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 premoduli 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 SasakiEinstein 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 TønnesenFriedman, and Otto van Koert.

Gao Chen 
Title : Classification of gravitational instantons
Abstract:
Gravitational instanton is a complete noncompact hyperk\"ahler 4manifold, i.e. CalabiYau 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.

CheolHyun Cho 
Title : Noncommutative 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 noncommutative algebra A, together with a central element W, which provides a noncommutativeLandauGinzburg model (A,W). The construction comes with a natural Ainfinity functor from the Fukaya category to the category of matrix factorizations of the constructed LandauGinzburg model. In particular it recovers and strengthens several interesting results of Seidel, EtingofGinzburg, 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 SiuCheong 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 : FanoRicci limit spaces and spectral convergence
Abstract:
We study the behavior under GromovHausdorff 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 anticanonical
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 (UCIrvine), 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 4d, in which this construction has some interesting applications to the fully nonlinear Yamabe problem and other geometric variational problems.

Oscar GarciaPrada 
Title : KaehlerYangMills equations and gravitating vortices
Abstract:
In this talk we first introduce the KaehlerYangMills
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 HermitianYangMills 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. AlvarezConsul and M. GarciaFernandez).

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 BandoMabuchi's bifurcation technique in the conic setting.

Tianjun Li 
Title : Symplectic CalabiYau surfaces
Abstract:
Symplectic CalabiYau surfaces are closed symplectic 4manifolds with torsion canonical class.
They are the exactly the minimal symplectic 4manifolds with Kodaira dimension zero.We will discuss what is known about the classification.

Toshiki Mabuchi 
Title : Extremal K\"ahler versions of the YauTianDonaldson Conjecture
Abstract:
In this talk, we discuss extremal K\"ahler versions of the YauTianDonaldson Conjecture by looking at their relationships to the asymptotic relative Chowstability.We focus on what is the right choice of the Kstability 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 CartanFubini extension (2001), according to which a germ of VMRTpreserving 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. HongMok (2010) extended CartanFubini extension to the nonequidimensional setting under a relative nondegeneracy condition on second fundamental forms on VMRTs. Recently MokZhang 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 subVMRT structure defined by intersections of VMRTs with tangent spaces. Assuming that subVMRTs satisfy new nondegeneracy conditions and that the distribution spanned by subVMRTs is bracketgenerating, 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 Lietheoretic approach. The method relies heavily on the structure theories of the local solutions and of the Winvariants 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 $(n1)H$, 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) (LedrappierWang) 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.

JyhHaur 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 noncollapsed CalabiYau 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 : MongeAmpere equations on complex and almost complex manifolds
Abstract:
Yau's Theorem on the complex MongeAmpere 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 nonKahler settings. In each case, a MongeAmpere 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 coauthors 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). UmeharaYamada called a conical elliptic metric reducible if the monodromy lies in U(1). Otherwise, they call it irreducible. We call a conical elliptic metric quasireducible if the monodromy lies in the semidirect 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 oneform on the surface with simple poles such that all its periods are purely imaginary. We call such a oneform unitary. By using techniques of translation surfaces, we prove some general existence theorems of (unitary) oneforms 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 quasireducible metrics to that of JenkinsStrebel differentials whose periods are all real. We also observe that a quasireducible 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 quasireducible. Moreover, we could draw the critical graphs of the associated JenkinsStrebel differentials of them and could write down the explicit forms of part of them. At last, we propose an existence proplem of quasireducible metrics. This is a joint work with Qing Chen, Bo Li, Santai Qu and JiJian Song.

ShingTung Yau 
Title : NonKahler CalabiYau Mirror Symmetry and Symplectic Structures.
