邱宇

清华大学

数学科学中心

2024年9月12日，16：00，数学学院东409

Deformed 3-Calabi-Yau categories and partial compactifications with orbifolding

We introduce a new family of quivers with potential for triangulated marked surfaces with punctures. We show that the deformation of the associated 3-Calabi-Yau categories corresponds to the partial compactification (with orbifolding) of the associated moduli spaces. As an application, we calculate the fundamental groups of these moduli spaces (of framed quadratic differentials), which in particular produces Euclidean Artin braid groups of type A, B, C and D.

张笑婷

首都师范大学交叉科学中心

2024年9月12日，15：00，数学学院东409

Geometric classification of total stability conditions and Reineke's conjecture

We construct a geometric model for the root category associated to any Dynkin diagram Q and classify the space of total stability conditions on the bounded derived category of Q. As an application, we prove Reineke's conjecture, that for any Dynkin quiver, there is a stability function on its module category such that any indecomposable is stable. This is based on joint works with Wen Chang and Yu Qiu.

余君

北京大学

2024年7月6日，16:00，数学院西109

Subgroup structure and applications

In this talk, we present some studies of elementary abelian p-subgroups of algebraic groups and applications of these results in the Inverse Galois problem, the classification of acceptable Lie groups, the Alperin weight conjecture, etc.

王起

清华大学

2024年7月4日，16:00，数学院东302

On brick finiteness of finite-dimensional algebras

A finite-dimensional algebra over an algebraically closed field admits exactly one of three representation types: representation-finite, tame, wild. While the module category can be extensively studied for representation-finite and tame algebras, exploring better-behaved modules or subcategories for wild algebras yields more challenges and fewer insights. In this talk, we consider the finiteness of bricks for a given algebra, which exhibits nice properties, particularly in the class of wild algebras. This talk will intersect with several areas in representation theory, such as τ-tilting theory and silting theory.

马力

北京科技大学

2024年6月28日，15:00，西南中心516

Gradient Ricci solitons and Dimension reductions

We first recall the Maximum principle of Omori-Yau in Riemannian geometry. Then we consider some properties of gradient steady Ricci solitons. We discuss some basic tools from geometric analysis in the study of the gradient steady Ricci solitons. We also study weighted Riemannian space and the modified distance functions in Riemannian geometry. Some recent progress about steady gradient Ricci solitons will be given.

赵健强

The Bishop's School (La Jolla)

2024年6月14日，15:00，西南中心518

Motivic Euler Sums and Motivic Multiple Mixed Values

Euler sums are alternating multi-variable generalizations of the Riemann zeta values. In this talk we first describe an approach to proving some families of Euler sum identities by lifting them to their motivic avatar. Then we consider a few variants of the Euler sums by restricting the summation indices in the definition of multiple zeta values to some fixed parity patterns, which are called multiple mixed values. These include Hoffman's multiple t-values, Kaneko and Tsumura's multiple T-values, and the multiple S-values studied previously by Prof. C. Xu and the speaker. By applying Brown and Glanois's descent theory on the motivic versions of these values we shall derive some criterion for when these values are unramified (i.e., are rational linear combinations of multiple zeta values). Further, we are able to generalize a result of Charlton to more families of unramified multiple t-values with unit components (i.e. components equal to 1). This work is joint with Prof. C. Xu.

许明

首都师范大学

2024年5月10日，16:00，西南中心516

齐性喷射几何的理论框架

首先，回顾芬斯勒和喷射几何中的基本概念；然后介绍齐性芬斯勒和齐性喷射几何；再后，定义齐性喷射流形的喷射向量场和联络算子，并讨论如何用它们刻画测地性、平行移动和曲率；最后，介绍上述理论的一些应用。

张子宇

上海科技大学

2024年5月8日，15:00，西南中心518

Examples of stable bundles on hyperkähler manifolds of higher dimensions

Stable sheaves on K3 surfaces have been extensively studied. However, it is a challenging question to construct non-trivial examples of stable sheaves on hyperkähler manifolds of higher dimensions. In this talk, I will present several constructions, which produce irreducible components of the moduli spaces of such sheaves. Based on joint work with Fabian Reede.

