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{\bf Kazuo Akutagawa }
{\bf Title}: Yamabe constants of infinite coverings and a positive mass theorem
{\bf Abstract}: The {\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold
$(M, C)$ is defined by the infimum of the normalized total-scalar-curvature functional $E$ among all metrics in $C$.
The study of the second variation of this functional $E$ led O.Kobayashi and Schoen to independently introduce a natural differential-topological invariant $Y(M)$, which is obtained by taking the supreme of $Y(M, C)$ over the space of all conformal classes. This invariant $Y(M)$ is called the {\it Yamabe invariant} of $M$. For the study of the Yamabe invariant, the relationship between $Y(M, C)$ and those of its conformal coverings is important, the case when $Y(M, C)> 0$ particularly. When $Y(M, C) \leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$, the desired relation is clear. When $Y(M, C) > 0$, such a uniqueness does not hold. However, Aubin proved that $Y(M, C)$ is strictly less than the Yamabe constant of any of its non-trivial {\it finite} conformal coverings, called {\it Aubin's Lemma}. In this talk, we generalize this lemma to the one for the Yamabe constant of any $(M_{\infty}, C_{\infty})$ of its {\it infinite} conformal coverings, under a certain topological condition on the relation between $\pi_1(M)$ and $\pi_1(M_{\infty})$.
For the proof of this, we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities. \\
\par
{\bf Werner Ballmann}
{\bf Title}: Boundary value problems for Dirac operators \\
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{\bf Gerard Besson}
{\bf Title}: An alternative proof of the geometrization for aspherical manifolds
{\bf Abstract}: In this lecture, we discuss Perelman's work. We will explain our alternative proof of the geometrization for aspherical manifolds assuming results on the Ricci flow. \\
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{\bf Mingliang Cai}
{\bf Title}: A positive mass theorem and some rigidity results for asymptotically hyperbolic manifolds
{\bf Abstract}: We will present some scalar curvature and Ricci curvature rigidity results for asymptotically hyperbolic manifolds and a positive mass theorem for such manifolds. \\
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{\bf Shu-Cheng Chang}
{\bf Title}: On the CR analogue of Li-Yau's eigenvalue estimate of a sublaplacian on a pseudohermitian 3-manifold
{\bf Abstract}: In this talk, joint with H.-L. Chiu, we study a lower bound estimate of the first positive eigenvalue of the sublaplacian on a $3$-dimensional pseudohermitian manifold with nonnegative CR Paneitz operator. By using the Li-Yau gradient estimate, we are able to get an effect lower bound estimate under a general curvature condition. Moreover, one can have the CR analogue of M. Obata's theorem on a closed pseudohermitian $3$-manifold with free pseudohermitian torsion. The key step is a discovery of new CR version of Bochner formula which involving the CR Paneitz operator. \\
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{\bf Jih-Hsin Cheng}
{\bf Title}: The mass and the Paneitz operator in 3-dimensional CR geometry
{\bf Abstract}: I will report on an on-going project about the study of the mass in CR geometry. We will define an analogue of the ADM mass for an asymptotically Heisenberg 3-manifold, called p-mass. For a closed 3-dimensional pseudohermitian manifold M, we consider the Green function of the CR Laplacian. We can identify the first nontrivial coefficient of the Green function expansion with the p-mass of the blow-up asymptotically Heisenberg 3-manifold at a point. Through such a connection we can prove the existence of a minimizer in a positive CR Yamabe class if the p-mass is nonnegative and positive unless M is the standard pseudohermitian 3-sphere. Solving the ¡õb equation with decay boundary value at ¡Þ on an asymptotically Heisenberg 3-manifold, we can then express the p-mass as an integral formula. There is an extra term involving the CR Paneitz operator P, which does not appear in the formula of the ADM mass. So in addition to the condition that the Tanaka-Webster scalar curvature is nonnegative, we need nonnegativity of P to get nonnegativity of the p-mass. We will also discuss the situation of vanishing p-mass. \\
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{\bf Yuxin Dong}
{\bf Title}: Chern type Theorems for graphs with parallel mean curvature
{\bf Abstract}: In 1965, Chern proved that the only entire graphic hypersurface of constant mean curvature in Euclidean space must be minimal. In this talk, we will discuss some results which generalize Chern type result both for graphic submanifolds in Euclidean space and for spacelike submanifolds in pseudo-Euclidean space. \\
