\infty$. We also study the Laplacian on noncollapsed metric cones; these provide a unified treatment of the asymptotic behavior of heat kernels and Green's functions on noncompact manifolds with nonnegative Ricci curvature and Euclidean volume growth. In particular, we get a unified proof of the asymptotic formulae of Colding-Minicozzi, Li and Li-Tam-Wang.