Morningside Center of Mathematics

Chinese Academy of Sciences

MCM Members Seminar

9:30am, Each Wednesday

**Speaker**: Dr. Hongbo Yin (MCM)

**Title**: Cube sum problems

**Time**: 9:30, May 23, 2018

**Place**: N818

**Abstract**: In this talk, I will explain my recent joint work with Jie Shu and Xu Song which says that if p is a prime congruent to 2 or 5 mod 9, then at least one of 3p and 3p^2 is cube sum. I will introduce some background of the cube sum problem first and then focus on the proof of our results. As comparison, if time permit, I will also explain the proof of Satge's classical result that if p is a prime congruent to 2 mod 9 then 2p is a cube sum and also the proof of Dasgupta and Voight's work on Sylvester conjecture.

**Speaker**: Prof. Binyong Sun (MCM)

**Title**: Cohomological test vectors

**Time**: 9:30, May 16, 2018

**Place**: N818

**Abstract**: Various types of modular symbols provide a powerful tool to study arithmetic of special values of L-functions. The Archimedean behaviors of the modular symbols are captured by certain restriction maps of relative Lie algebra cohomology spaces. We call these restriction maps modular symbols at infinity. The modular symbols are non-zero and of arithmetic interest only when the associated modular symbols at infinity are non-zero. Moreover, the latter holds if and only if certain invariant linear functionals on cohomological representations do not vanish on the minimal K-types (in the sense of Vogan). We will give some examples of invariant linear functionals on cohomological representations which does not vanish on the minimal K-types, including the Rankin-Selberg case GL(n)xGL(n-1).

**Speaker**: Prof. Xiaokui Yang (MCM)

**Title**: Positivity notions in complex differential and algebraic geometry

**Time**: 9:30, May 9, 2018

**Place**: N818

**Abstract**: In this presentation, we will describe the relationship between various positivity notions in complex differential geometry and complex algebraic geometry. We derive several new vanishing theorems for partially positive vector bundles. As applications, we characterize uniruled and rationally connected projective manifolds by using partially positive curvature tensors. In particular, we confirm a conjecture of Yau that a compact Kahler manifold with positive holomorphic sectional curvature is projective algebraic and rationally connected.

**Speaker**: Prof. Song Wang (MCM)

**Title**: Langlands and Multiplicities

**Time**: 9:30, May 2, 2018

**Place**: N818

**Abstract**: In this talk, we will survey current status on Langlands program, including recent breakthrough made by V. Lafforgue on Langlands parametrizations. Also, topics on multiplicities will also be addressed, and some ideals on constructing cusp forms of multiplicity greater than $1$ on $SO (2 N)$ over function fields.

**Main reference:** [Laf18] 1803.03791 SHTUKAS FOR REDUCTIVE GROUPS AND LANGLANDS CORRESPONDENCE FOR FUNCTION FIELDS

**Speaker**: Prof. Ye Tian (MCM)

**Title**: p-converse theorem for elliptic curves with complex multiplication

**Time**: 9:30, April 25, 2018

**Place**: N818

**Abstract**: Let E be an elliptic curve defined over rationals and p a prime. A theorem of Gross-Zagier and Kolyvagin says that if the L-function of E has vanishing order one at the center then the corank of its p-Selmer group is also one. In a joint work with Burungale, we show the converse to the above result in the case that E has complex multiplication and p>3 is ordinary for E. We will introduce the related concepts via the congruent number problem.

**Speaker**: Prof. Xu Shen (MCM)

**Title**: The Fargues-Rapoport conjecture

**Time**: 9:30, April 18, 2018

**Place**: N818

**Abstract**: Complex Shimura varieties arise as discrete quotients of certain Griffiths period domains. But things are quite different in the p-adic world. In this talk, we explain some ideas in the recent joint work with Miaofen Chen and Laurent Fargues on the structure of some p-adic period domains. More precisely, we will sketch a proof of the Fargues-Rapoport conjecture: for a basic local Shimura datum (G,b,μ), the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G, μ) is fully Hodge-Newton decomposable.

**Speaker**: Prof. Wenwei Li (MCM)

**Title**: On spherical spaces

**Time**: 9:30, April 11, 2018

**Place**: N818

**Abstract**: Spherical varieties are expected to be a general vehicle for extending Langlands program to a relative setting. Over an algebraically closed field, they have been classified in combinatorial terms by Luna-Vust, extending the well-known theory of toric varieties. In this talk, I plan to survey Wedhorn's recent work on spherical algebraic spaces. If possible, I will also sketch some possible mini-projects in this direction.

**Speaker**: Prof. Yongquan Hu (MCM)

**Title**: Asymptotic growth of the cohomology of Bianchi groups？

**Time**: 9:30, April 4 (Wednesday), 2018

**Place**: N817

**Abstract**: Given a level N and a weight k, we know the dimension formula of the space of classical modular forms. This turns out to be unknown if we consider Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In this talk, we consider the asymptotic behavior of the dimension when the level is fixed and the weight grows. I will first explain the background of this problem, and an upper bound obtained by Simon Marshall using Emerton's completed cohomology. Then I will explain how to improve this bound using the mod p representation theory of GL2(Qp).

**Speaker**：Prof. Baohua Fu（MCM）

**Title**：Equivariant compactifications of vector groups

**Time**：9:30am, March 28, 2018

**Place**：N817

**Abstract**: In 1999, Hassett-Tschinkel considered the equivariant version of this propblem obtained the classification up to dim. 3. I'll report recent progress on this (equivariant) problem. In particular, we obtain the classification up to dimension 5.

**Speaker**：Dr. Qinbo Chen（MCM）

**Title**：Gevrey genericity of Arnold diffusion

**Time**：9:30am, March 21, 2018

**Place**：N817

**Abstract**: It is well known that Arnold diffusion is a typical phenomenon for generic a priori unstable Hamiltonian systems. In this talk, by using variational method, we will show that under generic Gevrey-smooth perturbations, Arnold diffusion still exists in a priori unstable Hamiltonian systems.

**Speaker**：Prof. Xin Wan（MCM）

**Title**：BSD conjecture and generalizations

**Time**：9:30am, March 14, 2018

**Place**：N817

**Abstract**: We first give a survey on BSD conjecture and what is known about it, especially focus on some recent work on the full BSD formula in the rank 0 and 1 cases. Then we also discuss some generalizations to modular forms.