Members and postdoctors of Morningside Center of Mathematics will present their current work

in this seminar on each Thursday, 10:30-11:30am.

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**Speaker**: Dr. Arnaud Plessis (MCM)

**Title**: On Lehmer's problem

**Time**: 10:30-11:30am, March 9, 2023

**Place**: Online (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Let x be an algebraic number. Lehmer's problem predicts that the product of the degree of x and the Weil height of x is bounded by an absolute constant. The basic idea to tackle this problem is to study the points of small Weil height (or short, small points) in an algebraic closure of Q. The case where small points of an algebraic field L are either 0 or a root of unity is intensively studied and we will provide a few explicit examples. Then we will explain why it is complicated to precisely locate small points of L if the latter contains more small points than 0 and the roots of unity. Time permitting, we will discuss about Lehmer's problem on elliptic curves.

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**Speaker**: Dr. Dongming She (MCM)

**Title**: Stability of Rankin-Selberg gamma factors of classcial groups

**Time**: 10:30-11:30am, December 22, 2022

**Place**: Online (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Analytic stability of local gamma-factors is a crucial property for representations of reductive groups over local fields. It can be used to prove the equality of the arithmetic and analytic local epsilon and L-factors via local Langlands correspondence. And it is also very powerful in the study of local converse theorems. We will construct the Rankin-Selberg local gamma factors of a split classical group with a general linear group by Langlands-Shahidi method, and sketch the proof of the stability of their corresponding local gamma factors over p-adic fields. This is a joint work with Taiwang Deng.

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**Speaker**: Dr. Xiaozong Wang (MCM)

**Title**: Smoothing of 1-cycles over finite fields

**Time**: 10:30-11:30am, December 15, 2022

**Place**: Online (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: It is known by Hironaka's work in 1968 that on a smooth projective variety defined over an infinite field, any algebraic 1-cycle is rationally equivalent to a smooth one. In this talk, I will show that the result is also true when the variety is defined over a finite field. In this setting, several results on the density of smooth divisors satisfying certain conditions are needed to construct the rationally equivalent smooth 1-cycle. I will also discuss these density results.

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**Speaker**: Dr. Cong Ding (MCM)

**Title**: Rigidity for admissible pairs of deletion type

**Time**: 10:30-11:30am, December 8, 2022

**Place**: Online (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Schubert rigidity/flexibility for smooth Schubert varieties in Flag varieties is to find integral varieties for the Schubert differential systems. Motivated by this a rigidity for admissible pairs is formulated. When the ambient space is an irreducible Hermitian symmetric space of compact type, the rigidity holds for smooth Schubert varieties with trivial exceptions, where the admissible pairs coming from smooth Schubert varieties are associated to a subdiagram of of the marked Dynkin diagrams. When the admissible pair comes from a diagram by deleting chains in the marked Dynkin diagrams, the rigidity no longer holds. We will add a natural condition and discuss a weaker rigidity on these admissible pairs.

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**Speaker**: Dr. Yaqing Hu (MCM)

**Title**: Reflecting numbers of various types

**Time**: 10:30-11:30am, December 1, 2022

**Place**: Online (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: A nonzero integer n is called a reflecting number of type (k,m) if n - t^m = u^k, n + t^m = v^k have a rational solution (t,u,v) ∈ Q* × Q × Q. In particular, reflecting numbers of type (2,2) are all congruent numbers and thus will be called reflecting congruent numbers. We can show that all prime numbers p ≡ 5 mod 8 are reflecting congruent and in general for any integer k ≥ 0 there are infinitely many square-free reflecting congruent numbers in the residue class of 5 modulo 8 with exactly k+1 prime divisors. Moreover, we conjecture that all prime congruent numbers p ≡ 1 mod 8 are reflecting congruent. In addition, we show that there are no reflecting numbers of type (k,m) if gcd(k,m) ≥ 3.

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**Speaker**: Dr. Renjie Lyu (MCM)

**Title**: Degeneration of Hodge structure and cubic hypersurfaces

**Time**: 10:30-11:30am, November 24, 2022

**Place**: Online (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: The degeneration of Hodge structures is related to how a smooth projective variety degenerate. And it provides a Hodge-theoretic perspective to compactify moduli spaces. In this talk, I will focus on a particular degeneration of cubic hypersurfaces and study the associated limiting mixed Hodge structure. It generalizes part of results in Radu Laza’s and Brendan Hassett’s works. This is a joint work with Zhiwei Zheng.

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**Speaker**: Dr. Zhangchi Chen (MCM)

**Title**: Hodge-Riemann property of Griffiths positive matrices of (1,1) forms

**Time**: 10:30-11:30am, November 17, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: The Hard Lefschetz theorem (HLT), Hodge-Riemann bilinear relation theorem (HRR) and Lefschetz decomposition theorem (LD) are stated for a power of a K\"ahler class on a compact K\"ahler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative.

Gromov (1990), Timorin (1998), Dinh-Nguy\^en (2006, 2013) proved the mixed HLT, HRR and LD for a product of arbitrary K\"ahler classes. Instead of products, Dinh-Nguy\^en (2013) conjectured that determinants of Griffiths positive $k\times k$ matrices with $(1,1)$ form entries in $\mathbb{C}^n$ satisfies these theorems in the linear case.

Assume that the matrix only has diagonalized entries, for $k=2$ and $n\geqslant 4$, the determinant satisfies HLT for bidegrees $(n-2,0)$, $(n-3,1)$, $(1,n-3)$ and $(0,n-2)$. In particular, Dinh-Nguy\^en's conjecture is true for $k=2$ and $n=4,5$ with this extra assumption.

The key idea is to notice that the determinant of certain Lefschetz map has the form of Heron formula which calculates the area of a triangle with given side length. Thus HLT, HRR and LD are proved by checking the strict triangle inequality.