刘文飞

厦门大学

2024年4月12日，10:00，西南中心516

On numerically trivial automorphisms of properly elliptic surfaces

Numerically trivial automorphisms of a compact complex manifold are those biholomorphic automorphisms inducing the trivial action on the cohomology groups with rational coefficients. While projective surfaces form a nontrivial but well studied case, several natural questions remain open. In this talk, after reviewing some known results, I will focus on the case of properly elliptic surfaces. In contrast to the case of general type surfaces, we will use examples to show that the group of numerically trivial automorphisms can have arbitrary large (finite) order. If the Euler characteristic is 0, it is possible to list all the groups. Time permitting, I will also report on some results about automorphisms inducing trivial action on the cohomology with integral coefficients. The talk is based on work in progress with Fabrizio Catanese and Matthias Schütt.

杜荣

华东师范大学

2024年4月10日，15:00，西南中心518

Algebraic vector bundles on rational homogeneous spaces
I will provide an overview of the background concerning algebraic vector bundles on rational homogeneous spaces, along with some open problems within algebraic geometry associated with them. Specifically, I will concentrate on two types of algebraic vector bundles, uniform bundles and homogeneous bundles, on rational homogeneous spaces, and present our recent advancements in these areas.

刘裕

陕西师范大学

2024年4月5日 ，15:30，数学学院东302（大）

Muations of Relative Rigid Subcategories
Let C be a triangulated category with shift functor [1] and R be a rigid subcategory of C. We are concerned with almost complete two-term weak R[1]-cluster tilting subcategories which can be completed to a two-term weak R[1]-cluster tilting subcategories by one indecomposable object. Our main result shows that such an almost complete two-term weak R[1]-cluster tilting subcategory has exactly two complements. Moreover, we offer explicit constructions for these two complements via mutations. As an application, our main results can be applied to the functorial version of tau-tilting theory.

李灵光

同济大学

2024年3月27日，15:00，西南中心518

Moduli spaces of vector bundles in positive characteristic

Let X be a smooth projective curve of genus g(X)>1 over an algebraically closed field k of characteristic p>0 and F_X:X->X be the absolute Frobenius morphism. Let M^s_X(r,d) be the moduli space of stable vector bundles of rank r and degree d on X. Firstly, we will review some results about the stability of vector bundles under Frobenius pull back and Frobenius direct image. Secondly, we will talk about the Frobenius stratification of M^s_X(r,d) in terms of Harder-Narasimhan polygons of Frobenius pullbacks of stable vector bundles in the cases (p,g,r)=(3,2,3) and (2,2,4) with even degree d. Finally, we will talk about the rational curves in Frobenius strata.

张伟楠

香港大学

2024年3月19日，16:00，西南中心516

Drinfeld type presentation for twisted Yangians

Yangians, introduced by Drinfeld, are deformations of the current algebras. It is well-known that Yangians admit two presentations: the R-matrix presentation and the Drinfeld presentation. Twisted Yangians are certain coideal subalgebras of Yangians, and they are closely related to the theory of symmetric pairs. Twisted Yangians were originally introduced via a R-matrix presentation, and finding a Drinfeld type presentation for twisted Yangians has been an open problem for a long time.

In this talk, I will present our recent construction for the Drinfeld type presentation for twisted Yangians of type AI using the Gauss decomposition. We also show that the twisted Yangians can be viewed as a degenerate version of the affine i-quantum groups, which are coideal subalgebras of affine quantum groups arising from quantum symmetric pairs. This is joint with Kang Lu and Weiqiang Wang.

马家骏

厦门大学

2024年3月14日，16:00，数学院东409

在Lean中形式化Coxeter 群的理论的探索

Lean定理证明器是由微软领头开发的一种先进的证明检验工具。自2017年以来，Kevin Buzzard领导的mathlib项目致力于构建一个涵盖广泛数学定义和定理的Lean标准库，目的是促进数学理论的形式化与验证。Buzzard在ICM2022上分享了他们的工作。这个项目的成果吸引了包括Peter Scholze在内的多位著名数学家的注意，并发起和完成了The Liquid Tensor Experiment项目。最近Terence Tao也加入了数学形式化的研究，并主导完成了Polynomial Freiman-Ruzsa猜想的验证。

Coxeter群，如对称群，是数学中一个在各个分支中广泛应用的重要概念。本报告将介绍我们团队在形式化Coxeter群理论方面的工作。报告内容将涵盖Lean定理证明器的基础原理，我们项目的最新进展，以及我们在形式化过程中遇到的主要挑战。报告仅假设听众具有最基本的群论知识，旨在为听众提供一个关于数学形式化的概览。

杨中维

西南交通大学

2024年3月14日，15:00，数学院东409

Lean简介

Lean是一个交互式定理证明器，也是一门通用函数式编程语言。它最初是由Leonardo de Moura于2013年在微软研究院开发的一个实验性项目。Lean经过十来年的更迭、进化并结合近两年人工智能的飞速发展，现在已然成为数学形式化、机器证明的最强有力的语言（工具）。在此次讲座中，我们将对Lean进行简要介绍。

熊仪睿

谢菲尔德

2023年11月30日，15:00，西南中心516

What is a Bridgeland stability condition and how can I construct one?

