Eric Bedford


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.

Gao Chen




gravitational instantons with faster than quadratic curvature decay

In this paper, we study gravitational instantons (i.e., complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay).
We prove three main theorems:
1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH.
2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in ALG and ALH cases.
3.In ALF-D_k case, it must have an O(4)-multiplet.

Wenxiong Chen









Direct methods of moving planes, moving spheres, and blowing-ups 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 non-localness, 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.
However, due to technical restrictions, sometimes one needs to impose extra conditions when studying the extended problems in higher dimensions, and these conditions may not b e necessary if we investigate the original nonlocal problems directly.
In this talk, we will introduce direct methods of moving planes , moving spheres , and blowing-up and re-scaling arguments for the fractional Laplacian. By an elementary approach, we will first show the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space, such as strong maximum principles for anti-symmetric functions , narrow region principles , and decay at infinity . Then, using simple examples, semi-linear equations involving the fractional Laplacian, we will illustrate how this new method of moving planes can be conveniently employed to obtain symmetry and non-existence of positive solutions, under much weaker conditions than in the previous literatures.

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.

Young-Jun  Choi



Positivity of fiberwise Kahler-Einstein metrics

Let p : X —> Y be a family of Kahler manifold, i.e., a surjective holo-morphic mapping between complex manifolds such that every fiber  is a Kahler manifold. If every fiber admits a Kahler-Einstein metric, then it induces a fiberwise Kahler-Einstein   metric, which is defined by a real (1,1)-form on X which is Kahler-Einstein  when it is restricted on each fiber.

In this talk, we discuss the positivity (on the total space X ) of fiberwise Kahler-Einstein   metrics on families of bounded strongly pseudoconvex domains, canonically polarized compact Kahler manifolds and Calabi-Yau manifolds.

Lorenzo Foscolo





Exotic nearly Kähler structures on the 6-sphere and the product of two 3-spheres

Compact 6-dimensional nearly Kähler manifolds are the cross-sections of Riemannian cones with $G_2$ holonomy. Viewing $\mathbb{R}^7$ as the cone over $S^6$ endows the 6-sphere with a nearly Kähler structure which coincides with the standard $G_2$-invariant almost complex structure induced by octonionic multiplication.

A long-standing problem has been the question of existence of complete nearly Kähler 6-manifolds 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.

Qianqiao Guo

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.

Kota Hattori




The Nonuniqueness of the tangent cone at infinity of Ricci-flat manifolds

For a complete Riemannian manifold (M,g), the pointed Gromov-Hausdorff limit of (M, r^2g) as r to 0 is called the tangent cone at infinity.
By the Gromov's Compactness Theorem, there exists tangent cone at infinity for every complete Riemannian manifolds with nonnegative Ricci curvatures. Moreover, if it is Ricci-flat, with Euclidean volume growth and having at least one tangent cone at infinity with a smooth cross section, then it is uniquely determined by the result of Colding and Minicozzi.
In this talk I will explain that the assumption of the volume growth is essential for their uniqueness theorem.

Shouhei Honda


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 Gromov-Hausdorff 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.

Renjin Jiang



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.
In the second part, I will explain how to use Yau's gradient estimate for harmonic functions to establish the Li-Yau inequality for heat flows on MMS with Ricci curvature bounded from below.

Dano Kim



Congming Li




Analysis of solutions to Hardy-Littlewood-Sobolev type systems

This presentation is centered around the well-known Hardy-Littlewood-Sobolev type systems: gongshi

These are also sometimes called Lane-Emden type systems.
We give a brief survey on some important known results, a short introduction of some of our recent results, and also a brief description of some basic problems that we are interested.
Beyond the basic qualitative properties such as the existence, non-existence,
and classification of positive solutions, we are also interested in the integrability, asymptotic at infinite, and symmetries of positive solutions.

Mike Lock




Scalar-flat K\"ahler ALE metrics on minimal resolutions

Scalar-flat 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 scalar-flat 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 non-existence result for Ricci-flat metrics on certain spaces, which is related to a conjecture of Bando-Kasue-Nakajima. If time permits, I will also present some new examples self-dual metrics on connected sums of complex projective planes. This is joint work with Jeff Viaclovsky.

Zhiqin Lu

On the decay of the off-diagonal Bergman kernel on complete Kahler manifold

We give an Agmon-type exponential decay of the Bergman kernel for non-compact manifolds where only the Ricci curvature lower bound of the Kahler manifolds is assumed.

Yoshihiko Matsumoto



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 Eells-Sampson 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.

Mihai Paun

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.

Vamsi Pingali



A generalised Monge-Ampere equation

In this talk, I will introduce a fully nonlinear PDE (that includes the usual Monge-Ampere equation as a special case) which arises from Chern-Weil theory. Versions of this PDE also appear in other geometric and physical contexts (like the J-flow, 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.

Thomas Richard




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.

Xiaochun Rong




Modulo join rigidity in Alexandrov geometry

Alexandrov geometry was introduced by Burago-Gromov-Perelman 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.
A significant advantage is that many natural operations, such as gluing, quotient by isometries, join (e.g. cones and suspensions) are preserved. In this talk, we will discuss rigidity aspects of finite quotients of join structures. This is a joint work with Yusheng Wang of Beijing Normal University.

Bing Wang


K\”ahler Ricci flow on Fano manifold

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 theory is a generalization of the structure theory of non-collapsed K\"ahler Einstein manifolds.As applications, we prove the Hamilton-Tian 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,
which is a governing equation for the macroscopic continuum description of evolution of the material under the influence of both fluid velocity field and the macroscopic average of the microscopic orientation of rod-like liquid crystal molecules. I will indicate two ways to construct the finite time singularities for such a flow equation for suitably chosen initial data. This is a joint work with Tao Huang, Chun Liu, and Fanghua Lin.

Micah Warren




Krylov-Evans type theorem for twisted Monge-Amp\'ere equations.


Motivated by the pluriclosed flow of Streets and Tian, we establish Evans-Krylov type estimates for parabolic "twisted" Monge-Ampere 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 Evans-Krylov 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.

Guoyi Xu 




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.

Xiaokui Yang



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 Wu-Yau. This is joint work with V. Tosatti.

Huichun Zhang


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 Xi-Ping Zhu.
On the one hand, we will introduce some results on analysis of linear equations, including harmonic functions, the lower bounded estimate for the first eigenvalue, and some geometric rigidity results. On the other hand, we consider the harmonic maps from an Alexandrov space with curvature bounded from below into an Alexandrov space with non-positive curvature. These harmonic maps are locally Lipschitz continuous. It affirms a problem of Fanghua Lin.

Qi Zhang




Some local regularity properties of Ricci flows.

Under the assumption of bounded scalar curvature we show that the Ricci flow possesses
a number of regularity properties such as
distance continuity, backward pseudo locality,
Gaussian heat kernel bound, existence of good
cut off function, some of which were conjectured or sought after. Applications to 4 dimensonal Ricci flows and others will be described. This is a joint work with Richard Bamler.

Kai Zheng

Singular constant scalar curvature Kahler metrics

The talk is aiming to present some progress on singular constant scalar curvature Kahler metrics.