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**Speaker**: Dr. Xun Zhang (MCM)

**Title**: On Deligne-Beilinson cohomology of the universal K3 surface

**Time**: 10:30-11:30am, November 10, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: O'Grady's generalized Franchetta conjecture is concerned with algebraic cycles of codimension 2 on the universal polarized K3 surface of genus g. This conjecture has been confirmed only for certain small values of g. A variant of this conjecture, which replaces Chow groups with Betti cohomology groups, is confirmed for all g by Nicolas Bergeron and Zhiyuan Li.

In this talk, I will discuss another variant which replaces Chow groups with Deligne-Beilinson cohomology groups. Besides a sketch of proof, I will mainly focus on two technical points: the fisrt one is to extend the notion of Deligne-Beilinson cohomology to the case of Deligne-Mumford stacks, and the second one is a computation of the numbers of cuspidal and binodal members in certain two-dimensional families of polarized K3 surfaces with ADE singularities. This is joint work with Zhiyuan Li.

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**Speaker**: Dr. Ruishen Zhao (MCM)

**Title**: Special cycles on Shimura varieties with norm relations

**Time**: 10:30-11:30am, November 3, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk I will introduce a general method to establish suitable norm relations for special cycles on Shimura varieties. I will use unitary cycles in odd orthogonal Shimura varieties as a guiding example. Under suitable ordinary conditions, the resulting special cycles with norm relations can be used to construct a norm compatible family of Galois cohomology classes.

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**Speaker**: Prof. Qingtang Su (MCM)

**Title**: Wellposedness, singularity, and the stability of the water wave equations

**Time**: 10:30-11:30am, October 27, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk I will give a brief survey to the cauchy problem of the water wave equations, including the local wellposedness, long time wellposedness, the study of non-smoooth solutions as well as the formation of singularity, and (if time permitted) the stability of solutions.

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**Speaker**: Prof. Jinping Zhuge (MCM)

**Title**: Unique continuation, spectral theory, nodal geometry and homogenization

**Time**: 10:30-11:30am, October 20, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: I will give an introductory presentation on quantitative unique continuation and its connection to spectral thory and nodal geometry. Particularly, recent progress on periodic ellitpic operators will be reported and many important open questions will be mentioned.

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

**Title**: Equidistribution of translated measures in homogeneous spaces and Dirichlet improvability

**Time**: 10:30-11:30am, October 13, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Dynamics of subgroup action on homogeneous spaces of Lie groups has many applications to Diophantine approximation. After reviewing some of these applications, we will focus on equidistribution of translated measures and its application to Dirichlet improvability in Diophantine approximation.**——————————————————————————————————————————————————————**

**Speaker**: Prof. Zicheng Qian (MCM)

**Title**: On Breuil-Schraen L-invariants for GLn

**Time**: 10:30-11:30am, September 29, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: We quickly review some known results on (higher) L-invariants including Breuil and Schraen's definition using extensions between locally analytic generalized Steinberg representations for GL2(Qp) and GL3(Qp). Then we state our generalization of their results to GLn(Qp) based on a recent computation of such (higher) extension groups.

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

**Title**: On the cohomology of classifying spaces

**Time**: 10:30-11:30am, September 22, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this mostly expository talk, I'll try to explain a conjecture/question of Totaro, which is later resolved by Kubrak--Prikhodko. Time permitting, I'll explain a “short-cut” proof by Bhatt and myself.

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**Speaker**: Dr. Lin Zhou (MCM)

**Title**: Algebraic cycles on Gushel-Mukai varieties

**Time**: 10:30-11:30am, September 15, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: GM varieties are a kind of Fano varieties, which share rich geometry with cubic fourfolds and have close connections to HK varieties. In this talk, I will compute all integral Chow groups of complex GM varieties with dimension at least 3, except for the two infinite-dimensional cases, via unramified cohomology and the methods of Bloch-Srinivas. This is a part of a recent work by Fu and Moonen (arXiv: 2207.01118). If time permits，I will introduce Chow-Künneth decompositions of the Chow motives of GM varieties and other interesting contents of their paper.

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**Speaker**: Dr. Weibiao Wang (MCM)

**Title**: Extendable periodic automorphisms of closed surfaces over the 3-sphere

**Time**: 10:30-11:30am, September 8, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: An automorphism of a surface is said to be extendable over a manifold if there exists an embedding such that the surface map extends to an automorphism of the manifold. Given a periodic map on a closed surface, how can we tell whether it is extendable over the 3-sphere? This is completely solved recently. I will give a brief introduction on this topic.

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**Speaker**: Prof. Weizhe Zheng (MCM)

**Title**: Decomposition theorem in integral l-adic cohomology

**Time**: 10:30-11:30am, May 12, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: I will review the decomposition theorem in l-adic cohomology and present an integral refinement for l large enough. This is based on joint work with Anna Cadoret on ultraproduct cohomology.

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**Speaker**: Prof. Siqi He (MCM)

**Title**: Multi-valued harmonic 1-forms and related problems

**Time**: 10:30-11:30am, April 28, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Multi-valued harmonic 1-forms could be considered as a generalization of quadratic differentials to real manifolds. In the first half the talk, we will explains the motivations and interesting problems that related to the multi-valued harmonic 1-forms. In the second half of the talk, we will discuss how to use multi-valued harmonic 1-forms to construct branched deformations of special Lagrangian submanifolds.

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

**Title**: Parallel transport for Higgs bundles over p-adic curves

**Time**: 10:30-11:30am, April 21, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Faltings conjectured that under the p-adic Simpson correspondence, finite dimensional p-adic representations of the geometric étale fundamental group of a smooth proper p-adic curve X are equivalent to semi-stable Higgs bundles of degree zero over X. We will talk about an equivalence between these representations and Higgs bundles whose underlying vector bundle admits potentially a strongly semi-stable reduction of degree zero. These Higgs bundles are semi-stable of degree zero and we will investigate some evidence for Faltings' conjecture.

