北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

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关于我们
院长致辞
理事会
协作机构
参观来访
人员
管理层
科研人员
博士后
来访学者
行政团队
学术研究
研究团队
公开课
讨论班
招生招聘
教研人员
博士后
学生
会议
学术会议
工作坊
论坛
学院生活
住宿
交通
配套设施
周边旅游
新闻
新闻动态
通知公告
资料下载
清华大学 "求真书院"
清华大学丘成桐数学科学中心
清华三亚国际数学论坛
上海数学与交叉学科研究院
BIMSA > AMSS-YMSC-BIMSA拓扑及应用进展联合讨论班 Parametric toric topology of complex Grassmann manifolds
Parametric toric topology of complex Grassmann manifolds
组织者
段海豹 , Fei Han , Yong Lin , 潘建中 , 魏国卫 , 吴杰 , 夏克林 , 丘成栋 , Chao Zhou
演讲者
Victor Buchstaber
时间
2022年04月28日 17:00 至 18:30
地点
1110
线上
Zoom 388 528 9728 (BIMSA)
摘要
The talk will present the results obtained jointly with S. Terzi\'c in the cycle of works 2010 - 2022. The central object of the parametric toric topology is a smooth manifold $M^{2m}$ with an effective action of the compact torus $T^q,\;1\leqslant q \leqslant m$. These actions must satisfy the conditions under which any point of the manifold $M^{2m}$ receives local coordinates of three types: angular, moment, and parametric. This makes it possible to develop classical methods and methods of toric topology in problems on the equivariant structure of the manifold $M^{2m}$ and the structure of the orbit space $M^{2m}/T^q$. The complexity of the $T^q$-action on $M^{2m}$ is the number $m-q$. In toric geometry and topology, actions of complexity 0 are studied. The key example in these theories is the complex projective space $\mathbb{C}P^m$ with the canonical $T^m$-action and a moment map $\mathbb{C}P^m \to \Delta^m$, where $\Delta^m$ is the standard simplex in $\mathbb{R}^{m+1}$. The key example in problems of parametric toric topology is the complex Grassmann manifold $G_{n,2},\, n\geqslant 4$, with the canonical effective $T^{n-1}$-action, i.e. with the action of complexity $n-3 > 0$, and with the moment map $G_{n,2} \to \Delta_{n,2}$ where $\Delta_{n,2}$ is the standard hypersimplex in $\mathbb{R}^{m+1}$. We will introduce the universal parameter space $\mathcal{F}_n$ for $ G_{n,2}$ and show that this space is a smooth closed manifold of dimension $2(n-3)$, which can be identified with the Chow quotient $G_{n,2} /\!/(\mathbb{C}^*)^n$. We will construct a continuous projection $G_{n} : U = \Delta_{n,2}\times\mathcal{F}_n \to X_n = G_{n,2}/\mathbb{T}^n$, give an explicit construction of a closed set $\text{\rm Sing}X_n \subset X_n$, and show that: 1) the spaces $Y_n = X_n\setminus \text{\rm Sing}X_n$ and $V_n = G_{n}^{-1}(Y_n)\subset U_n$ are open dense manifolds in $X_n$ and $U_n$, respectively; 2) the mapping $G_{n} : V_n \to Y_n$ is a diffeomorphism. We will describe sets $G_{n}^{-1}(x)$ for all $x\in \text{\rm Sing}X_n$. The main goal of the talk is to show fruitful connections of the results of parametric toric topology with the fundamental results of algebraic and differential topology, geometric combinatorics, algebraic and symplectic geometry.
演讲者介绍
Prof Victor Matveevich Buchstaber is a famous expert in algebraic topology, homotopy theory, and mathematical physics. He received his PhD in 1970 under the supervision of Sergei Novikov and Dr. Sci. in Physical and Mathematical Sciences in 1984 from Moscow State University. In 1974 he was an invited speaker in the International Congress of Mathematicians in Vancouver (Canada). He became a corresponding member of the Russian Academy of Sciences In 2006 . Now he is a principle researcher in the Department of Geometry and Topology of the Steklov Mathematical Institute. Prof Buchstaber is a leading expert of the new-born area of toric topology.
北京雁栖湖应用数学研究院
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