ÍâÍøÌìÌÃ

Zoom link:Ìý

Passcode: 114700

The problems related to the standard Ìýaction of the compact torus $T^{n}$ on a complex Grassmann manifold $G_{n,2}$, $n\geq 3$ Ìýare widely known in algebraic topology, algebraic geometry and mathematical physics. ÌýThis action has the complexity $n-3$ for a given $n$. The talks are devoted to toric topology of the family $\{(G_{n,2}, T^n)\}$ whose members are connected by the natural equivariant embeddings. Ìý

ÌýIn the seminal papers of Gel'fand, Serganova, Goresky, MacPherson, it was studied the action of the algebraic torus $(\mathbb{C} ^{\ast})^{n}$ on $G_{n,2}$ using Ìýthe canonical moment map $\mu : G_{n,2} \to \Delta_{n,2}$, where $\Delta_{n,2}$ is the hypersimplex. Ìý Their results Ìýwere formulated Ìýin Ìýterms of Ìýthe strata $\{W_{\sigma}\}$ for Ìýthe $(\mathbb{C} ^{\ast})^{n}$ Ìý-action Ìýon $G_{n,2}$ and the decomposition of $\Delta _{n,2}$ into the chambers.

ÌýIn the first talk (October 22) it will be given the description of the Ìýorbit space $G_{n,2}/T^n$ in Ìýthe new notions: an universal space of parameters $\mathcal{F}_{n}$; virtual spaces Ìý Ìýof parameters $\widetilde{F}_{\sigma}\subset \mathcal{F}_{n}$ of Ìýthe strata $W_{\sigma}$;Ìýthe correspondence which Ìýto Ìýthe set of the strata definingÌýa Ìýchamber Ìýassigns the decomposition of the space $\mathcal{F}_{n}$ Ìýinto the corresponding virtual spaces of parameters.

ÌýIn modern algebraic geometry it isÌýknown the notion of the wonderful compactification Ìýbased on the arrangement of smooth Ìýsubvarieties in a smooth algebraic variety. In the second talk (November 5) Ìý we describe Ìý our smooth manifolds $\mathcal{F}_{n}$ Ìý in terms of the wonderful compactification and Ìýshow that the Ìý family ${\mathcal{F}_{n}\} $ can be identified with the family Ìý $\{\overline{M(0,n)}\}$, where $\overline{M(0,n)}$ is Ìý the Deligne-Mumford compactification Ìý of Ìýthe moduli space $M(0,n)$, Ìýwhich plays an important role in Ìýknown Ìýproblems of modern Ìýmathematical physics. In toric geometry and toric topology are obtained many results in terms of subspace arrangements. The proofs of the results presented in the first talkÌýessentially use our description of the chamber decomposition of $\Delta _{n,2}$ based on the special Ìýhyperplane arrangement.

This description will be also presented Ìýin the second Ìýtalk.