Stability serves as a fundamental concept in constructing moduli spaces of sheaves on varieties or modules over algebras. While classical stability conditions were well-defined for abelian categories, extending these definitions to derived categories had been an unresolved challenge. Inspired by physicists’ work on string theory, T. Bridgeland proposed a satisfactory definition of stability conditions for derived categories in the early 2000s.

In this talk, I aim to provide an overview of stability conditions, starting with classical stability and then Bridgeland stability conditions. I hope to show you why the space of Bridgeland stability conditions is an interesting space— a complex manifold — by looking at the baby examples. If time permits, I will talk about my recent work on the stability conditions on local $\mathbb{F}_0$.

2023年11月24日，16:00，西南中心501

Lusztig sheaves and integrable highest weight modules

We consider the localization $\mathcal{Q}_{\mathbf{V},\mathbf{W}}/\mathcal{N}_{\mathbf{V}}$ of Lusztig's sheaves for framed quivers, and define functors $E^{(n)}_{i},F^{(n)}_{i},K^{\pm}_{i},n\in \mathbb{N},i \in I$ between the localizations. With these functors, the Grothendieck group of localizations realizes the irreducible integrable highest weight modules $L(\Lambda)$ of quantum groups, and the nonzero simple perverse sheaves in localizations form the canonical bases of $L(\Lambda)$. Moreover, we also generalize our construction to $N$-framed quivers and provide a categorical realization of tensor products of $L(\Lambda)$. This is a joint work with Jiepeng Fang and Jie Xiao.

方杰鹏

北京大学

2023年11月24日，15:00，西南中心501

Lie algebras arising from two-periodic projective complex and derived categories

Let $A$ be a finite-dimensional $\mathbb{C}$-algebra of finite global dimension and $\mathcal{A}$ be the category of finitely generated right $A$-modules. By using of the category of two-periodic projective complexes $\mathcal{C}_2(\mathcal{P})$, we construct the motivic Bridgeland's Hall algebra for $\mathcal{A}$, where structure constants are given by Poincar\'{e} polynomials in $t$, then construct a $\mathbb{C}$-Lie subalgebra $\mathfrak{g}=\mathfrak{n}\oplus \mathfrak{h}$ at $t=-1$, where $\mathfrak{n}$ is constructed by stack functions about indecomposable radical complexes, and $\mathfrak{h}$ is by contractible complexes. For the stable category $\mathcal{K}_2(\mathcal{P})$ of $\mathcal{C}_2(\mathcal{P})$, we construct its moduli spaces and a $\mathbb{C}$-Lie algebra $\tilde{\mathfrak{g}}=\tilde{\mathfrak{n}}\oplus \tilde{\mathfrak{h}}$, where $\tilde{\mathfrak{n}}$ is constructed by support-indecomposable constructible functions, and $\tilde{\mathfrak{h}}$ is by the Grothendieck group of $\mathcal{K}_2(\mathcal{P})$. We prove that the natural functor $\mathcal{C}_2(\mathcal{P})\rightarrow \mathcal{K}_2(\mathcal{P})$ together with the natural isomorphism between Grothendieck groups of $\mathcal{A}$ and $\mathcal{K}_2(\mathcal{P})$ induces a Lie algebra isomorphism $\mathfrak{g}\cong\tilde{\mathfrak{g}}$. This is a joint work with Yixin Lan and Jie Xiao.

行田康晃

东京大学

2023年9月4日，16:00，数学学院东409

The Exchange Quiver of Cluster Algebras and the Exchange Quiver of Root Systems

In 2002, Fomin and Zelevinsky classified cluster algebras of finite type by using Dynkin root systems. In the proof of this classification, they established a bijection between cluster variables and almost positive roots, demonstrating that it induces a graph isomorphism between the exchange graph of clusters and that of c-clusters. This graph isomorphism now plays a crucial role in connecting cluster algebra theory with Lie theory.

In recent times, the exchange graph of clusters (and c-clusters, respectively) has been given a "natural" orientation in cluster algebra theory (and Lie theory). In this presentation, I will demonstrate that these orientations are preserved by the graph isomorphism provided by Fomin and Zelevinsky.

