Beijing Institute of Mathematical Sciences and Applications

Haibao Duan , Fei Han , Yong Lin , Jianzhong Pan , Guo-Wei Wei , Jie Wu , Kelin Xia , Shing Toung Yau , Chao Zhou

Victor Buchstaber

Thursday, April 28, 2022 5:00 PM - 6:30 PM

1110

Zoom 388 528 9728 (BIMSA)

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.

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.