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.

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.

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.

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.

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.

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.

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.

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.

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 $p$.

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.

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.

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.

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.

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.