覃帆

上海交通大学

2023年7月7日，16:00，数学学院东409

Stability scattering diagrams and quiver coverings

In this talk, we introduce nice gradings associated to quiver coverings. We explain that Bridgeland’s stability scattering diagrams respect quiver coverings if nice grading exist. In particular, our results apply to once punctured closed surfaces.

曹培根

香港大学

2023年6月29日，15:30，西南中心501

Cluster additive functions and Ringel’s conjectures

Let A be a symmetrizable generalized Cartan matrix of size r. A cluster-additive function associated to A is a map from Z times [1,r] to Z satisfying certain mesh type relations. Such functions were introduced by Ringel, which are closely related with additive functions in representation theory. In the case that the Cartan matrix is of finite type, Ringel conjectured that cluster additive functions admit some certain periodicity and any cluster additive function is a non-negative linear combination of cluster-hammock functions, which are a class of “elementary cluster additive functions”.

In this talk, we will give some link between cluster additive functions and cluster algebras. Ringel’s conjectures are easy consequences of our results. This talk is based on a work in progress with Antoine de St. Germain and Prof. Jiang-Hua Lu.

段冰

兰州大学

2023年6月9日，15:30，数学学院东409

Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras

Let $\mathscr{C}$ be the category of finite-dimensional modules over a simply-laced quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. For any height function $\xi$ and $\ell\in \mathbb{Z}_{\geq 1}$, we introduce certain subcategories $\mathscr{C}^{\leq \xi}_\ell$ of $\mathscr{C}$, and prove that the quantum Grothendieck ring $K_t(\mathscr{C}^{\leq \xi}_\ell)$ of $\mathscr{C}^{\leq \xi}_\ell$ admits a quantum cluster algebra structure. Using $F$-polynomials and monoidal categorifications of cluster algebras, we classify all real simple modules in $\mathscr{C}^{\leq \xi}_1$ in terms of their highest $l$-weight monomials, among them the families of type $D$ and type $E$ are new. For any $\ell$, inspired by Hernandez and Leclerc's work, we propose two conjectures for the study of real simple modules, and prove them for the subcategories $\mathscr{C}^{\leq \xi}_\ell$ whose Grothendieck rings are cluster algebras of finite type. This is a joint work with Professor Ralf Schiffler.

李江涛

中南大学

2023年6月8日，16:00，西南中心516

Unit cyclotomic multiple zeta values

Recently, inspired by Zhao’s conjecture about unit cyclotomic multiple zeta values, the author showed that the cyclotomic multiple zeta values are generated by unit cyclotomic multiple zeta values for $N=2,3,4$. Furthermore, the author calculated the motivic Galois actions for the unit cyclotomic multiple zeta values. As an application, the explicit relations among unit cyclotomic multiple zeta values are given in substantial cases. The reference is: arXiv: 2007.00173v2.

生云鹤

吉林大学

2023年6月8日，15:00，西南中心516

Rota-Baxter groups, post-groups and related structures

Rota-Baxter operators on Lie algebras were first studied by Belavin, Drinfeld and Semenov-Tian-Shansky as operator forms of the classical Yang-Baxter equation.

As a fundamental tool in studying integrable systems, the factorization theorem of Lie groups by Semenov-Tian-Shansky was obtained by integrating a factorization of Lie algebras from solutions of the modified Yang-Baxter equation. Integrating the Rota-Baxter operators on Lie algebras, we introduce the notion of Rota-Baxter operators on Lie groups and more generally on groups. Then the factorization theorem can be achieved directly on groups. As the underlying structures of Rota-Baxter operators on groups, the notion of post-groups was introduced. The differentiation of post-Lie groups gives post-Lie algebras. Post-groups are also related to braces and Lie-Butcher groups, and give rise to solutions of Yang-Baxter equations.

The talk is based on the joint work with Chengming Bai, Li Guo, Honglei Lang and Rong Tang.

鲍涣辰

新加坡国立大学 

2023年6月2日，14:30，西南中心516

Symmetric subgroups schemes, Frobenius splittings, and quantum symmetric pairs

Let G be a connected reductive group over an algebraically closed field. Such groups are classified via root data and can be parameterised via Chevalley group schemes over integers. In this talk, we shall first recall the construction of Chevalley group schemes by Lusztig using quantum groups. Then we shall discuss the construction of (quasi-split) symmetric subgroup schemes parameterising symmetric subgroups K of G using quantum symmetric pairs. The existence of such group schemes allows us to apply characteristic p methods to study the geometry of K-orbits on the flag variety of G, which we shall discuss as well. This is based on joint work with Jinfeng Song (NUS).

曾昊智

华中科技大学

王丽涵

California State University