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

**Title**: Siegel Zeros

**Time**: 10:30-11:30am, April 14, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Generalized Riemasnn Hypothesis assertes that for various L-functions (for example, Riemann zeta functions, Dirichlet L-functions, Dedekind Zeta functions, Hecke L-series, L-fuctions of modular forms etc...) all non-trivial zeros lie in the critical vertical line Re(s)=1/2. However, we can only get much weaker unconditional results: (Almost) all zeros are far from s=1 in some sense. The zeros that near s = 1 if exist are called Siegel zeros. We will survey Siegel Zeros. First, we explain certain results on Dirichlet L-functions related to such zeros, leading to an explicit formula for Dirichlet prime number theorem. Then we explain the exact definition of Siegel zeros for L-functions families, and also a criterion for Siegel zeros. Also some examples and further related topics such as zero-free regions will be explained.

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**Speaker**: Prof. Xin Wan (MCM)

**Title**: Tamagawa number conjecture and p-adic Langlands

**Time**: 10:30-11:30am, April 7, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: We propose a local definition of the p-part of the Tamagawa number at p for 2-dimensional deRham representations of the absolute Galois group of Qp (allowing arbitrary ramification and Hodge-Tate weights) in terms of p-adic Langlands functor of Colmez, and explain the compatibility with Bloch-Kato's definition in the Fontaine-Laffaille case, and deduce the corresponding Tamagawa number conjecture in the rank 0 case

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**Speaker**: Dr. Zhihao Zhao (MCM)

**Title**: Affine Grassmannians and local models for triality groups

**Time**: 10:30-11:30am, March. 31, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, I will give an explicit description of affine Grassmannians for triality groups as functors classifying suitable lattices in a fixed space. These triality groups are of type $^3 D_4$ and can be constructed by certain twisted composition algebras. Further, I will briefly introduce global affine Grassmannians for triality groups. I combine this description with the Pappas-Zhu construction, to obtain corresponding local models; the singularities of these local models are supposed to model the singularities of certain orthogonal Shimura varieties.

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

**Title**: On quadratic twists of elliptic curves

**Time**: 10:30-11:30am, March. 24, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: L-function and Selmer groups are two arithmetic invariants of elliptic curves over number fields. In this talk, we discuss their behaviour under quadratic twists.

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

**Title**: Rigidity of automorphic conjugate self-dual Galois representations

**Time**: 10:30-11:30am, March. 17, 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Let F/F^+ be an imaginary quadratic extension of a totally real number field. In this talk, I will explain the notion of rigid conjugate self-dual residue Galois representations of F with coefficients in a finite field of characteristic l. In a joint work with Yifeng Liu, Liang Xiao, Wei Zhang and Xinwen Zhu, we show that, under some technical conditions, if such a residue Galois representation comes from a given conjugate self-dual automorphic cuspidal representation of GL_n(A_F), then it is rigid for almost all prime l. This kind of results can be used to prove some (weak) R=T theorems for automorphic forms on unitary groups.

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

**Title**: On the Hodge-Tate period maps for Shimura varieties

**Time**: 10:30-11:30am, March. 10 , 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: We explain a construction of the Hodge-Tate period maps for Shimura varieties. We will also discuss some applications to the l-adic cohomology.

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

**Title**: Gelfand-Kirillov dimension and the mod p Jacquet-Langlands correspondence for GL_2(Qp)

**Time**: 10:30-11:30am, March. 3 , 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: I will report some recent results on mod p Jacquet-Langlands correspondence for GL_2(Qp). Precisely, we prove that some admissible smooth representations of the (non-split) quaternion algebra over Qp coming from mod p cohomology have Gelfand-Kirillov dimension 1. As an application, we prove that the degree two Scholze’s functor vanishes on (generic) supersingular representations of GL2(Qp). This is joint work with Haoran Wang.

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**Speaker**: Prof. Baohua Fu (MCM)

**Title**: An introduction to Hodge Conjecture

**Time**: 10:30-11:30am, Jan. 6 , 2022

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: As one of the seven Millennium Prize Problems, Hodge Conjecture aims to establish a connection between the topology and the subvarieties of a smooth projective complex algebraic variety. I'll give a gentle and historical introduction to this conjecture.

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**Speaker**: Dr. Yupeng Wang (MCM)

**Title**: On the Hodge-Tate crystals over $\mathcal{O}_K$

**Time**: 10:30-11:30am, Dec. 30 , 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: The prismatic theory is established by Bhatt-Scholze and plays an important role in the integral p-adic Hodge theory. In this talk, we will focus on the Hodge-Tate crystals; that is, vector bundles with coefficients in $\bar{\mathcal{O}}_\Delta$ over $\mathcal{O}_K$. I will show that a Hodge-Tate crystal is determined by a "nilpotent" matrix and related to a $\mathbb{C}_p$-representation. Also, I will compute its absolute prismatic cohomology. This is a joint work with Yu Min.

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**Speaker**: Dr. Jun Wang (MCM)

**Title**: On the exceptional case of Sharifi's conjectures

**Time**: 10:30-11:30am, Dec. 23 , 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, I will discuss some results on the exceptional eigenspace of Sharifi's conjectures. If time permits, I will briefly mention some applications on Iwasawa theory of Eisenstein series.

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**Speaker**: Dr. Dongming She (MCM)

**Title**: Reductive monoids and multiplicativity of gamma-factors

**Time**: 10:30-11:30am, Dec. 16 , 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: The Braverman-Kazhdan-Ngo-Lafforgue program proposed an approach utilizing reductive monoids to construct the Langlands L-functions, Schwartz spaces and Fourier transforms in general, providing a vast generalization of the Godement-Jacquet theory for GL(n). I will briefly talk about some recent developments on this topic and sketch the proof of the multiplicativity of local gamma factors under some assumptions on the conjectural \rho-Fourier transforms following the recent work of Shahidi and Sokurski.

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**Speaker**: Dr. Arnaud Plessis (MCM)

**Title**: On small points of a given algebraic set

**Time**: 10:30-11:30am, Dec. 9 , 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: After recalling the definition of the Weil (absolute, logarithmic) height, I will claim several statements concerning the fields having the so-called Bogomolov Property. Next, I will explain why it is difficult to localize small points of a given algebraic field which does not have this property. Finally, I will conclude my talk by presenting a new idea aiming to handle this case.

