|
Xu Wang
|
Title: A flat Higgs bundle structure on the complexified Kahler cone
Abstract: We shall construct a natural flat Higgs bundle structure on the complexified Kahler cone and use it to study Wilson's conjecture. In the proof of the flatness, we found a variation formula of the Hodge Star operator. Some applications of it will also be given.
|
|
| |
| |
|
|
Andriy Haydys
|
Title:Isolated singularities of affine special Kaehler metrics in two dimensions.
Abstract:I will show that there are just two types of isolated singularities of special Kaehler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities. I will also present examples of such metrics.
|
|
| |
| |
|
|
Shin-ichi Matsumura
|
Titile: A transcendental approach to injectivity theorem for log canonical pairs
Abstract: In this talk, I expain transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs. As a result, in the case of purely log terminal pairs, I give an analytic proof of the injectivity theorem
originally proved by the Hodge theory. Our method is based on the theory of harmonic integrals and the $L^2$-method for the $¥dbar$-equation, and it enables us to generalize the injectivity theorem to the complex analytic setting.
|
|
| |
| |
|
|
Takayuki Koike
|
Title:Complex K3 surfaces containing Levi-flat hypersurfaces
Abstract:We show the existence of a complex K3 surface X which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such X by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.
|
|
| |
| |
|
|
Thibaut Delcroix
|
Title: K-stability of Fano spherical varieties
Abstract: The resolution of the Yau-Tian-Donaldson conjecture for Fano
manifolds, that is, the equivalence of the existence of Kähler-Einstein
metrics with K-stability, raises the question of determining when a given Fano manifold is K-stable.
I will present a combinatorial criterion of K-stability for Fano spherical manifolds. These form a very large class of almost-homogeneous manifolds,containing toric manifolds, homogeneous toric bundles, and classes of manifolds for which the Kähler-Einstein existence question was not solved yet, for example equivariant compactifications of (complex) symmetricspaces.
|
|
| |
| |
|
|
Gang Liu
|
Title: On Yau's uniformization conjecture
Abstract: Let M be a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth. We prove M is biholomorphic to C^n. This confirms Yau's uniformization conjecture when M has maximal volume growth.
|
|
| |
| |
|
|
Ruadhai Dervan
|
Title: Relative K-stability for Kähler manifolds
Abstract: We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. We also prove a general L^p lower bound on the Calabi functional, generalising Donaldson's work. Both of these results improve the known results for projective manifolds.
|
|
| |
| |
|
|
Hugues Auvray
|
Title: Bergman kernels on punctured Riemann surfaces
Abstract: In a joint work with X. Ma (Paris 7) and G. Marinescu (Cologne), we obtain refined asymptotics for Bergman kernels computed from singular data on Riemann surfaces.More precisely, we work on the complement of a finite set of points, seen as singularities, on a compact Riemann surface, that we endow with a metric extending Poincaré's cusp metric around the singularities. As for the polarization line bundle, it comes equipped with a positively curved Hermitian metric, whose curvature is the base metric near the singularities.I shall thus explain how an advanced description of the model geometry (given by Poincaré's metric on the punctured unit disc), and localization techniques in the spirit of Bismut-Lebeau in a weighted analysis context, allow us to describe the Bergman kernels attached to these punctured Riemann surfaces, up to their singularities. If time allows, I shall also mention an arithmetic interpretation of these results, in terms of modular forms.
|
|
| |
| |
|
|
Guoyi Xu
|
Title: When the fundamental group of a Riemannian manifold is finitely generated?
Abstract: For every compact Riemannian manifold, it is well known that the fundamental group is finitely generated. For complete non-compact Riemannian manifolds, the fundamental group possibly is not finitely generated. A natural question is: which complete Riemannian manifolds have finitely generated fundamental group? We will survey the progress in this question from Bieberbach, Cheeger-Gromoll, Gromov to more recent work by Kapovitch and Wilking, and my recent work will also be presented. No technical proofs in the talk, some elementary topology and Riemannian geometry knowledge is enough to understand most of the talk.
|
|
| |
| |
|
|
Kento Fujita
|
Title: Uniform K-stability and plt blowups of log Fano pairs
Abstract: We show relationships between uniform K-stability and plt blowups of log Fano pairs.
|
|
| |
| |
|
|
Ryosuke Takahashi
|
Title:Smooth approximation of the modified conical K\"ahler-Ricci flow
Abstract:In this talk, we discuss the conical K\"ahler-Ricci flow modified by a holomorphic vector field on Fano manifolds, whose stationary points are exactly K\"ahler-Ricci solitons with cone singularities along a simple normal crossing divisor.By using the approximation method established by Liu-Zhang and Y.Wang, we construct a long-time solution as the limit of a sequence of smooth K\"ahler-Ricci flows.
|
|
| |
| |
|
|
Changzheng Li
|
Title: On a conjectural Peterson isomorphism in quantum K-theory
Abstract: The Peterson's isomorphism says that the torus-equivariant of quantum cohomology of a complete flag variety is essential isomorphic to the torus-equivariant homology of the associated affine Grassmannian. In this talk, we will formulate a precise conjecture of the Peterson isomorphism on the level of quantum K-theory, and will provide some evidence. This is my on-going joint work with Thomas Lam, Leonardo Mihalcea and Mark Shimozono.
|
|
| |
| |
|
|
Haotian Wu
|
Title: Degenerate neckpinches in mean curvature flow and Ricci flow
Abstract: We survey results concerning the formation of degenerate neckpinches that model Type-II singularities in mean curvature flow and Ricci flow. We then explain recent work with James Isenberg on mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up.
|
|
| |
| |
|
|
Jiakun Liu
|
Title: Schauder estimates and Stochastic PDEs
Abstract: In this talk, we first introduce a new proof of Schauder estimates by Xu-Jia Wang’s perturbation argument, including applications to uniform elliptic and parabolic equations (Wang), Monge-Ampere equations (Jian-Wang), and optimal transportation equations (L.-Trudinger-Wang). More recently, we extend this idea to establish a sharp Schauder theory for stochastic PDEs of parabolic type, which answers an open problem proposed by Krylov in 1999.
|
|
| |
|
|
|
|
|
Yu Li
|
Title: Ricci flow on asymptotically Euclidean manifolds
Abstract: In this talk, we prove that if an asymptotically Euclidean manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent proof of positive mass theorem.
|
|
