# Differential Geometry Seminar Schedule for Spring 2013

## Fridays 3:00 - 3:50pm, SH 6635

### 4/5 Rafe Mazzeo, Stanford University "Ricci flow on conic surfaces"

Abstract: Ricci flow on surfaces with conic singularities presents several analytic and geometric difficulties, ranging from nonuniqueness of the flow to a somewhat complicated convergence theory. I will describe how to resolve these issues for surfaces with cone angles less than $2\pi$. (joint with Rubinstein and Sesum).

### 4/12 Ken-Ichi Yoshikawa, Kyoto University "Analytic torsion of log-Enriques surfaces"

Abstract: Log-Enriques surfaces are singular rational surfaces that arise as degenerations of Enriques surfaces. Even though they are rational, they carry Ricci-flat orbifold Kaeher metrics. Then one can construct an invariant of log-Enriques surfaces by using analytic torsion and volume. The goal of my talk is to explain that, as a function on the moduli space, this analytic torsion invariant is expressed as a nice automorphic form on the Kaehler moduli of a Del Pezzo surface.

### 4/26 Chiung-Jue Sung, National Tsing Hua University "Harmonic forms on complete manifolds"

Abstract: In this talk, we will mention some results and questions concerning harmonic forms on complete manifolds. In particular, we will explain a finite dimensionality result for polynomial growth harmonic forms.

### 5/17 Li-Sheng Tseng, UC Irvine "Cohomologies on Symplectic Manifolds"

Abstract: Cohomologies of differential forms provide some of the most basic invariants on manifolds. In this talk, I will discuss some new cohomologies of forms on symplectic manifolds. These cohomologies can be defined simply in terms of linear differential operators that are intrinsically symplectic. Of interest, they have standard Hodge theoretical properties and provide useful invariants especially for non-Kahler symplectic manifolds. Moreover, they also encode the data of Lefschetz maps between de Rham cohomologies. This is joint work with C.-J.Tsai and S.-T. Yau.

# Differential Geometry Seminar Schedule for Winter 2013

## Fridays 3:00 - 3:50pm, SH 6635

### 1/18 Yuanqi Wang, UCSB "Bessel Functions, Heat Kernel and the Conical Kahler-Ricci Flow"

Abstract: Inspired by Donaldson's program, we introduce the Kahler Ricci flow with conical singularities. The main part of this talk is to show that the conical Kahler Ricci flow exists for short time and for long time in a proper space. These existence results are highly related to heat kernel and Bessel functions. We will also discuss some easy applications of the conical Kahler Ricci flow in conical Kahler geometry.

### 2/1 Yuguang Zhang, Capital Normal University, China, visiting UCSD "Degenerations of Calabi-Yau manifolds"

Abstract: In this talk, we study the Gromov-Hausdorff convergence of Ricci-flat metrics under degenerations of Calabi-Yau manifolds. More precisely, for a family of polarized Calabi-Yau manifolds degenerating to a singular Calabi-Yau variety, we prove that the Gromov-Hausdorff limit of Ricci-flat kahler metrics on them is unique.

### 2/15 Lee Kennard, UCSB "Betti number bounds for positively curved manifolds with symmetry"

Abstract: After reviewing a few topological classification results (up to diffeomorphism, homeomorphism, or homotopy equivalence) for manifolds admitting metrics with positive sectional curvature and large isometry groups, I will discuss a couple of new results on weaker invariants (e.g., Betti numbers and Euler characteristics) of such manifolds. In one case, we obtain a sharp calculation of the Euler characteristic. This is joint work with Manuel Amann (KIT).

