Date: 09/20, 16:00-17:30 五教5107
Speaker: 李逸（东南大学）
Title: Scalar Curvature along Ricci flow
Abstract: In this talk I will discuss two interesting geometric flows, the classical Ricci flow and the coupled Kahler-Ricci flow introduced recently by Yuan Yuan, Yugang Zhang and myself. For the Ricci flow, a very recent work on the scalar curvature that is related with Hamilton's conjecture was derived, which generalizes X. D. Cao's result. For the coupled Ricci flow, besides some basic properties, I will propose two conjectures on long time existence. |
Date: 09/27, 16:00-17:30 五教5107
Speaker: 林明辉（华中师范大学）
Title: Growth of Mordeil-Weil ranks in p-adic Lie extensions
Abstract: A well-known theorem of Mordeil-Weil tells us that the group of F-rational points of an abelian variety is a finitely generated (abelian) group, where F is a number field. One study of the interest lies in the variation of the Mordell-Weil ranks of an abelian variety in a p-adic Lie extension, where here p is a prime. Mazur initiated Iwasawa theory of Selmer groups for abelian varieties with good ordinary reduction at all primes above p, and applied it to obtain an upper bound for the growth of Mordell-Weil ranks in a cyclotomic Zp-extension. In this talk, we will discuss higher analog of such upper bound for the Mordell-Weil rank of an abelian variety with good ordinary reduction at all primes above p in a p-adic Lie extension. Finally, if time permits, we mention the situation of an elliptic curve with supersingular reduction at p. This is joint work with Pin-Chi Hung. |
Date: 10/11, 16:00-17:30 五教5107
Speaker: 郑恺（同济大学）
Title: Singular cscK metrics
Abstract: Singular cscK metrics are served as singularity models of the Yau-Tian-Donaldson conjecture for cscK metrics. In this talk, we will present recent progress on existence of singular cscK metrics. |
Date: 10/15, Tuesday, 16:00-17:30 五教5506
Speaker: Christian Blohmann ( MPIM, Bonn）
Title: Hamiltonian actions: from integrable systems to hamiltonian Lie algebroids
Abstract: I will give a tour of the notion of hamiltonian actions from basic concepts to recent developments. In the first hour I will give a self contained review of the following topics: conserved quantities in classical mechanics; integrability; hamiltonian group actions; toric manifolds, Guillemin-Sternberg convexity theorem, Delzant reconstrucion; symplectic reduction; Bochner-Weinstein linearizaton at critical points. In the last half hour I will describe a recent generalization of hamiltonian actions to Lie algebroids and Lie groupoids, which is motivated by a study of the symmetries of General Relativity. This is joint work with Alan Weinstein. |
Date: 10/18, 10:00-11:30 五教5504
Speaker: 涂君武（上海科技大学）
Title: On the foundations of categorical enumerative invariants
Abstract: We clarify some of the missing details in Costello’s construction of categorical enumerative invariants back in 2005. Conjecturally, these categorically defined invariants should: — Recover the Gromov-Witten invariants when applied to the Fukaya categories. — Give a mathematical construction of the BCOV invariants when applied to the derived categories of Calabi-Yau’s. — Give a mirror theory of the FJRW theory, when applied to the category of (G-equivariant) matrix factorizations. |
Date: 10/25, 15:00-16:30 五教5107
Speaker: Giulio Ruzza （SISSA, UCLouvain）
Title: Random matrices, integrable hierarchies and isomonodromic deformations
Abstract: I will give a basic introduction to random matrices, focusing in particular on the Gaussian Unitary Ensemble (GUE), Wigner Semicircle Law, and combinatorics of the correlators. Depending on time I will move towards more recent results about computation of correlators of the GUE and of related models. |
Date: 10/25, 16:30-18:00 五教5107
Speaker: 林兴君（武汉大学）
Title:On constructions and classification of rational vertex operator algebras
Abstract: In this talk, we first talk about some basic results about vertex operator algebras. We then talk about the classical classification of even unimodular lattices of rank 24. Finally, we talk about the classification of holomorphic vertex operator algebras of central charge 24. |
Date: 10/30,Wednesday, 16:00-17:30 五教5207
Speaker: 扶磊（清华大学）
Title: $\ell$-adic tautological systems
Abstract: Tautological systems of differential equations are introduced by Lian and Yau in order to describe the differential equations satisfied by the period integrals of hyperplane sections of complex projective homogeneous Calabi-Yau varieties. We introduce the $\ell$-adic tautological systems and study their relation with exponential sums associated to representations of reductive groups. |
