MCM Members Seminar

9:30am, Each Wednesday

Speaker: Prof. Song Wang (MCM)

TitleCuspidality Criterion

Time: 15:30-16:30, Jan 17, 2019

Place: N817

AbstractIn this talk we will survey the known modularity and cuspidality cases up to now. In particular, we will survey one of our old work on cuspidality criterion on GL(2) \times GL(3).

Speaker: Prof. Xin Wan (MCM)

TitleIwasawa theory and Bloch-Kato conjecture for unitary groups

Time: 15:30-16:30, Jan 10, 2019

Place: N817

AbstractWe present some recent work on Iwasawa theory for motives corresonding to Galois representations associated to cusp forms on unitary groups over totally real fields twisted by Hecke characters, and some consequences for Bloch-Kato conjectures. We prove that if the central critical value is 0, then the Selmer group has positive rank.

Speaker: Prof. Ye Tian (MCM)

TitleIntroduction to Heegner Points

Time: 15:30-16:30, Jan 3, 2019

Place: N817

AbstractWe introduce the basic arithmetic theory of Heegner points and some applications.

Speaker: Prof. Yongquan Hu (MCM)

TitleMop p cohomology of Shimura curves

Time: 15:30-16:30, Dec 20, 2018

Place: N817

AbstractAt present, the mod p (and p-adic) local Langlands correspondence is only well understood for the group GL2(Qp), but remains mysterious even for GL2 of an unramified extension of Qp. However, the Buzzard-Diamond-Jarvis conjecture and the mod p local-global compatibility for GL2/Q suggest that this hypothetical correspondence may be realized in the cohomology of Shimura curves with characteristic p coefficients, cut out by some modular residual global representation. In the talk, I will report some results on the mod p cohomology of Shimura curves from the point of view of the mod p Langlands program. This is joint work (in progress) with Haoran Wang.

Speaker: Prof. Xu Shen (MCM)

TitleNewton strata for good reductions of Shimura varieties of orthogonal type

Time: 15:30-16:30, Dec 13, 2018

Place: N817

AbstractIn this talk, we will study the arithmetic geometry of the GSpin and SO Shimura varieties, which are special examples of Shimura varieties of abelian type. Over complex numbers, these Shimura varieties are closely related to moduli of hyperkaehler manifolds. Thanks to the works of Kisin and Vasiu, we can talk about smooth reductions of these varieties at good primes. It turns out the geometry over characteristic p is much finer, in the sense that these exist some natural stratifications for the reductions. We will describe all the Newton strata. Joint work with Chao Zhang.

Speaker: Prof. Baohua Fu (MCM)

TitleOn Fano complete intersections in rational homogeneous varieties

Time: 15:30-16:30, Dec 6, 2018

Place: N817

AbstractComplete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. We first classify these Fano complete intersections which are locally rigid.  It turns out that most of them are hyperplane sections.  We then classify general hyperplane sections which are quasi-homogeneous. This is a joint work with Chenyu Bai and Laurent Manivel.

Speaker: Dr. Qinbo Chen (MCM)

Title: Convergence of viscosity solutions in the vanishing contact structure problem

Time: 15:30-16:30, Nov 29, 2018

Place: N817

Abstract: I will present a joint work with Hitoshi Ishii, Wei Cheng and Kai Zhao on the vanishing contact structure problem, which focuses on the asymptotic behavior of the viscosity solutions uε of Hamilton-Jacobi equation H (x, Du(x), ε u(x)) =c, as the factor ε goes to zero. It is a natural generalization of the vanishing discount problem which was first studied in a general framework by P.-L. Lions, G. Papanicolaou and S. Varadhan. In this talk, I will first briefly introduce some basic notations and results in Aubry-Mather theory and weak KAM theory. Then I will explain how to characterize the limit solution in terms of Peierls barrier functions and Mather measures from a dynamical viewpoint.

Speaker: Dr. Bin Zhao (MCM)

Title: Slopes of modular forms

Time: 15:30-16:30, Nov 22, 2018

Place: N817

Abstract: In this talk, I will first explain the motivation to study the slopes of modular forms. It has an intimate relation with the study of the geometry of eigencurves. On the boundary of the weight disc, the previous work by Liu-Wan-Xiao gives an almost complete answer to this question. I will then explain a recent joint work in progress with Rufei Ren on the generalization of their result to eigenvarieties for certain Hilbert modular forms.

Speaker: Dr. Hao Zhang (MCM)

Title: The p-adic Gelfand-Kapranov-Zelevinsky Hypergeometric system

Time: 15:30-16:30, Nov 15, 2018

Place: N817

Abstract: To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. In this talk, we will talk about the p-adic counterpart of the GKZ hypergeometric system, and show how it gives the hypergeometric function over the finite field introduced by Gelfand and Graev.

