Schedule

Saturday (August 8)

Breakfast at Commons Room  (8:00 am -- 9:00 am)

Andrzej Derdzinski  (9:00 am-9:50 am)

Einstein Metrics

The talk outlines classical existence and nonexistence theorems for (pseudo)Riemannian Einstein metrics on compact manifolds. This is followed by a more detailed presentation of recent results and open questions concerning homogeneous Riemannian Einstein manifolds and curvature-homogeneous pseudo-Riemannian Einstein four-manifolds.

Harold Donnelly  (10:00 am-10:50 am)

Positive Ricci curvature and eigenfunctions of the Laplacian

Abstract: Let M denote a complete noncompact Riemannian manifold having non-negative Ricci curvature. It was conjectured that the Laplacian of M admits no square integrable eigenfunctions. A number of positive results were proved under additional hypotheses, starting with Escobar's thesis in 1985. Recently, counterexamples to the general conjecture have been discovered. The lecture will review these developments and suggest directions for future work.

Guofang Wei (11:00am-11:50am)

Ricci curvature for 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.

Shin-ichi Ohta (2:00pm-2:50pm)

Weighted Ricci curvature and heat flow on Finsler manifolds

We introduce the new notion of weighted Ricci curvature of a Finsler manifold equipped with an arbitrary measure. Bounding this curvature from below is equivalent to the curvature-dimension condition recently introduced and developed by Lott, Sturm and Villani. Among applications, we see that such spaces satisfy several functional/geometric inequalities.

We also consider heat flow on Finsler manifolds and the relation between its behavior and our weighted Ricci curvature. This is joint work with Karl-Theodor Sturm in Bonn.

Stephanie Alexander  (3:00pm - 3:50pm)

Alexandrov geometry in Lorentz spaces

Abstract:  We say a semi-Riemannian manifold satisfies  R ≥ K  if spacelike sectional curvatures are  ≥ K  and timelike ones are ≤ K  (and the reverse for  R ≤  K). Examples of Lorentz spaces with  R ≥ 0  include "big-bang" Robertson-Walker  spaces, and timelike convex hypersurfaces of Minkowski space.

We give the equivalence of  R ≥  K (R ≤ K)  with each of the following: (a) other algebraic conditions on the curvature tensor, (b) function comparisons, (c) triangle comparisons.  This is joint work with Richard Bishop.

Further, we discuss globalization theorems and related conjectures.

Sunday (August 9)

Jianguo Cao & Jian Ge (9:00am-9:50am)

We will simplify the earlier proofs of Perelman's collapsing theorem of 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian.  Among other things, we use Perelman's semi-convex analysis of distance functions to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verifcation of Perelman's collapsing theorem is the last step of Perelman's proof of Thurston's Geometrization Conjecture. Our proof of Perelman's collapsing theorem is almost self-contained. We believe our proof of this collapsing theorem is accessible to non-experts and advanced graduate students.

Christina Sormani  (10:00am - 10:50am )

A New Convergence for Riemannian Manifolds

Abstract: We define the intrinsic flat distance between Riemannian manifolds and describe their limits: integral current spaces.  This work is based on Ambrosio-Kirchheim's theory of integral currents on metric spaces.  It is joint work with Stefan Wenger.  Stefan Wenger has also proven a compactness theorem in this setting.  See http://comet.lehman.cuny.edu/sormani/research/talkintrinsicflat.html

Xiaochun Rong  ( 11:00am-11:50 am)

Alexandrov spaces of maximal volumes

Abstract: We will report a recent progress on a classification of Alexandrov spaces with curvature bounded from below whose volumes are (relatively) maximal. This is a joint work with Nan Li.

Xinyue Cheng (2:00pm-2:50pm)

The Ricci Curvature In Finsler Geometry

The Ricci curvature plays a very important role in Finsler geometry. In this talk, we will discuss the Ricci curvature and its some applications in Finsler geometry, including the role of the Ricci curvature in Finsler projective geometry, the volume comparison in Finsler geometry and Ricci curvature of Randers metrics.

Ying You & Yaoyong Yu (3:00pm-3:50pm)

On Einstein mth root metrics

In this talk,  we will discuss a special class of Finsler metrics --- mth root metrics. We show that if a mth root metric is a (weak) Einstein metric, then it must be Ricci-flat.