Harbin Seminar on Arithmetic and Algebraic Geometry
组织机构 (Organizing institute): 哈工大数学研究院 (IASM of HIT)
Date: 2025.03.14
Time: 16:00-17:00
Venue: Mingde Building B201-1
Speaker: Galyna Dobrovolska (Ariel University)
Title: Combinatorial wall-crossing and the Mullineux involution
Abstract: The rational Cherednik algebra H_c is an algebra of interest in modern representation theory, which is a degeneration of the double affine Hecke algebra, introduced by Cherednik to prove Macdonald's conjectures about properties of Macdonald polynomials. For values of c lying in chambers separated by walls, representations of H_c are labeled by partitions. Combinatorial wall-crossing is a bijection from the set of irreducible representations of H_c to the set of irreducible representations of H_c', where c and c' lie in adjacent chambers separated by a wall. Combinatorial wall-crossing across one wall was proven by Losev to be equal in large positive characteristic to an extension of the Mullineux involution from modular representation theory of the symmetric group. We will exhibit interesting patterns in combinatorial wall-crossing, both proven and observed in computer experiments, and use them to prove and refine parts of a conjecture of Bezrukavnikov.
Date: 2025.01.09
Time: 14:30-15:30
Venue: Mingde Building B201-1
Speaker: Xiaoyu Zhang (Duisburg-Essen University)
Title: Modularity of Galois representations valued in dual group of SO(n) and applications
Abstract: One of the basic ingredients in Wiles’ proof of Fermat’s Last Theorem is modularity lifting theorems of Galois representations valued in GL(2). This was further generalised by Clozel-Harris-Taylor to definite unitary groups. In this talk, I will discuss the case of definite special orthogonal groups and then talk about some potential applications in p-adic Langlands program.
Date: 2025.01.03
Time: 14:30-15:30
Venue: GeWu building 201
Speaker: Caucher Birkar (Tsinghua University)
Title: Geometry and Integers
Abstract: In this talk we will discuss some recent interesting connections between properties of sets of non-negative integers with different kinds of geometries. Starting with a primitive integer vector, we examine certain associated functions and relate the setting to statements in convex, toric, and birational geometries.
Date: 2024.10.26
Time: 10:00-12:00
Venue: 明德楼 B201-1 报告厅
Speaker: 李烨暄 (中北大学)
Title: The relative Auslander-Reiten theory over an infinite-dimensional coalgebra
Abstract: In this talk, we will cover several aspects: Firstly, we will delve into the higher-dimensional Auslander-Reiten theory over an infinite-dimensional coalgebra; the finite-dimensional case by duality reduces to that of finite-dimensional algebra. We will introduce the n-transpose of a finitely n-copresented comodule and n-Auslander-Reiten translations, and then prove the n-Auslander-Reiten formula on n-cluster-tilting subcategories of comodule categories. Secondly, we will introduce the Gorenstein transpose via a minimal Gorenstein injective copresentation of a quasi-finite comodule, and explore the relations between the Gorenstein transpose of comodules and the transpose of the same comodule. As an application, we will construct the almost split sequences in terms of Gorenstein transpose. Finally, we will generalize the notion of the Auslander transpose of comodules to that of the transpose with respect to a semidualizing bicomodule T and showcase some nice homological properties of T-transpose. Additionally, we will provide a Foxby equivalence of comodule categories and investigate a characterization of T-reflexive comodules. This work is joint with Prof. Hailou Yao.
Date: 2024.10.10
Time: 10:30-12:00
Venue: 明德楼 B201-1 报告厅
Speaker: 陈升 (长春理工大学)
Title: Arithmetic purity of strong approximation for complete toric varieties
Abstract: In this talk, I will discuss arithmetic purity of strong approximation: motivations and related results. Finally, for complete toric varieties, I will briefly explain my work on this topic.
Date: 2024.10.09
Time: 14:30-16:00
Venue: 明德楼 B201-1 报告厅
ZOOM Online: 592 203 0941
Password: 7Zj4xf
Speaker: 阳煜 (京都大学数理解析研究所)
Title: Fundamental groups of curves and local moduli
Abstract: In 1996, A. Tamagawa discovered a surprising phenomenon: anabelian geometry also exists for curves over algebraically closed fields of characteristic p>0 (i.e., curves in positive characteristic can possibly be determined by their geometric fundamental groups without relying on Galois actions). However, after 28 years, only a few results have emerged in this field.
In this talk, I want to explain the following insight of the speaker about fundamental groups of curves in positive characteristic:
The (admissible or geometric log etale) fundamental groups of pointed stable curves over algebraically closed fields of characteristic p can be regarded as an analogue of local moduli spaces of the curves.