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**Speaker**: Dr. Yu Min (MCM)

**Title**: Relative (phi, Gamma)-modules and prismatic F-crystals

**Time**: 10:30-11:30am, Dec. 2 , 2021

**Place**: MCM410 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, I will show that for a smooth p-adic formal scheme, the category of prismatic F-crystals on it is equivalent to the category of relative (phi,Gamma)-modules on its generic fiber. Then I will compare the cohomology of the corresponding coefficient objects. This is joint work with Yupeng Wang.

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**Speaker**: Dr. Lizao Ye (MCM)

**Title**: Automorphic Sheaves with given unipotent Arthur parameter.

**Time**: 10:30-11:30am, Nov. 25 , 2021

**Place**: ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Let X be a complete smooth algebraic curve, G be a reductive group, BunG be the moduli stack of G-bundles on X. A unipotent Arthur parameter is essentially a unipotent orbit in the Langlands dual group G^ of G. We are interested in Hecke-Eigensheaves on BunG with Eigenvalue given by such parameter. Before, the only known examples of such sheaves are: 1. constant sheaf, corresponding to the regular orbit; 2. Compatified Eisenstein series of Braverman-Gaitsgory, corresponding to regular orbits in Levi-subgroups of G^. (Of course the first case is a special case of the second one.) What about other unipotent orbits? Lafforgue-Lysenko considered subregular orbit in even orthogonal groups; later Ye considered subregular orbit in the exceptional group of type G2. However these attempts, using mainly Fourier transform, are unsatisfactory and can describe the corresponding sheaf only up to some ambiguity. I will report on my work which, in the setting of D-modules and using completely different method (chiral homology), constructs such Hecke-Eigensheaves corresponding to many unipotent orbits not considered before, including all subregular ones, and regardless of the type of G or the curve X.

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**Speaker**: Dr. Zhangchi Chen (MCM)

**Title**: Directed harmonic currents near non-hyperbolic linearized singularities

**Time**: 10:30-11:30am, Nov. 18 , 2021

**Place**: ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Let (D^2,F,{0}) be a singular holomorphic foliation on the bidisc D^2 defined by z∂z + λw ∂w with λ ∈ C^∗. This foliation has a non-degenerate linearizable singularity at 0. Let T be a harmonic current directed by F which does not give mass to any of the separatrices (z=0) and (w=0). In 2014, Nguyên proved that if 0 is a hyperbolic singularity, i.e. λ not real, then the Lelong numebr of T at 0 vanishes. Suppose the trivial extension of T across 0 is dd^c-closed. For the non-hyperbolic case λ ∈ R∗, we prove that the Lelong number at 0: 1) is strictly positive if λ > 0; 2) vanishes if λ ∈ Q<0; 3) vanishes if λ < 0 and T is invariant under the action of some cofinite subgroup of the monodromy group.

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**Speaker**: Dr. Renjie Lyu (MCM)

**Title**: Lines on the secant cubic hypersurfaces of Severi varieties

**Time**: 10:30-11:30am, Nov. 11 , 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: The secant variety of Severi varieties provide special examples of (singular)cubic hypersurfaces. An interesting question is when a given cubic hypersurface is projectively equivalent to a secant cubic. A result of F.~Charles says any smooth cubic hypersurface is determined by the Hilbert scheme of lines. Inspired by this we describe the Hilbert scheme of lines on secant cubic hypersurfaces. And we prove that a cubic hypersurface is isomorphic to a secant cubic if and only if the associated Hilbert schemes of lines are isomorphic.

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**Speaker**: Dr. Yaqing Hu (MCM)

**Title**: Waring's problem for locally Nilpotent groups: the case of discrete Heisenberg groups

**Time**: 10:30-11:30am, Nov. 4 , 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Kamke solved an analog of Waring's problem with $n$th powers replaced by integer-valued polynomials. Larsen and Nguyen explored the view of algebraic groups as a natural setting for Waring's problem. In this talk, we will develop a theory of polynomial maps from nonempty commutative semigroups to arbitrary groups, prove that it has desirable formal properties when the target group is locally nilpotent, and apply it to solve an analog of Waring's problem for the general discrete Heisenberg groups $H_{2n+1}(\mathbb{Z})$ for any integer $n\ge 1$.

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**Speaker**: Dr. Cong Ding (MCM)

**Title**: Special birational transformations on irreducible compact Hermitian symmetric spaces

**Time**: 10:30-11:30am, October 28, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: I will geometrically explain how to transform an irreducible compact Hermitian symmetric space of rank r to a projective space of the same dimension by successive blow-ups for r-1 times and successive blow-downs for r-1 times along smooth centers.

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**Speaker**: Associate Prof. Zhengyi Zhou (MCM)

**Title**: A hierarchy of contact manifolds from rational symplectic field theory

**Time**: 10:30-11:30am, October 21, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Symplectic field theory as a functor from the symplectic cobordism category is a natural tool to study contact manifolds and their cobordisms. In this talk, I will explain how to phrase the rational symplectic field theory as bi-Lie infinity algebras and use it to define several numerical invariants, which give a coarse classification of contact manifolds in the (exact) symplectic cobordism category. Time permitting, I will explain the invariants for the contact boundary of $\mathbb{CP}^n$ minus multiple hyperplanes in a generic position. This is joint work with Agustin Moreno.

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**Speaker**: Assistant Prof. Jingren Chi (MCM)

**Title**: Harmonic analysis in bad reduction of Shimura varieties

**Time**: 10:30-11:30am, October 14, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: I will explain the attempts of Rapoport, Scholze and Scholze-Shin to extend the Langlands-Kottwitz method of describing etale cohomology of Shimura varieties to cases of bad reduction (in which the level structure could be arbitrary and the local group could be non-quasi-split). In their approach, the problem is reduced to certain conjectures on orbital integrals and characters of local test functions that are of independent interest. Then I will explain how to prove these conjectures for inner forms of GL(n). This is based on ongoing joint work with Thomas Haines.