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{\bf Hao Fang}
{\bf Title}: On new conformal curvature invariants \\
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{\bf Kenji Fukaya}
{\bf Title}: Lagrangian Floer theory
{\bf Abstract}: In this talk I want to report on joint work with Y.-G. Oh, H. Ohta and K. Ono on Floer homology of Lagrangian submanifold. I want to explain how the method of homological algebra and pseudo holomorphic curve together with chain level intersection theory provides a machinery to study Lagrangian submanifold of symplectic manifold. \\
{\bf Akito Futaki}
{\bf Title}: Complete Ricci-flat Kaehler metrics on the canonical bundles of toric Fano manifolds
{\bf Abstract}: We prove the existence of complete Ricci-flat Kaehler metrics on the canonical bundles of toric Fano manifolds. This is an application of an existence result of toric Sasaki-Einstein metrics. \\
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{\bf Yuxin Ge}
{\bf Title}: On the $\sigma_2$-scalar curvature and its application
{\bf Abstract}: In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{ \sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature from conformal geometry. As application, we prove if a compact $3$-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int \sigma_2>0$, then topologically $M$ is a quotient of sphere. This is a joint work with G. Wang et C-S. Lin. \\
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{\bf Ursula Hamenstaedt}
{\bf Title}: Minimal Reeb orbits on boundaries of convex domains in $\mathbb{C}^2$
{\bf Abstract}: Using minimal surfaces we show that a periodic Reeb orbit of self-linking number $-1$ on the boundary of a compact convex domain in $\mathbb{C}^2$ is unknotted and bounds an embedded symplectic disc. \\
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{\bf Qing Han}
{\bf Title}: Isometric embedding of 2-dim Riemannian metric in 3-space
{\bf Abstract}: It is a classical problem whether a 2-dimensional Riemannian manifold admits an isometric embedding in Euclidean 3-space. In this talk, we will review known results and present some new ones. \\
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{\bf Kengo Hirachi}
{\bf Title}: Q-curvature in CR geometry
{\bf Abstract}: For 3-dimensional CR manifolds, the Q-curvature appears in the logarithmic term in the Szego kernel. I will generalize this relation to higher dimensions by generalizing the notion of Q- curvature. This family of Q-curvatures is associated with non-linear CR invariant differential operators and is constructed by using the ambient metric. \\
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{\bf Bernhard Leeb}
{\bf Title}: On convex sets in symmetric spaces of higher rank \\
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{\bf John Lott}
{\bf Title}: Dimensional reduction and long-time behaviour of Ricci flow
{\bf Abstract}: Perelman used Ricci flow to prove the geometrization conjecture. However, the precise long-time behaviour of a three-dimensional Ricci flow is largely unknown. I will give some convergence results under the assumption that the sectional curvatures decay as least as fast as the inverse of the time. \\
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{\bf Zhiqun Lu}
{\bf Title}: On the DDVV conjecture
{\bf Abstract}: We proved an algebraic inequality in submanifold geometry which relates the scalar curvature, the normal scalar curvature, and the second fundamental form of a 3-fold immersed in a space form. \\
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{\bf Xiaohuan Mo}
{\bf Title}: On the non-Riemannian quantity H of a Finsler metric
{\bf Abstract}: In my lecture, I will present non-Riemannian quantity H on a Finsler manifold in terms of flag curvature and Ricci scalar. In particular, I will show that all R-quadratic Finsler metrics have vanishing non-Riemannian invariant H generalizing result previously only known in the case of Randers metric. \\
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{\bf Libin Mou}
{\bf Title}: Nonsmooth analysis on metric spaces and its applications
{\bf Abstract}: This talk presents a nonsmooth analysis on metric spaces with convex tangent cones (such as Busemann G-spaces with curvatures bounded from above). Applications and unsolved problems will be discussed. \\
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{\bf Frank Pacard}
{\bf Title}: Constant mean curvature surfaces in Riemannian manifolds
{\bf Abstract}: I will report some results concerning the existence of constant mean curvature surfaces in Riemannian manifolds. These results give a partial description of some boundaries of the moduli space of constant mean curvature surfaces in a fixed Riemannian manifold. \\
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{\bf Shengliang Pan}