Speaker: Dr. Alexandre Pyvovarov (MCM)

Title: Some new cases of the Breuil-Schneider conjecture

Time: 15:30-16:30, Nov 8, 2018

Place: N817

Abstract: Let F and E be two finite extensions of Qp such that E is large enough. Let r : Gal(F_bar/F) -> GL_n(E) be a Galois representation. In 2013 Caraiani, Emerton, Gee, Geraghty, Paskunas and Shin have constructed an E -Banach representation V(r) of GL_n(F). The authors have hypothesized that the representation V(r) corresponds to Galois representation r under hypothetical p-adic Langlands correspondence. In this work, we show that, under certain assumptions on r, the locally algebraic vectors of V(r) are isomorphic to an irreducible locally algebraic representation. This locally algebraic representation can be determined explicitly via the classical local Langlands correspondence and the knowledge of the Hodge-Tate weights of the Galois representation. From this we can derive new cases of the Breuil-Schneider conjecture.

Speaker: Dr. Bingyu Xia (MCM)

Title: Hilbert scheme of twisted cubics as simple wall-crossings

Time: 15:30-16:30, Nov 1, 2018

Place: N817

Abstract: Hilbert scheme is introduced by Grothendieck and it played an important role in algebraic geometry. Hilbert scheme of twisted cubics in the projective space P^3 is one of the easiest but nontrivial Hilbert scheme, its geometric structure was first described by Piene and Schlessinger in 1985. In this talk, I will introduce Bridgeland stability conditions on the derived category of the projective space P^3, and use wall-crossing phenomena of stability conditions to reprove Piene and Schlessinger's result.

Speaker: Dr. Yewon Jeong (MCM)

Title: Moduli of second fundamental forms of a nonsingular intersection of two quadrics

Time: 15:30-16:30, Oct 25, 2018

Place: N817

Abstract: In 1979, Griffiths and Harris raised a question on the moduli of second fundamental forms of a projective complex submanifold of codimension two. We will report on our study of the question for complete intersections of two quadrics.

Speaker: Dr. Jie Liu (MCM)

Title: Quasi-polarized Calabi-Yau threefolds

Time: 15:30-16:30, Oct 18, 2018

Place: N817

Abstract: A pair (X,L) is called a quasi-polarized Calabi-Yau threefold if X is Calabi-Yau threefold with at worst Gorenstein canonical singularities and L is a nef and big line bundle. In this talk, I will start by introducing the notion of canonical singularities, and then I will discuss the Fujita type results for (X,L) and their applications to the birational geometry of Fano manifolds with coindex four.

Speaker: Dr. Ka-Fai Li (MCM)

Title: The Kahler-Ricci flow on non-compact manifolds

Time: 15:30-16:30, Oct 11, 2018

Place: N817

Abstract: The Ricci flow was introduced by Hamilton in 1982, it is an intrinsic geometric flow that deforms the metric of a Riemannian manifold according to its Ricci curvature. While the existence and uniqueness of the solution is well-known on compact manifolds, we often need to impose some strong conditions in order to arrive at the same conclusion on non-compact manifolds. In this presentation, we will first discuss the background of the Ricci flow, then we will talk about some existence and uniqueness results on non-compact Kahler manifolds. If time is permitted, we will also discuss their applications.

Speaker: Dr. Peng Yu (MCM)

Title: CM Values Associated to Special Cycles on Shimura Varieties

Time: 9:30, June 6, 2018

Place: N818

Abstract: Kudla has a program inspired by theta functions and the work of Hirzebruch and Zagier in 1976. Its main goal is to prove that the generating functions arisen from certain arithmetic cycles on orthogonal Shimura varieties are actually modular. As part of the program, Kudla also conjectured a formula that relates the height pairing of the arithmetic cycles with some proper cycles with the central derivative of some Siegel Eisenstein series. The infinite part of the height turns out to be CM values of special functions on Shimura varieties. In this talk, I will present the background of the problem and known results on CM values. If time permits, I will further talk about my work on its applications to Siegel 3-fold case.

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).

SpeakerProf. Baohua FuMCM

TitleEquivariant compactifications of vector groups

Time9:30am, March 28, 2018

PlaceN817

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.

SpeakerDr. Qinbo ChenMCM

TitleGevrey genericity of Arnold diffusion

Time9:30am, March 21, 2018

PlaceN817

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.

SpeakerProf. Xin WanMCM

TitleBSD conjecture and generalizations

Time9:30am, March 14, 2018

PlaceN817

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.

Speaker: Prof. Weizhe Zheng (MCM)

Title: Around l-independence

Time: 9:30, May 30, 2018

Place: N818

Abstract: Serre and Tate proposed a number of conjectures on the l-independence of l-adic cohomology. I will survey some old and new results around these conjectures.