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**Speaker**: Dr. Yangyu Fan (MCM)

**Title**: Relatively supercuspidal spectra

**Time**: 10:30-11:30am, September 30, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Let E/F be a quadratic field extension of p-adic fields. In this talk, we will classify the relatively supercuspidal spectra for the pair (GL_n(E), GL_n(F)). This is a joint work with Cai Li.

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**Speaker**: Dr. Yue Xu (MCM)

**Title**: Distributions of abelian ramification groups of quadratic fields

**Time**: 10:30-11:30am, September 23, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, we will review the classical Cohen-Lenstra heuristics for ideal class groups of quadratic field, and propose some similar conjectures for abelian ramification groups. As an application, we will give some distribution results about the fundamental units for certain real quadratic fields. This is a joint work with Jianing Li and Yi Ouyang.

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**Speaker**: Dr. Xiaozong Wang (MCM)

**Title**: On the Bertini regularity theorem for arithmetic varieties

**Time**: 10:30-11:30am, September 16, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, we are interested in the existence of regular projective subschemes of a regular projective arithmetic variety. Let $\mathcal{X}$ be a regular projective arithmetic variety equipped with an ample Hermitian line bundle $\overline{\mathcal{L}}$. We show that the proportion of global sections $\sigma$ with $\left\lVert \sigma \right\rVert_{\infty}<1$ of $\overline{\mathcal{L}}^{\otimes d}$ whose divisor does not have a singular point on the fiber $\mathcal{X}_p$ over any prime $p\leq e^{\varepsilon d}$ tends to $\zeta_{\mathcal{X}}(1+\dim \mathcal{X})^{-1}$ as $d\rightarrow \infty$.

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**Speaker**: Prof. Weizhe Zheng (MCM)

**Title**: Ultraproduct cohomology and the decomposition theorem

**Time**: 10:30-11:30am, April 15, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Ultraproducts of étale cohomology provide a large family of Weil cohomology theories for algebraic varieties. Their properties are closely related to questions of l-independence and torsion-freeness of l-adic cohomology. I will present recent progress in ultraproduct cohomology with coefficients, such as the decomposition theorem, and applications to l-adic cohomology. This talk is based on joint work with Anna Cadoret.

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

**Title**: Minkwoski-Siegel Formula

**Time**: 10:30-11:30am, April 8, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk we will survey the Minkowski-Siegel Formula which interprets the average number of the integral solutions of ternery quadratic forms among a genus class in terms of various quantities such as local densities.

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

**Title**: An introduction to diamonds

**Time**: 10:30-11:30am, April 1, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: We give a brief and example-based introduction to Scholze's theory of diamonds, which plays a fundamental role in the recent work of Fargues-Scholze on the geometrization of local Langlands correspondences.

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

**Title**: Distribution problem on elliptic curves

**Time**: 10:30-11:30am, Mar. 25, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, we review distribution problems of elliptic curves over a fixed number field.

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**Speaker**: Prof. Xin Wan (MCM)

**Title**: BSD formula for general weight modular forms of rank 0 and 1

**Time**: 10:30-11:30am, Mar. 18, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: We present a joint work with Jetchev and Skinner to prove BSD formula for modular forms of general weight and analytic rank 0 or 1. This generalizes our previous work for elliptic curves. We also explain the new techniques used in the argument. The rank 0 case is deduced from rank 1 case without using cyclotomic Iwasawa theory.

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

**Title**: On a generalization of Colmez's functor

**Time**: 10:30-11:30am, Mar. 11, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In 2005, Colmez defined an exact functor from the category of finite length admissible smooth representations of GL_2(Q_p) over a field of characteristic p to the category of finite length continuous representations of the absolute Galois group of Q_p. This functor has played a crucial role in the p-adic Langlands program for GL_2(Q_p). In this talk, I will review the construction of Colmez’s functor, and discuss a generalization due to Breuil. If time permits, I will explain the proof of the exactness of this generalized functor. This is a joint work with Breuil, Herzig, Morra and Schraen.

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**Speaker**: Prof. Baohua Fu (MCM)

**Title**: Isolated symplectic singularities

**Time**: 10:30-11:30am, Jan. 21, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In 2000, Beauville introduced symplectic singularities, as an analogue of hyperkahler among rational singularities. Basic examples include singularities in finite symplectic quotients and in nilpotent orbit closures. As the simplest case, isolated symplectic singularities are very interesting and connected to the Lebrun-Salamon conjecture, while there are very few such examples. Are there any new examples? I will report some recent progress.

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**Speaker**: Dr. Yewon Jeong (MCM)

**Title**: Several types of dual defective cubic hypersurfaces

**Time**: 10:30-11:30am, Jan. 14, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Given a hypersurface X=V(f) in a complex projective space, we say X is dual defective if the Gauss map of X, the restriction of the gradient map of f on X, has positive dimensional fibers. Especially for cubics, there is an interesting classification of them. We will study several types of dual defective cubic hypersurfaces and the relation between them.

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**Speaker**: Dr. Bin Zhao (MCM)

**Title**: Slopes of modular forms

**Time**: 10:30-11:30am, Jan. 7, 2021

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, I will first explain the motivation to study the slopes of modular forms. Then I will explain a conjecture raised by Bergdall and Pollack which gives an effective algorithm to compute the slopes of modular forms and give some important consequences of this conjecture. I will talk about some strategies to prove this conjecture if time permits. This is a joint work in progress with Ruochuan Liu, Nha Truong and Liang Xiao.

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**Speaker**: Dr. Hao Zhang (MCM)

**Title**: The p-adic Gelfand-Kapranov-Zelevinsky hypergeometric complex

**Time**: 10:30-11:30am, Dec. 31, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**:In this talk, I will go over our construction of p-adic GKZ-hypergeometric complex. Itis a twisted logarithmic de-Rham complex describing the variation of a family of exponential sums. Using rigid cohomology and arithmetic D-module theory, we study finiteness properties and holonomicity of this complex.