{\bf Title}: On a new curve evolution problem in the plane \\
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{\bf Xiaochun Rong}
{\bf Title}: Positively curved manifolds with abelian symmetry
{\bf Abstract}: We will survey a recent development in the study of positively curved manifolds which admits an isometry abelian group action. \\
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{\bf Viktor Schroeder}
{\bf Title}: Hyperbolic rank of metric spaces
{\bf Abstract}: We discuss the hyperbolic rank, a quasi-isometry invariant of metric spaces first introduced by Gromov. The hyperbolic rank measures in some sense the amount of hyperbolicity of a metric space. One can compute the hyperbolic rank of certain standard spaces, e.g. symmetric spaces. It seems to be very difficult to compute the hyperbolic rank of certain singular spaces as for example euclidean buildings. We present some new results in this direction and discuss open problems. \\
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{\bf Ravi Shankar}
{\bf Title}: Conjugate points in length spaces
{\bf Abstract}: In this talk we present at least three new notions of conjugate point for complete, proper, compact inner metric spaces whose curavture is bounded above by a fixed real number $k$. We do this in order to generalize known classical theorems in Riemannian geometry like: the Klingenberg long homotopy lemma, the Rauch comparison theorem and an estimate on the injectivity radius in terms of a shortest closed geodesic. We will also present several open problems/questions for further study. This talk reports on joint work in progress with Christina Sormani (CUNY). \\
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{\bf Yi-Bing Shen}
{\bf Title}: On a class of critical Riemannian-Finsler metrics
{\bf Abstract}: As is well known, among Riemannian metrics on a compact $n$-manifold $M$ there is an important class of metrics called Einstein metrics, which are the critical points of the normalized Einstein-Hilbert functional
$$\frac{1}{{\mathrm{Vol}}^{1-2/n}(M)}\int_M R\ d{\mu}_M,$$
where $R$ is the scalar curvature of the Riemannian metric, $d{\mu}_M$ is the volume element of $M$. An analogous functional in Finsler geometry is defined and its Euler-Lagrange equation is given. Some examples of critical metrics are given, where non-Riemannian critical metrics are related to Calabi-Yau metrics. Moreover, some rigidity theorems for a Finsler metric to be Riemannian are obtained. \\
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{\bf Changping Wang}
{\bf Title}: Submanifolds in Lie sphere geometry and its sub-geometries \\
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{\bf Guofang Wang}
{\bf Title}: Analytic Aspects of Sasakian geometry \\
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{\bf Xue-Ping Wang}
{\bf Title}: Threshold resonances on manifolds with conical end
{\bf Abstract} In this talk, I will give some properties of resonance at zero for Schroedinger operators on Riemannian manifolds with conical end. As an application, I will describe the asymptotic expansion in large time of the Schroedinger group without any cut-off in energy. \\
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{\bf Shihshu Walter Wei}
{\bf Title}: p-Harmonic geometry, variational problems and quasi-regular mappings
{\bf Abstract}: We'll study variational problems by using p-harmonic maps as catalysts that do not seem to work by employing ordinary harmonic maps (in which p=2). This approach is naturally connected to many other branches of mathematics. In particular, some results on topology, PDEs, minimal varieties, quasi-regular and quasi-conformal mappings will be discussed. \\
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{\bf Jyh-Yang Wu}
{\bf Title}: Energy, variance and scalr curvature
{\bf Abstract}: In this talk, we shall discuss two meanings about the scalar curvatute. Following the idea of the path integral and the priciple of classical-quantum duality, we shall derive a Newton eqation fo motion on curved space with quantum effect. The concept of variance of a random variable on a Riemannian manifold will also roduced. It will be related with the scalar curvature. \\
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{\bf Yuan-Long Xin}
{\bf Title}: Curvature estimates for minimal sub-manifolds of higher co-dimension
{\bf Abstract}: We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau's results and Ecker-Huisken's results are generalized to higher codimension. In this way we improve Hildebrandt-Jost-Widman's result for Bernstein type theorem. \\
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{\bf Yu Zheng}
{\bf Title}: On the uniform bounded estimations of some $\sigma_k$ curvature flow of hypersurface
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