Date: 11/08, 16:00-17:30 五教5107
Speaker: 杜荣（华东师范大学）
Title: Vector Bundles on Flag varieties
Abstract: We will introduce the background of algebraic vector bundles on projective spaces and Grassmannians and their related problems in algebraic geometry. In addition, the generalization of the Grauert-Mulich-Barth theorem on semistable bundles on flag varieties will also be discussed. This is a joint work with Xinyi Fang and Yun Gao. |
Date: 12/04,Wednesday, 16:00-17:30 五教5107
Speaker: 赵永强（西湖大学）
Title: Scrollar syzygy and Galois representation
Abstract: The theory of scrollar syzygy resolution was introduced by Schreyer on his work for Green's conjecture for canonical curves. Besides its applications in syzygy theory, it is also closely related to the parametrization theory of curves of small gonality and has important applications in the theory of Hurwitz space. However, it is still an open problem on how to determine all scrollar syzygies. In this talk, we will discuss its relations with Galois theory. We will give a (conjectural) complete description of all syzygy bundles through representation theory. This is a joint work with Wouter Castryck. |
Date: 12/06, 16:00-17:30 五教5206
Speaker: 姚成建（上海科技大学）
Title: Gravitating Vortices with positive curvature on Riemann sphere
Abstract: Gravitating vortex equation is a system of PDEs coupling constant scalar curvature problem and Hermitian-Yang-Mills problem on a line bundle on a Riemann surface in the presence of a Higgs field. We prove that gravitating vortex equation on P^1 with positive curvature is solvable if and only if the divisor, i.e. the zeros of the Higgs field, is polystable under the canonical SL(2,C)-action. Our method uses a continuity path starting from Yisong Yang's solution with zero curvature (in relation to some gravitating problem in theoretical physics) and deforming the coupling constant towards 0. This is a joint work with Mario Garcia-Fernandez and Vamsi Pingali. |
Date: 12/13, 16:00-17:30 五教5107
Speaker: 邵国宽（中山大学）
Title: Morse inequalities on complex/CR manifolds
Abstract: In this talk, we will present recent progress on Morse inequalities. Firstly we give an introduction to the classical result with an outline of a pure analytic proof. Then we talk about one version on CR manifolds by Hsiao-Li. In particular, we deduce Demailly's holomorphic Morse inequalities. Finally we discuss our result on complex manifolds with boundary, in which the upper bounds depend only on the boundary. |
Date: 12/19, Thursday,10:30-12:00 五教5401
Speaker: 江瑞奇（湖南大学）
Title: On harmonic maps
Abstract: In this talk, we first give a brief introduction to the regularity and existence of harmonic maps and related topics, and then present our recent progress on the existence of extrinsic polyharmonic maps. |
Date: 12/27, 16:00-17:30 五教5107
Speaker: 吴鹏（上海数学中心）
Title: Complex structures on Einstein four-manifolds of positive scalar curvature
Abstract: In this talk we will discuss the relationship between complex structures and Einstein metrics of positive scalar curvature on four-dimensional Riemannian manifolds. One direction, that is, when a four-manifold with a complex structure admits a compatible Einstein metric of positive scalar curvature has been answered by Tian, LeBrun, respectively. We will consider the other direction, that is, when a four-manifold with an Einstein metric of positive scalar curvature admits a compatible complex structure. We will show that if the determinant of the self-dual Weyl curvature is positive then the manifold admits a compatible complex structure. Our method relies on Derdzinski's proof of the Weitzenbock formula for self-dual Weyl curvature. |
Date: 1/3, 2020, 10:30-12:00, 五教5406
Speaker: Zhongshan An （University of Connecticut）
Title: Ellipticity of the Bartnik Boundary Conditions
Abstract: The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics. In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, derive a local result to the existence question. |
Date: 1/3, 2020, 10:00-11：30, 五教5301
Speaker:谢松晏 （中科院数学所）
Title: Introduction to complex hyperbolicity
Abstract: In the year 1970, Kobayashi introduced an intrinsic canonical metric for every complex manifold $X$, and when this obtained metric is everywhere non-degenerated, $X$ is said to be complex hyperbolic. He conjectured that, in all dimensions, "most" manifolds should be complex hyperbolic. In this talk we shall present the history and some recent developments in this subject, including the Green-Griffiths-Lang conjecture. |
