7 (Mon) 
8 (Tue) 
9 (Wed) 
10 (Thu) 
11 (Fri) 

D. Salamon (9:3010:30) 
O. Biquard (9:3010:30) 
C. Arezzo (9:3010:30) 
Y. Sano (9:3010:30) 

J. Song (11:0012:00) 
N.C. Leung (11:0012:00) 
S. Paul (11:0012:00) 
K. Ono (11:0012:00) 

Lunch  Lunch  
X.X. Chen (14:0015:00) 
S. Honda (14:0015:00) 
N. Mok (13:3014:30) 

S. Sun (15:3016:30) 
T. Yamaguchi (15:3016:30) 
K. Fukaya (15:0016:00) 

G. Tian (16:3017:30) 

Reception (18:00) 
This is a joint work with B. Wang. Based on the compactness of the moduli space of noncollapse Calabi Yau space with mild singularities. Our theory is a generalization of the well known theory of noncollapsed Kähler Einstein manifolds. As an application, we prove the Hamilton Tian conjecture on the Kähler Ricci flow, as well as the complete partial \(C^0\) estimate of Tian for Kähler metrics with Ricci bounded from below. DonaldsonSun proved the partial \(C^0\) estimate for Kähler Einstein metrics.
I will talk about the algebraic structure of GromovHausdorff limits of KählerEinstein manifolds, and its relation with the existence theory of KählerEinstein metrics on Fano manifolds, as well as the theory of moduli spaces.
In this lecture I will explain the momentweight inequality, and its role in the proof of the HilbertMumford numerical criterion for stability. The setting is Hamiltonian group actions on closed Kähler manifolds. The momentweight inequality relates the Mumford numerical invariants to the norm of the moment map on the complexified group orbit. Key ingredients in the proof are the negative gradient flow of the moment map squared and the KempNess function. In the analogy between finitedimensional GIT and CSCK metrics the Mumford numerical invariants correspond to the Futaki invariants and the KempfNess function corresponds to the Mabuchi functional. This is joint work with Valentina Georgoulas and Joel Robbin, based on conversations with Xiuxiong Chen, Song Sun, and Sean Paul.
We study Riemannian geometry of canonical KählerEinstein currents on projective CalabiYau varieties and canonical models of general type with crepant resolutions. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original projective variety with well defined tangent cones. A number of applications are given for degenerations of CalabiYau manifolds and the KählerRicci flow on smooth minimal models of general type.
In this talk we introduce a Rellich type compactness with respect to the GromovHausdorff topology. This compactness has several applications. For example we can give an alternative simplified proof of Fukaya's conjecture on the behavior of the eigenvalues of the Laplacian with respect to the GHtopology which was proved by CheegerColding. In this talk we particularly discuss a new relationship between Cheeger constant, \(p\)Laplacian, and GromovHausdorff convergence obtained by this compactness.
Alexandrov spaces with curvature bounded below typically appear as GromovHausdorff limits of Riemannian manifolds with a lower curvature bound, and the study of such spaces have been an important subject in Riemannian geometry. It sometimes brings us significant information of global structure of manifolds. Local homeomorphism structure of Alexandrov spaces is known as Perelman's topological stability theorem. However Lipschitz structure is not known at this stage, although Perelman claimed it long time ago. In this lecture, I will talk about Lipshitz homotopy structure based on recent joint works with Ayato Mitsuishi. We make use of gradient curves for distance functions to establish Lipschitz homotopy version of critical point theory. Several applications will also be presented.
I will discuss the obstruction I found some time ago for desingularizing Einstein 4orbifolds, which leads to a wall crossing phenomenon in the Dirichlet problem for Asymptotically Hyperbolic Einstein metrics.
We studied Witten deformation of Laplacian operators twisted by Morse functions. In the semiclassical limit, we show that the \(A_{\infty}\) structure on small eigenforms converges to the one in Morse category defined by counting gradient flow trees. This is a joint work with K.L. Chan and Z.M. Ma. This research work is supported by RGC grant CUHK 2130314.
I will describe a new gluing result for Kähler constant scalar curvature metrics obtained in collaboration with R. Lena and L. Mazzieri. This construction generalised to Kcsc metrics of any sign the celebrated Kummer construction for CalabiYau manifolds discovered by LeBrun and Singer.
The speaker will give an account of his recent work on the algebraic geometry that lies behind Mabuchi's Kenergy map.
With J.M. Hwang the speaker has developed a geometric theory of uniruled projective manifolds \(X\) modeled on varieties of minimal rational tangents \(\mathcal{C}_x(X) \subset \Bbb PT_x(X)\), alias VMRTs.
Generalizing works of HwangMok, HongMok considered pairs \((X_0;X)\) of uniruled projective manifolds, and established a nonequidimensional CartanFubini Extension Principle (2010) in terms of a certain nondegeneracy condition on the second fundamental form for a pair \((\mathcal B \subset\mathcal A)\) consisting of a VMRT \(\mathcal A \,\) and a linear section \(\mathcal B \,\) of \(\mathcal A\).
The latter has led to the characterization of standard embeddings \(i: G_0/P_0 \hookrightarrow G/P\) between rational homogeneous manifolds of Picard number 1 by HongMok (2010) in the longroot and nonlinear cases and by HongPark (2011) in the shortroot cases and in the cases of linear subspaces with identifiable exceptions.
The argument therein involving parallel transport of VMRTs has also been applied by HongMok (2013) to establish homological rigidity for certain smooth Schubert cycles.
Recently in a joint work with Y. Zhang we have established a stronger rigidity phenomenon for subVMRT structures, where in place of a germ of mapping \(f: (X_0;0) \to (X;0)\) we consider a germ of submanifold \((S;0) \subset (X;0)\) for a uniruled projective manifold \(X\) equipped with a minimal rational component \(\mathcal K\).
Defining a subVMRT structure by taking intersections \(\mathcal C_x(X) \cap \Bbb PT_x(S)\) we have obtained sufficient conditions for \(S\) to extend to a rationally saturated projective subvariety \(Z \subset X\).
In the rational homogeneous case the method yields a strengthening of the results of HongMok and HongPark.
For instance, if a germ of submanifold \((S;0) \subset (X;0)\) inherits by intersecting VMRTs with projectivized tangent subspaces a Grassmann structure of rank \(\ge 2\), then \(S\) in fact extends to a subGrassmannian in its standard embedding.
In this talk, I will give a version of Matsushima Theorem for normal varieties. I will explain its proof following the original approach of Matsushima and its generalization in my 2012 paper.
On a compact Kähler manifold, Futaki and Mabuchi showed that the complex gradient vector field of the projection of the scalar curvature to the space of the normalized Hamiltonian functions of holomorphic vector fields is independent of the choice of Kähler metrics. Such vector field is called the extremal vector field. In this talk, I will explain about a computation on the extremal vector fields on smooth toric Fano manifolds.
Arnold's conjecture for fixed points of Hamiltonian diffeomorphisms has stimulated the development of symplectic geometry. It was a motivation of Floer to initiate what is now called Floer cohomology. Although the Betti number version of the conjecture was settled, the original form of the conjecture is still open. In particular, whether the nontriviality of the fundamental group implies some additional lower bound or not. Recently, there are some progress, though not so much. I will explain an approach based on a joint work with Andrei Pajitnov. If time allows, I may touch another direction of study.