speaker 
title 
abstract 

Dynamics Complex Dimension 2 
The iteration of polynomials and rational functions has been successful and beautiful in complex dimension 1. We will discuss analogous questions and results in complex dimension 2. We will give an outline of the current state of the subject and some of the future challenges. 

gravitational instantons with faster than quadratic curvature decay 
In this paper, we study gravitational instantons (i.e., complete hyperk\"aler 4manifolds with faster than quadratic curvature decay). 

Direct methods of moving planes, moving spheres, and blowingups for the fractional Laplacian 
Many conventional approaches on partial diﬀerential operators do not work on the nonlocal fractional operator. To overcome this diﬃculty arising from nonlocalness, Caﬀarelli and Silvestre introduced the extension method to reduced the problem into a local one in one higher dimensions, which has become a powerful tool in studying such nonlocal problems and has yielded a series of fruitful results. We firmly believe that these ideas and approaches can be eﬀectively applied to a wide range of nonlinear problems involving fractional Laplacians or other nonlocal operators. 
YoungJun Choi

Positivity of fiberwise KahlerEinstein metrics 
Let p : X —> Y be a family of Kahler manifold, i.e., a surjective holomorphic mapping between complex manifolds such that every fiber is a Kahler manifold. If every fiber admits a KahlerEinstein metric, then it induces a fiberwise KahlerEinstein metric, which is defined by a real (1,1)form on X which is KahlerEinstein when it is restricted on each fiber. In this talk, we discuss the positivity (on the total space X ) of fiberwise KahlerEinstein metrics on families of bounded strongly pseudoconvex domains, canonically polarized compact Kahler manifolds and CalabiYau manifolds. 

Exotic nearly Kähler structures on the 6sphere and the product of two 3spheres 
Compact 6dimensional nearly Kähler manifolds are the crosssections of Riemannian cones with $G_2$ holonomy. Viewing $\mathbb{R}^7$ as the cone over $S^6$ endows the 6sphere with a nearly Kähler structure which coincides with the standard $G_2$invariant almost complex structure induced by octonionic multiplication. A longstanding problem has been the question of existence of complete nearly Kähler 6manifolds besides the four known homogeneous ones. We resolve this problem by proving the existence of an exotic (inhomogeneous) nearly Kähler structure on $S^6$ and on $S^3\times S^3$. This is joint work with Mark Haskins, Imperial College London. 
Blowup Analysis for a Nonlinear Equation with Negative Exponent 
We study the blowup behavior of the minimizing sequence to certain nonlinear ellipic equation with negative critical exponents. We analyze the location and the asymptotic behavior of solutions near the blowup point. This is joint work with Meijun Zhu. 


The Nonuniqueness of the tangent cone at infinity of Ricciflat manifolds 
For a complete Riemannian manifold (M,g), the pointed GromovHausdorff limit of (M, r^2g) as r to 0 is called the tangent cone at infinity. 

Elliptic PDEs on compact Ricci limit spaces and applications 
In this talk we discuss the behavior of solutions of several elliptic PDEs with respect to the GromovHausdorff topology and applications. Applications include the study of second order differential structure on limit spaces of Riemannian manifolds with lower Ricci curvature bounds. This talk is based on arXiv:1410.3296. 

Harmonic function and heat flow on metric measure spaces (MMS) 
This talk is divided into two parts. In the first part, I will talk about how to get quantitative Lipshcitz regularity (Yau's gradient estimate) of harmonic functions via heat flows on MMS with Ricci curvature bounded from below. 




Analysis of solutions to HardyLittlewoodSobolev type systems 
This presentation is centered around the wellknown HardyLittlewoodSobolev type systems: These are also sometimes called LaneEmden type systems. 

Scalarflat K\"ahler ALE metrics on minimal resolutions 
Scalarflat K\"ahler ALE surfaces have been studied in a variety of settings since the late 1970s. All previously known examples have group at infinity either cyclic or contained in SU(2). I will describe an existence result for scalarflat K\"ahler ALE metrics with group at infinity G, where the underlying space is the minimal resolution of C^2/G, for all finite subgroups G of U(2) which act freely on S^3. I will also discuss a nonexistence result for Ricciflat metrics on certain spaces, which is related to a conjecture of BandoKasueNakajima. If time permits, I will also present some new examples selfdual metrics on connected sums of complex projective planes. This is joint work with Jeff Viaclovsky. 
On the decay of the offdiagonal Bergman kernel on complete Kahler manifold 
We give an Agmontype exponential decay of the Bergman kernel for noncompact manifolds where only the Ricci curvature lower bound of the Kahler manifolds is assumed. 


