## Speeches

 Charles Boyer Title : Fundamental Problems in Sasakian Geometry Abstract: I want to focus my talk on several foundational problems in Sasakian geometry. (1) Given a manifold determine how many inequivalent contact structures of Sasaki type there are. (2) Given an isotopy class of contact structures determine the pre-moduli space of compatible Sasaki classes. (3) Determine the Sasakian structures within a fixed underlying strictly pseudoconvex CR structure. (4) Determine those Sasakian structures with extremal representatives. (5) Given extremal Sasakian structures when do they have constant scalar curvature? (6) In the monotone case discuss the moduli space of Sasaki-Einstein structures. Of course we can give only partial answers to these problems and only in special cases. My talk is based in part on joint work with several colleagues: Hongnian Huang, Eveline Legendre, Leonardo Macarini, Justin Pati, Christina Tønnesen-Friedman, and Otto van Koert. Gao Chen Title : Classification of gravitational instantons Abstract: Gravitational instanton is a complete non-compact hyperk\"ahler 4-manifold, i.e. Calabi-Yau surface with fast enough curvature decay near the ends. In this talk, I will discuss the classification of gravitational instanton with faster than quadratic curvature decay condition. It's a joint work with Xiuxiong Chen. Cheol-Hyun Cho Title : Non-commutative homological mirror functor Abstract: We give a constructive homological mirror formalism using formal Lagrangian Floer deformation theory. Given a symplectic manifold X and a choice of a reference Lagrangian submanifold L, our formalism provides a possibly non-commutative algebra A, together with a central element W, which provides a non-commutativeLandau-Ginzburg model (A,W). The construction comes with a natural A-infinity functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. In particular it recovers and strengthens several interesting results of Seidel, Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results. Applying the mirror construction to an elliptic curve quotient, we also obtain a deformation quantization of an affine del Pezzo surface. This is based on joint works with Hansol Hong and Siu-Cheong Lau. Kenji Fukaya Title : Exponential decay estimate in the gluing analysis of pseudoholomorphic curve Abstract: In this talk I will explain one of the analytic points in the story of pseudoholomorhpic curve.The problem is an exponential decay estimate with respect to the gluing parameter.This is the problem how the moduli space of pseudoholomorphic curve behave near the point represented by a map from nodal curve. Such a gluing problem is studyed from long time ago. However the decay estimate with respect to the gluing parameter was never written up in detail. Recently with Oh,Ohta,Ono we wrote a detail of this estimate. In this talk I will explain the main analytic point of the proof of this decay estimate and where it is used in the study of the moduli space of pseudoholomorphic curve. Akito Futaki Title : Fano-Ricci limit spaces and spectral convergence Abstract: We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $¥overline{¥partial}$-Laplacian on compact K¥"ahler manifolds. This situation typically occurs for a sequence of Fano manifolds with the anti-canonical K¥"ahler class. This talk is based on a joint work with S. Honda and S. Saito Matthew J. Gursky Title : A formal Riemannian structure on the space of conformal metrics and some applications Abstract: In this talk I will present some results from project with J. Streets (UC-Irvine), in which we define a formal Riemannian metric on the set of metrics in a conformal class with positive (or negative) curvature. In the case of surfaces, this metric corresponds to the metric defined in the Kahler cone. I will then talk about extensions to higher dimensions, especially 4-d, in which this construction has some interesting applications to the fully nonlinear Yamabe problem and other geometric variational problems. Oscar Garcia-Prada Title : Kaehler-Yang-Mills equations and gravitating vortices Abstract: In this talk we first introduce the Kaehler-Yang-Mills equations on a holomorphic bundle over a compact complex manifold. They emerge from a natural extension of the theories for constant scalar curvature Kaehler metrics and Hermitian-Yang-Mills connections. We construct solutions to these equations by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, that we call gravitating vortex equations, describe abelian vortices on the Riemann surface coupled to a metric (joint work with L. Alvarez-Consul and M. Garcia-Fernandez). Long Li Title: Convexity and Uniqueness in conic Kaehler geometry Abstract: In this talk, we will discuss conic constant scalar curvature Kaehler(cscK) metrics, and it turns out that convexity plays a crucial role in the study of uniqueness problems. We first prove that conic Mabuchi's functional is convex along any conic geodesic in the space of all Kaehler potentials. And then the regularities and reductivity for conic cscK metrics are established. Finally, we will mention how to prove uniqueness by following Bando-Mabuchi's bifurcation technique in the conic setting. Tianjun Li Title : Symplectic Calabi-Yau surfaces Abstract: Symplectic Calabi-Yau surfaces are closed symplectic 4-manifolds with torsion canonical class. They are the exactly the minimal symplectic 4-manifolds with Kodaira dimension zero.We will discuss what is known about the classification. Toshiki Mabuchi Title : Extremal K\"ahler versions of the Yau-Tian-Donaldson Conjecture Abstract: In this talk, we discuss extremal K\"ahler versions of the Yau-Tian-Donaldson Conjecture by looking at their relationships to the asymptotic relative Chow-stability.We focus on what is the right choice of the K-stability for extremal K\"ahler cases. Ngaiming Mok Title : Geometric substructures and uniruled projective subvarieties of Fano manifolds of Picard number 1 Abstract: In the late 1990s Hwang and Mok initiated a geometric theory of uniruled projective manifolds modeled on their varieties of minimal rational tangents (VMRTs), proving later on Cartan-Fubini extension (2001), according to which a germ of VMRT-preserving biholomorphism f : (X; x0) → (Y ; y0) between two Fano manifolds of Picard number 1 extends necessarily to a biholomorphism F : X → Y whenever their VMRTs are of positive dimension and Gauss maps on VMRTs are generically