Abstract: There are only very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions larger 24 the known examples are diffeomorphic to locally rank 1 symmetric spaces. We will construct metrics with positive sectional curvature on open and dense sets of points on the projective tangent bundles of RP^n, CP^n and HP^n. The so called deformation conjecture says that these kind of metrics can be deformed into metrics with positive sectional curvature everywhere. However, the simplest new example within our class, the projective tangent bundle of RP^3, is diffeomorphic to the product RP^3 x RP^2. This non-orientable manifold is known not to admit a metric with positive sectional curvature. Thus the construction provides a counterexample to the deformation conjecture.

Abstract: We describe the decomposition of the zeta-determinant (which apriori is non-local object) onto contribution coming from different part of the manifolds.

Abstract: In 1970, Atiyah and Hirzebruch proved the following vanishing theorem: If a spin manifold admits a non-trivial $S^1$ action, then its $\widehat{A}$ (A-hat) genus vanishes. In the middle of 80, trying to generalize the AH vanishing theorem, Landweber, Stong, Ochanine introduced "elliptic genus", and they defined a new cohomology theory so call "ellitpic cohomology" by using their "elliptic genus". Witten rigidity theorem says that their elliptic genus is rigid with respect to $S^1$ action (i.e. constant on $g\in S^1$). In my talk, I will explain various generalizations of Witten rigidity theorem. To get them, we use the index theory and the modular invariance of certain Jacobi forms.

Abstract: Let $M^m$ be a $m$-dimensional Riemannian manifold and $N^n$ be a $n$-dimensional Riemannian manifold which is isometrically embedded in $\RR^K$ for some integer $K>0$. We say $u:M \rightarrow N \subset \RR^K$ is a weakly intrinsic biharmonic map if it is a critical point of the functional $F(v)=\int |\tau(v)|^2$ where $\tau(v)$ is the tension field of $v$. In talk, we consider the special case: $m=4$ and $N$ is the standard $K-1$-sphere in $\RR^K$. In this case, interior and boundary regularity of $u$ are proved.

Abstract: A noncompact Riemannian manifold is said to have quadratic curvature decay if its sectional curvatures decay at least quadratically with respect to the distance from a basepoint. This is a natural and scale-invariant condition. I'll give topological restrictions for a manifold to admit quadratic curvature decay, along with some additional constraints. The first case is when the additional constraint is one of having volume growth which is slower than that of the Euclidean space of the same dimension. (This is joint work with Zhongmin Shen.) The second case is when the volume growth is Euclidean and the quadratic decay constant is sufficiently small.

Abstract: There is a family of nonlinear equations that arises in conformal geometry. The Yamabe equation is a particular example of these equations. The other equations in this family are actually fully nonlinear and involves the Ricci tensor. I will discuss an existence result for one such equation on a 4-manifold. It turns out to have strong implication for the Ricci tensor. In particular it is possible to construct on the basis of natural conditions, metrics of positive Ricci tensor in a given conformal class of metrics.

Abstract: Using a simple ineqality,it will be shown that conic sections can be classified (as parabolic,elliptic or hyperbolic) by having a precise "snip" of a curve without having any formula for its equation. It is assumed that the snip is small so that the curve is not recognizable. All that is needed is a planimeter that measures areas accurately.

Abstract: According to the soul theorem of Cheeger and Gromoll, any complete open manifold of nonnegative sectional curvature is diffeomorphic to the normal bundle of its compact totally geodesic submanifold which is called the soul. It is a natural question what kind of normal bundles can occur. I shall describe a joint work with Vitali Kapovitch relating this question to a conjecture of Halperin in rational homotopy theory. As an application we can often reduce the question to the case of a simply-connected soul.

Abstract: In this talk I will describe invariants such as the L^2 Betti numbers of a foliation endowed with a transverse measure. Such invariants are defined by means of Alain Connes' "noncommutative measure theory", and can be interpreted as traces in a noncommutative von Neumann algebra. They can be used to formulate a foliated Atiyah-Singer index theorem and foliated Morse inequalities.

Abstract: In this talk I will go over the proof of the main result of my thesis, where I construct a foliated, L^2 version of Morse homology and show that it is isomorphic (in a "noncommutative" sense) to the foliated L^2 cohomology of Connes. I will also mention current work on the symplectic version of this result, i.e. the "foliated Arnold conjecture".

Abstract: A Stein-fillable contact manifold is a contact manifold which is the boundary of a Stein domain and whose contact structure is the induced maximal complex subbundle. Stein-fillable contact manifolds can be constructed by attaching contact handles. In this talk I will discuss the Reeb dynamics and contact homology of a subcritical Stein-fillable contact manifold via its contact handle construction.

Abstract: The theory of harmonic maps of Riemann surfaces M into Lie groups G has several specific features which lead to an especially rich theory. Such harmonic maps equations can be formulated as certain "gauge-theoretic equations" for a pair of a connection and a Higgs field on the principal bundle over M with the structure group G. To investigate of the moduli space of such solutions is advantageous for the study of moduli space of harmonic maps itself. I would like to describe some geometric structures on the moduli space. We can show that every uniton solution is a critical point of the energy function on the moduli space, moreover, give a characterization of the fixed point set by S^1-action in terms of a flag transform. This is a joint work with Yoshihiro Ohnita.

Abstract: In this talk, we consider the following Cauchy problem of nonlinear singular PDE: $$ t\frac{\partial u}{\partial t}=F(t,x,u,\frac{\partial u}{\partial x}), \eqno(1) $$ where the classical Cauchy-Kowalewski theorem has been extended. We shall discuss two situations: (I). The equation (1) has regular singularity near $(0,0)\in{\bf C}_t\times {\bf C}_x$, in this case we can prove the equation (1) has a unique holomorphic solution near $(0,0)\in{\bf C}_t\times{\bf C}_x$. (II). The equation (1) has irregular singularity near $(0,0)\in{\bf C}_t\times {\bf C}_x$, in this case we can prove the equation (1) has a unique formal Gevrey class solution $u(t,x)\in G\{t,x\}_{(s,\sigma)}$, also we can give a formula to calculate the formal Gevrey index for the equation (1).