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**Speaker**: Dr. Bingyu Xia (MCM)

**Title**: Bridgeland stability condition on surfaces with curves of negative self-intersection

**Time**: 10:30-11:30am, Dec. 24, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: I will explain some background motivation in birational geometry, and construct stability conditions for surfaces containing a curve whose self-intersection is negative. This is joint work with R. Tramel.

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**Speaker**: Dr. Alexandre Pyvovarov (MCM)

**Title**: Generic smooth representations

**Time**: 10:30-11:30am, Dec. 17, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: Let $F$ be a non-archimedean local field. In this talk we will explore genericity of irreducible smooth representations of $GL_n(F)$ by restriction to a maximal compact subgroup $K$ of $GL_n(F)$. Let $(J, \lambda)$ be a Bushnell--Kutzko type for a Bernstein component $\Omega$. The work of Schneider--Zink gives an irreducible $K$-representation $\sigma_{min}(\lambda)$, which appears with multiplicity one in $\mathrm{Ind}_J^K \lambda$. Let $\pi$ be an irreducible smooth representation of $GL_n(F)$ in $\Omega$. We will prove that $\pi$ is generic if and only if $\sigma_{min}(\lambda)$ is contained in $\pi$, in which case it occurs with multiplicity one.

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**Speaker**: Dr. Renjie Lyu (MCM)

**Title**: Cubic hypersurfaces: Chow groups, motives and rationality problems

**Time**: 10:30-11:30am, Dec. 10, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: The Chow group of algebraic varieties is extensively used to study the rationality problem. In this talk, I will talk about two results. One is called the universal generation of the Chow groups of smooth cubic hypersurfaces. It studies the cylinder homomorphism on the Chow group by the Fano correspondence. Another is a relation of Chow motives between a smooth cubic hypersurface and its Fano variety of lines. The proof is built upon an observation due to Galkin and Shinder to study the Grothendieck ring of varieties. Besides, I will show applications of the results to integral Hodge(Tate) conjectures.

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**Speaker**: Dr. Yu Min (MCM)

**Title**: Integral p-adic Hodge theory of formal schemes in low ramification

**Time**: 10:30-11:30am, Dec. 3, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, I will briefly review the theory of prismatic cohomology and talk about some results concerning the module structure of prismatic cohomology groups. Then I will discuss their applications in the study of the integral comparison theorem and the degeneration of the (integral) Hodge-to-de Rham spectral sequence.

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**Speaker**: Dr. Dongming She (MCM)

**Title**: Local Langlands correspondence for the twisted exterior and symmetric square epsilon-factors of GL(N)

**Time**: 10:30-11:30am, Nov. 26, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: We will introduce the local Langlands correspondence, the Langlands-Shahidi method, and sketch the proof of the equality of the twisted symmetric and exterior square local arithmetic and analytic L- and epsilon-factors of GL(N) over a p-adic field. We use GSpin groups to define these twisted local analytic factors via Langlands-Shahidi method. The proof uses some globalization method to reduce it to the stability of the corresponding analytic gamma-factors, whose proof is given by some analysis of the asymptotic behavior of certain partial Bessel functions.

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**Speaker**: Dr. Arnaud Plessis (MCM)

**Title**: On a conjecture of R\'emond.

**Time**: 10:30-11:30am, Nov. 19, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: A R\'emomd conjecture predicts that the points of small height in G(k(Gamma)), when G is either G_m^n or an abelian variety defined over a number field k and Gamma a subgroup of G(\overline{k}) of finite rank, lie in the saturated group of Gamma. This conjecture is only known in a few cases. I will state them by explaining why it is difficult to get more results concerning this conjecture. Finally, I will talk about my current research, which consists to find a new idea to deal with this problem.

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**Speaker**: Dr. Guhan Venkat (MCM)

**Title**: Rationality of Stark-Heegner cycles attached to base change Bianchi modular forms

**Time**: 10:30-11:30am, Nov. 05, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**:Stark–Heegner cycles for Bianchi modular forms, that is automorphic forms for GL(2) over an imaginary quadratic field F, were defined in an earlier joint work with C. Williams (U. of Warwick). These are local cohomology classes in the p-adic Galois representation associated to the Bianchi form. They are conjectured to be the restriction (at p) of global cohomology classes in the (semi–stable) Bloch–Kato Selmer group defined over ring class fields of a relative quadratic extension K/F. In this talk, I will report on an ongoing project about some degenarate cases where this conjecture holds.

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

**Title**: Hypergeometric sheaves for classical groups

**Time**: 10:30-11:30am, Oct. 29, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**:Recently, Jakob and Yun introduced a new class of p-adic representations called euphotic representations, generalizing simple supercuspidal representations and epipelagic representations. In this talk, we will talk about hypergeometric local systems for classical groups constructed by certain euphotic representations. It is based on my joint work in progress with Masoud Kamgarpour and Lingfei Yi.

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**Speaker**: Dr. Ka-Fai Li (MCM)

**Title**: U(n)-invariant metrics and positively curved complete Kahler manifolds.

**Time**: 10:30-11:30am, Oct. 22, 2020

**Place**: ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**:The U(n)-invariant Kahler metrics were studied by Wu-Zheng to construct examples of complete Kahler manifolds with positive holomorphic bisectional curvature. In this talk, we will review Wu-Zheng's construction and discuss the longtime behavior of the Kahler-Ricci flow solution starting from these metrics.

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**Speaker**: Dr. Yangyu Fan (MCM)

**Title**: *p*-adic Gross-Zagier formula

**Time**: 10:30-11:30am, Oct. 15, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**:In this talk, I will briefly introduce recent work of D. Disegni on the p-adic Gross-Zagier formula on Shimura curves.