Harmonic maps between asymptotically hyperbolic manifolds 
In 1990s, P. Li and L.F. Tam studied the asymptotic Dirichlet problem on proper harmonic maps between the hyperbolic spaces, and showed an existence and uniqueness result under the $C^1$ boundary regularity. We generalize it to asymptotically hyperbolic manifolds. Analogously to the classical theorems of EellsSampson and Hamilton, the unique existence in each relative homotopy class is shown under some assumption. This talk is based on a joint work with K. Akutagawa. 
On deformation of cscK and extremal metrics 
We will report on a joint work with X. Chen and Y. Zheng. In the unlikely event that the time permits, we will present some applications/speculations. 


A generalised MongeAmpere equation 
In this talk, I will introduce a fully nonlinear PDE (that includes the usual MongeAmpere equation as a special case) which arises from ChernWeil theory. Versions of this PDE also appear in other geometric and physical contexts (like the Jflow, Mirror symmetry etc). I will mention the existing results on the subject (notably the works of Wei Sun, Collins, and Szekelyhidi, among others). I will also describe some new applications of the existing results and also a few new results on a priori estimates. 

On Ricci flow invariant curvature cones. 
(joint with H. Seshadri, IISc Bangalore) When attacking geometric problems using Ricci flow, it is often an important step to find a condition on the curvature of the Riemannian manifold under consideration which is at the same time relevant to the investigated problem and preserved by the Ricci flow equation. Thanks to Hamilton’s maximum principle, showing the preservation of a curvature condition can be reduced to the study of an ODE on the finite dimensional space of algebraic curvature operators. Although numerous examples of preserved curvature conditions have been found, it is still not clear how weak those curvature conditions can. I will present some results which show that preserved curvature conditions cannot be to weak. 

Modulo join rigidity in Alexandrov geometry 
Alexandrov geometry was introduced by BuragoGromovPerelman in 1992, which is a synthetic geometry on length metric spaces satisfying the Toponogov triangle comparison. One might characterize this direction of research as trying to show that Alexandrov's world is just as good as Riemann's. 

K\”ahler Ricci flow on Fano manifold 
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 theory is a generalization of the structure theory of noncollapsed K\"ahler Einstein manifolds.As applications, we prove the HamiltonTian conjecture and the partial$C^0$conjecture of Tian. 
Changyou Wang

Existence of finite time singularity of nematic liquid crystal flow in dimension three 
In this talk, I will describe the simplified version of Ericksen and Leslie that models the hydrodynamic flow of nematic liquid crystals, 

KrylovEvans type theorem for twisted MongeAmp\'ere equations.

Motivated by the pluriclosed flow of Streets and Tian, we establish EvansKrylov type estimates for parabolic "twisted" MongeAmpere equations in both the real and complex setting. In particular, a boundon the second derivatives on solutions to these equations yields bounds on Holder norms of the second derivatives. These equations are parabolic but neither not convex nor concave, so the celebrated proof of EvansKrylov does not apply. In the real case, the method exploits a partial Legendre transform to form second derivative quantities which are subsolutions. Despite the lack of a bona fide complex Legendre transform, we show the result holds in the complex case as well, by formally aping the calculation. This is joint work with Jeff Streets. 

The width of Riemannian manifolds 
In 1917, Birkhoff brought Minimax method into studying geometry the first time, while solving the existence of the closed geodesics in compact manifolds. In this talk, following the history of Minimax method, I will first review the Minimax method applied in linear and nonlinear cases. The different concept of the width will be introduced in geometric contexts. Generally, the width is realized by the value of the functional at the ``eigenfunctions", which are found by the Minimax argument. I will present my recent work on the realization problem of the width with multiple parameters, which is closely related one question of Marques and Neves, originally goes back to Gromov’s philosophy. 

Holomorphic sectional curvature and the canonical bundle 
In this talk, we present the relationship between the holomorphic sectional curvature and the canonical line bundle over compact Kaehler manifolds. In particular, we confirm a conjecture of Yau, by using very recent ideas of WuYau. This is joint work with V. Tosatti. 

An introduction on generalized Ricci lower bounds on Alexandrov spaces 
In this talk, we shall introduce some results on generalized Ricci lower bounds of Alexandrov spaces, based on some works joint with Professor XiPing Zhu. 

Some local regularity properties of Ricci flows. 
Under the assumption of bounded scalar curvature we show that the Ricci flow possesses 
Singular constant scalar curvature Kahler metrics 
The talk is aiming to present some progress on singular constant scalar curvature Kahler metrics. 