This observation led to the speaker discovering some new kinds of anabelian phenomena of curves in characteristic p and to formulated numerious new conjectures. For example, the following highly non-trivial anbelian results of the speaker provide strong evidence supporting this insight:
• The homeomorphism conjecture holds for 1-dimensional moduli spaces (roughly speaking, this conjecture means that the moduli spaces of curves can be reconstructed group-theoretically as topological spaces).
• A new proof of Mochizuki’s famous result concerning (Isom-version) Grothendieck’s anabelian conjecture for curves over sub-p-adic fields without using Faltings’ p-adic Hodge theory.
• The group-theoretical specialization conjecture holds (roughly speaking, this conjecture means that the topological and group-theoretical degeneration of curves can be completely determined by open continuous homomorphisms of dmissible fundamental groups).
Date: 2024.09.21–09.22
Venue: 明德楼 B201-1 报告厅
Conference: 冰城2024算术几何研讨会
==================================================================================
Date: 2024.09.21
Time: 9:30-10:30
Speaker: 王善文（中国人民大学）
Title: Functions in one p-adic variable
Abstract: We extend the dictionary between Fontaine’s rings and p-adic functional analysis. As an application, we give a refinement of the p-adic local Langlands correspondence for principal series representations of GL_2(Q_p). This is joint work with Pierre Colmez.
Time: 11:00-12:00
Speaker: 金方舟（同济大学）
Title: Characteristic classes and conductors in motivic homotopy
Abstract: We define and study several characteristic classes in motivic homotopy theory, and use them to deduce some conductor formulas. This is a joint work with P. Sun and E. Yang.
Time: 14:00-15:00
Speaker: 曹阳（山东大学）
Title: Invariant Brauer subgroup for non-linear algebraic group
Abstract: For algebraic varieties with an action of an algebraic group, the invariant Brauer subgroup is a natural object to study the local-global principle of rational points. In this talk, I will focus on the case if the algebraic group is not necessary linear, and talk about the application of invariant Brauer subgroup to cohomological obstruction and descent method.
Time: 15:30-16:30
Speaker: 户亚青（山东大学）
Title: A Survey on Polynomial Maps in Groups
Abstract: We give a historical survey on some functional definitions of polynomial maps in abelian and nilpotent groups.
Time: 16:45-17:45
Speaker: 李加宁（山东大学）
Title: Kummer扩张的理想类群
Abstract: 设K=Q(μ_p,N^(1/p)), p,N是两个素数, 令G=Gal(K/Q). 我将讨论K的理想类群的p-部分, 尤其是其G-模结构与某些模形式L函数特殊值的关系. 这可视为Hebrand-Ribet定理(报告将回顾此定理内容)从分圆域到Kummer型扩张的类比. 在p=3的情形, 可得到些明确的结果.
==================================================================================
Date: 2024.09.22
Time: 9:00-10:00
Speaker: 吴峙佑（中科院晨兴数学所）
Title: Plectic weight filtration on cohomology of Hilbert modular varieties
Abstract: The plectic program is proposed by Jan Nekovar and Tony Scholl to uncover some hidden symmetries of Hilbert modular varieties. It would yield exciting new results on special values of L functions once established. In this talk, I will report my work on the construction of plectic weight filtration, providing evidence of the plectic conjectures.
Time: 10:30-11:30
Speaker: 申旭（中科院晨兴数学所）
Title: Bruhat–Tits buildings and p-adic period domains
Abstract: Rémy-Thuillier-Werner constructed embeddings of Bruhat-Tits buildings into the Berkovich spaces associated to suitable flag varieties (generalizing the work of Berkovich for split groups), and defined compactifications of BT buildings by taking closure inside these Berkovich flag varieties. We show that, in the setting of a local Shimura datum, the RTW embedding factors through the associated p-adic period domain (defined by the Fargues-Fontaine curve). This brings the question to compare these two p-adic candidates of symmetric spaces: topological and combinatorial vs p-adic analytic. We will compare the boundaries of the BT buildings and non basic Newton strata. Moreover, we will discuss some applications to cohomology and smooth representations. This is joint work in progress with Ruishen Zhao.
Date: 2024.09.12
Time: 14:30-15:30 (北京时间)
Venue: 明德楼 B201-1 报告厅
Speaker: 张旭成 (清华大学丘成桐数学中心)
Title: A stacky approach to identifying the stability condition
Abstract: For any reductive group we find a stacky interpretation of the stability condition for principal bundles over a curve: it is the unique maximal open locus that admits a schematic moduli space. Some applications and further progress will be discussed. This is a joint work with Dario Weissmann and Andres Fernandez Herrero.