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**Speaker**: Dr. Cong Ding (MCM)

**Title**: Complex submanifolds with splitting tangent sequences in rational homogeneous spaces of Picard number one

**Time**: 10:30-11:30am, Sep. 24, 2020

**Place**: MCM110 & ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: In this talk, we will present some properties of submanifolds with splitting tangent sequence in irreducible Hermitian symmetric spaces of compact type and more general rational homogeneous spaces of Picard number one.

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**Speaker**: Dr. Yupeng Wang (MCM)

**Title**: A p-adic Simpson correspondence for rigid analytic varieties

**Time**: 4:00-5:00pm, Sep. 17, 2020

**Place**: ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: The p-adic Simpson correspondence for proper smooth schemes over Spec(Z_p) was firstly studied by Faltings. Now, let k be a p-adic field. Assume K is the field of p-adic complex numbers. Then we can construct a p-adic Simpson correspondence for a rigid analytic variety X with a liftable good reduction $\mathfrak{X}$ defined over Spf($\mathcal(O)_K$) by constructing a new periods sheaf on X_{pro\'{e}t}.Firstly, we give a new description of Faltings'Extension by using the theory of cotangent complexes. Secondly, we construct the desired sheaf of periods. Nextly, we will prove a decompletion theorem which is used to compute cohomology groups. Finally, we establish the p-adic Simpson correspondence.

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**Speaker**: Dr. Jun Wang (MCM)

**Title**: Sharifi's conjectures and generalizations

**Time**: 4:00-5:00pm, Sep. 10, 2020

**Place**: ZOOM (Zoom ID: 466 356 2952 Password: mcm1234)

**Abstract**: R. Sharifi formulated remarkable conjectures which relate the arithmetic of cyclotomic fields to Eisenstein quotient of the homology groups of modular curves. In this talk, I will give a brief introduction to Sharifi's conjectures. Then I will talk about two generalizations of these conjectures. One is for exceptional eigenspace of modular curves, and the other is for a possible extension of Sharifi's conjecture to Bianchi manifold. It is work in progress with Sheng-Chi Shih and Emmanuel Lecouturier.

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

**Title**: Harder-Narasimhan strata and p-adic period domains

**Time**: 10:30-11:30am, Jan. 16, 2020

**Place**: MCM110

**Abstract**: In this talk, we will discuss the structures of certain moduli spaces of p-adic Hodge structures. More precisely, we will revisit the Harder-Narasimhan stratification on a p-adic flag variety by the theory of modifications of G-bundles on the Fargues-Fontaine curve. This allows us to compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of p-adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs.

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**Speaker**: Prof. Baohua Fu (MCM)

**Title**: Rigidity of wonderful group compactifications under Fano deformations

**Time**: 10:30-11:30am, Jan. 9, 2020

**Place**: MCM110

**Abstract**: For a complex connected simple linear algebraic group G of adjoint type, De Concini and Procesi constructed its wonderful compactification \bar{G}, which is a smooth Fano G\times G-variety enjoying many interesting properties. Assume G is not of type B3, it is shown that its wonderful compactification \bar{G} is rigid under Fano deformations. Namely, for any family of smooth Fano varieties over a connected base, if one fiber is isomorphic to \bar{G}, then so are all other fibers. This is a joint work with Qifeng Li (KIAS).

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**Speaker**: Dr. Peng Yu (MCM)

**Title**: CM value formula for orthogonal Shimura variety with application to lambda invariant

**Time**: 10:30-11:30am, Jan. 2, 2020

**Place**: MCM110

**Abstract**: In 1985, Gross and Zagier discovered a beautiful factorization formula for the norm of difference of singular moduli. This has inspired a lot of interesting work, one of which is the study of CM values of automorphic Green functions on orthogonal or unitary Shimura varieties. Now we generalize the definition of CM cycles beyond the ‘small’ and ‘big’ CM cycles and give a uniform formula in general using the idea of regularized theta lifts. Finally, as an application, we are able to give an explicit factorization formula for the norm of λ((d1+\sqrt{d1})/2) - λ((d2+\sqrt{d2})/2) with λ being the modular lambda invariant under the condition (d1, d2) = 1.

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**Speaker**: Dr. Alexandre Pyvovarov (MCM)

**Title**: Bernstein Centre

**Time**: 10:30-11:30am, Dec 26, 2019

**Place**: MCM110

**Abstract**: Let $F$ be a local non-archimedean field and $\mathcal{O}_F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of $GL_n(F)$, let $(J, \lambda)$ be a Bushnell-Kutzko $\Omega$-type, and let $\mathfrak{Z}_{\Omega}$ be the centre of the Bernstein component $\Omega$. We will explain how to compute $(\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} \lambda)\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m})$, where $\kappa(\mathfrak{m})$ is the residue field at maximal ideal $\mathfrak{m}$ of $\mathfrak{Z}_{\Omega}$, and the maximal ideal $\mathfrak{m}$ belongs to a Zariski-dense set in $\mathrm{Spec}\: \mathfrak{Z}_{\Omega}$.

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**Speaker**: Dr. Hao Zhang (MCM)

**Title**: An introduction to Dwork theory

**Time**: 10:30-11:30am, Dec 19, 2019

**Place**: MCM110

**Abstract**: In this talk, I will give an introduction of Dwork theory in studying zeta functions and L-functions. Especially on a comparison theorem between algebraic and analytic Dwork cohomology. And then talk about some applications.

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**Speaker**: Prof. Laurent Fargues (MCM)

**Title**: An Arithmetic Analog of the Abel Jacobi Morphism

**Time**: 4:00-5:00pm, Dec 13, 2019 (NOT Thursday!)

**Place**: N224

**Abstract**: Global class field theory for function fields can be deduced from the fact that, in high degree, the Abel Jacobi morphism of a proper smooth algebraic curve is a locally trivial fibration in simply connected varieties. I will explain a similar statement in the framework of my geometrisation conjecture of the local Langlands correspondence. Here the curve is the one I defined and studied in my joint work with Fontaine.