### 3/15, 3-3:50pm Regina Rotman, University of Toronto "Short geodesic segments on closed Riemannian manifolds"

Abstract: A well-known result of J. P. Serre states that for an arbitrary pair of points on a closed Riemannian manifold there exist infinitely many geodesics connecting these points. A natural question is whether one can estimate the length of the "k-th" geodesic in terms of the diameter of a manifold. We will demonstrate that given any pair of points p, q on a closed Riemannian manifold of dimension n and diameter d, there always exist at least k geodesics of length at most 4nk^2d connecting them. We will also demonstrate that for any two points of a manifold that is diffeomorphic to the 2-sphere there always exist at least k geodesics between them of length at most 22kd. (Joint with A. Nabutovsky)

### 3/15, 4-4:50pm, Wilderich Tuschmann, Karlsruhe Institute of Technology "Riemannian orbifolds, collapsing and minimal volumes"

Abstract: I will discuss recent work with Oliver Baues on characterizing the diffeomorphism type of certain Riemannian orbifolds, including solvmanifolds, by bounded curvature collapses and its applications to D-minimimal volumes of homotopy tori.

# Differential Geometry Seminar Schedule for Fall 2012

## Fridays 3:00 - 3:50pm, SH 6635

### 10/5 Lee Kennard, UCSB "An introduction to the Grove research program"

Abstract: There exist few families of examples of Riemannian manifolds with positive sectional curvature. In dimensions above 24, the only known examples are metrics on spheres and projective spaces. In searching for new examples, it is natural to study metrics which are invariant under a large group of symmetries. This has led to many topological obstructions to the existence of highly symmetric metrics of positive curvature. We will review some of the conjectures in the subject and results that provide evidence for these conjectures in the presence of symmetry.

### 10/12 Lee Kennard, UCSB "Tools from topology applicable to the Grove program"

Abstract: We will discuss techniques from geometry and topology which have been applied to problems in the Grove program. (This talk will be independent of last week's talk.) We will briefly discuss how, using Morse theoretic results, one uses the assumptions of positive sectional curvature and symmetry to reduce geometric problems in this setting to (curvature-free) topological ones. We will then describe some of the tools from topology, including cohomology operations and results from Smith theory and group cohomology, which provide answers to some of these problems. We will also discuss a few of the topological roadblocks, as well as conjectured detours and immediate applications to the Grove program.

### 10/19, Guofang Wei, UCSB "Comparison results for Ricci curvature"

Abstract: Ricci curvature occurs in the Einstein equation, Ricci ow, optimal transport, and is important both in mathematics and physics. Comparison method is one of the key tools in studying the Ricci curvature. We will start with Bishop-Gromov volume comparison, and then discuss some of its generalizations and their applications.

### 11/2 Richard Bamler, Stanford University "Long-time analysis of 3 dimensional Ricci flow"

Abstract: It is still an open problem how Perelman's Ricci flow with surgeries behaves for large times. For example, it is unknown whether surgeries eventually stop to occur and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$. In this talk, I will present new tools to treat this question after providing a quick review of Perelman's results. In particular, I will show that in the case in which the initial manifold satisfies a certain purely topological condition, surgeries do in fact stop to occur after some time and the curvature is globally bounded by $C t^{-1}$. For example, this condition is satisfied by manifolds of the form $\Sigma \times S^1$ where $\Sigma$ is a surface of genus $\geq 1$.

### 11/9 Bingyu Song, Central China Normal University, China, visiting UCSB "Laplacian theorem in Finsler spaces and its application"

Abstract: In this talk, I will introduce some definitions in the Finsler geometry and compare it to the Riemannian case. Then I will discuss the comparison theorem under the bound of the Ricci curvature.

### 11/16, Jianqing Yu, Chern Institute of Mathematics, visiting UCSB "On the Witten rigidity theorem"

Abstract: In this talk, I will first explain the basic idea of Witten's loop space construction of elliptic genus. Then I will briefly discuss the proof of Witten's rigidity theorem as well as its family generalizations. At last, I will talk about my joint work with Bo LIU, the Witten rigidity theorem on Z/k manifolds (arxiv 1104.3972) and String^c manifolds (arxiv 1206.5955).