finite. Hong-Mok (2010) extended Cartan-Fubini extension to the non-equidimensional setting under a relative nondegeneracy condition on second fundamental forms on VMRTs. Recently Mok-Zhang considered the analytic continuation of a germ of complex submanifold (S; x0) ,→ (X; x0) of a Fano manifold of Picard number 1 uniruled by lines, where S inherits a sub-VMRT structure defined by intersections of VMRTs with tangent spaces. Assuming that sub-VMRTs satisfy new non-degeneracy conditions and that the distribution spanned by sub-VMRTs is bracket-generating, we showed that S ⊂ Z for some irreducible subvariety Z ⊂ X, dim(Z) = dim(S) by constructing a universal family of chains of rational curves by an analytic process and proving its algebraicity Zhaohu Nie Title : Classification of solutions to the Toda system for a general simple Lie algebra Abstract: Toda systems are generalizations of the Liouville equation to other simple Lie algebras. They are examples of integral systems and have various applications in geometry and physics. For Lie algebras of type A, Lin, Wei and Ye classified their solutions with finite energy and singular sources at the origin. The speaker generalized their classification to Lie algebras of types B and C. In this talk, we will generalize the classification of solutions to Toda systems for all types of simple Lie algebras using a unified Lie-theoretic approach. The method relies heavily on the structure theories of the local solutions and of the W-invariants for the Toda system. The solution space is shown to be parametrized by a subgroup of the corresponding Lie group. We will also show the quantization result for the corresponding integrals. Mihai Paun Title : Singular Hermitian metrics and Kodaira dimension of algebraic fiber spaces Xiaochun Rong Title : Quantitative volume space form rigidity under lower Ricci curvature bound Abstract: Consider a compact $n$-manifold $M$ of Ricci curvature bounded below, normalized by $(n-1)H$, where $H=\pm 1$ or $0$. Then $M$ is isometric to a $H$-space form under either of the maximal volume conditions: (i) (Bishop) there is $\rho>0$ such that every $\rho$-ball on $M$ has the maximal volume i.e., the volume of a $\rho$-ball in a simply connected $H$-space form; (ii) (Ledrappier-Wang) For $H=-1$, the volume entropy of $M$ is maximal i.e., the volume entropy of any hyperbolic $n$-manifold. We will discuss quantitative version of the two `volume' rigidity for space form. This work is joint with Lina Chen and Shicheng Xu of Capital Normal University Min Ru Title : Holomorphic curves into projective varieties intersecting divisors Abstract: In this talk, I will discuss some recent works on holomorphic curves into projective varieties intersecting divisors. Their arithmetic properties will also be discussed. Dietmar A. Salamon Title : Symplectic structures, moment maps,and the Donaldson geometric flow. Li Sheng Title : Prescribed Scalar Curvatures for Homogeneous Toric Bundles. Abstract: We study the generalized Abreu equation on a Delzant ploytope $\Delta \subset R^2$ and use the method developed by Chen, Li and Sheng to study constant scalar curvatures on homogeneous toric bundles. Jyh-Haur Teh Title : Several differential cohomologies on complex manifolds. Abstract: I will introduce several new differential cohomologies on complex manifolds. The construction is based on a variant of spark complex introduced by Harvey and Lawson. Bing Wang Title :On the Kähler Ricci flow Abstract: Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized K¨ahler Ricci flows with proper geometric bounds. Our methods connects the structure theory of critical metrics with the structure of geometric flows. This is joint work with X. X. Chen. Ben Weinkove Title : Monge-Ampere equations on complex and almost complex manifolds Abstract: Yau's Theorem on the complex Monge-Ampere equation shows that one can prescribe the volume form of a Kahler metric on a compact Kahler manifold. I will describe extensions of this result to non-Kahler settings. In each case, a Monge-Ampere type equation is used to prescribe the volume form of a special metric on a complex or almost complex manifold. This talk is based on work with my co-authors Tosatti, Szekelyhidi and Chu. Bin Xu Title : Differentials and conical elliptic metrics on compact Riemann surfaces Abstract: A celebrated open problem asks whether there exists a conformal metric with positive constant curvature (conical elliptic metric) on a compact Riemann surface such that it has the prescribed finitely many conical singularities. A developing map of a conical elliptic metric has monodromy in SO(3). Umehara-Yamada called a conical elliptic metric reducible if the monodromy lies in U(1). Otherwise, they call it irreducible. We call a conical elliptic metric quasi-reducible if the monodromy lies in the semi-direct product of U(1) and Z/2. We observe that all the information of a reducible metric on a compact Riemann surface can be encapsulated into a one-form on the surface with simple poles such that all its periods are purely imaginary. We call such a one-form unitary. By using techniques of translation surfaces, we prove some general existence theorems of (unitary) one-forms with simple poles on compact Riemann surfaces and apply some of them to reducible metrics. In particular, for any genus g and n positive numbers given, we give a necessary and sufficient condition for them under which there exists a compact Riemann surface of genus g and a reducible metric on the surface with conical angles prescribed by these n positive numbers. We could also reduce the investigation of quasi-reducible metrics to that of Jenkins-Strebel differentials whose periods are all real. We also observe that a quasi-reducible metric is irreducible in general and can be thought of as a quotient of another reducible metric by a holomorphic involution. We observe that all the elliptic metrics on the Riemann sphere with four conical angles a*pi, b*pi, c*pi, d*pi, a, b, c and d being positive integers, must be quasi-reducible. Moreover, we could draw the critical graphs of the associated Jenkins-Strebel differentials of them and could write down the explicit forms of part of them. At last, we propose an existence proplem of quasi-reducible metrics. This is a joint work with Qing Chen, Bo Li, Santai Qu and Ji-Jian Song. Shing-Tung Yau Title : Non-Kahler Calabi-Yau Mirror Symmetry and Symplectic Structures.