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**Speaker**: Dr. Ka-Fai Li (MCM)

**Title**: Some curvature flows on compact complex manifolds

**Time**: 10:30-11:30am, Dec. 5, 2019

**Place**: MCM110

**Abstract**: In this talk, we will talk about the Kahler-Ricci flow and Anomaly flow on compact complex manifolds. We will discuss their relation to a parabolic Monge-Ampere equation and their limit behavior.

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**Speaker**: Dr. Bingyu Xia (MCM)

**Title**: A generalized Quot scheme for objects in derived category

**Time**: 10:30-11:30am, Nov. 28, 2019

**Place**: MCM110

**Abstract**: I will introduce a generalized version of Quot scheme for objects in derived category, and talk about its relation to husks and stable pairs.

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**Speaker**: Dr. Yewon Jeong (MCM)

**Title**: Several types of cubic hypersurfaces with degenerate Gauss map

**Time**: 10:30-11:30am, Nov. 21, 2019

**Place**: MCM110

**Abstract**: Given a hypersurface X = V ( f ) in a complex projective space, the Gauss map of X can be regarded as the restriction of the gradient map of f on X. We say, the hypersurface X has degenerate Gauss map if general fibers of the Gauss map have positive dimension. Especially for cubic hypersurfaces with degenerate Gauss map, there is an interesting classification of them. We will study several types of cubic hypersurfaces and the relation between them.

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**Speaker**: Dr. Bin Zhao (MCM)

**Title**: Spectral Halo of Eigencurves

**Time**: 10:30-11:30am, Nov. 14, 2019

**Place**: MCM110

**Abstract**: In a previous work of Ruochuan Liu, Daqing Wan and Liang Xiao, they proved that over the boundary of the weight disc, most components of the eigencurve are disjoint unions of spaces finite flat over the weight disc. In this talk, I will explain a joint work with Liang Xiao on a refinement of this result. A main ingredient in the proof is a down to earth computation of the projective envelope of a Serre weight.

**Speaker**: Dr. Jie Liu (MCM)

**Title**: Determine varieties via hypersurfaces

**Time**: 10:30-11:30am, Nov. 7, 2019

**Place**: MCM110

**Abstract**: It is a classical problem in algebraic geometry, especially in adjunction theory, to ask which properties of the ambient space can be determined by its ample divisors. In this talk, I will focus on some very explicit examples to show that how the VMRT (varieties of minimal rational tangents) theory can be applied to this problem.

**Speaker**: Dr. Guhanvenkat Harikumar (MCM)

**Title**: Stark-Heegner cycles for Bianchi modular forms

**Time**: 10:30-11:30am, Oct 31, 2019

**Place**: MCM110

**Abstract**: In his seminal paper in 2001, Henri Darmon came up with a systematic construction of p-adic points, viz. Stark-Heegner points, on elliptic curves over the rationals. In this talk, I will report on the construction of local (p-adic) cohomology classes in the Harris-Taylor-Soudry representation associated to a Bianchi cusp form, building on the ideas of Henri Darmon and Rotger-Seveso. These local cohomology classes are conjecturally the restriction of global cohomology classes in an appropriate Bloch-Kato Selmer group and have consequences towards the Bloch-Kato-Beilinson conjecture as well as Gross-Zagier type results. This is based on a joint work with Chris Williams (University of Warwick).

**Speaker**: Dr. Shinan Liu (MCM)

**Title**: Local model of Shimura varieties in $\Gamma_0(p)$ and $\Gamma_1(p)$ levels

**Time**: 10:30-11:30am, Oct 24, 2019

**Place**: MCM110

**Abstract**: Local models of Shimura varieties are objects defined by linear algebra, which describe singularities of Shimura varieties at bad primes. In this exipository talk, we first give an instroduction to Pappas-Zhu's group-theoretic definition of local model in$\Gamma_0(p)$ level, then we quickly review the work of Haines-Stroh and our work in $\Gamma_1(p)$ level.

**Speaker**: Dr. Yangyu Fan (MCM)

**Title**: Katz p-adic L-function

**Time**: 10:30-11:30am, Oct 17, 2019

**Place**: MCM110

**Abstract**: Let $E$ be an imaginary quadratic field. When the prime $p$ splits in $E$, Katz constructed a two variable $p$-adic L-function interpolating algebraic critical Hecke L-values over $E$ in 1970's. In this talk, we will introduce some recent results concerning the counter construction in the non-split case.

**Speaker**: Prof. Shigeru Mukai (MCM)

**Title**: Geometric realization of T-shaped root systems and the Jacobians of del Pezzo surfaces

**Time**: 10:30-11:30am, Oct 10, 2019

**Place**: MCM110

**Abstract**: It is well known that the blow-up of the projective plane has a Cremona symmetry of the Weyl group of the root system of type E. This was generalized to a T-shaped Cremona symmetry of higher dimensional multi-projective spaces by Coble and Mukai. Using this framework I will describe the “Jacobians” of del Pezzo surfaces dP_d, and construct several extremal elliptic fibrations over the d-dimensional projective space.

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

**Title**: Cuspidality Criterion

**Time**: 15:30-16:30, Jan 17, 2019

**Place**: N817

**Abstract**: In 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)

**Title**: Iwasawa theory and Bloch-Kato conjecture for unitary groups

**Time**: 15:30-16:30, Jan 10, 2019

**Place**: N817

**Abstract**: We 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)

**Title**: Introduction to Heegner Points

**Time**: 15:30-16:30, Jan 3, 2019

**Place**: N817

**Abstract**: We introduce the basic arithmetic theory of Heegner points and some applications.

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

**Title**: Mop p cohomology of Shimura curves

**Time**: 15:30-16:30, Dec 20, 2018

**Place**: N817

**Abstract**: At 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)

**Title**: Newton strata for good reductions of Shimura varieties of orthogonal type

**Time**: 15:30-16:30, Dec 13, 2018

**Place**: N817

**Abstract**: In 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)

**Title**: On Fano complete intersections in rational homogeneous varieties

**Time**: 15:30-16:30, Dec 6, 2018

**Place**: N817

**Abstract**: Complete 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).

**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.

**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.

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