Seminar Geometrija i primene, 22. februar 2024.

Naredni sastanak Seminara biće održan uživo u četvrtak, 22. februara 2024. u sali 301f Matematičkog instituta SANU sa početkom u 17.15.

Predavač: Ivan Limončenko, Matematički institut SANU

Apstrakt: In 1992 Pukhlikov and Khovanskii obtained a description of intersection ring of a nonsingular projective toric variety via the volume polynomial of a virtual polytope.

A topological generalization of a nonsingular projective toric variety was introduced and studied by Davis and Januszkiewicz in 1991 and is known in toric topology as a (quasi)toric manifold. They showed that cohomology rings of quasitoric manifolds are isomorphic to quotient rings of Stanley-Reisner algebras of simple polytopes by linear ideals. Since that time (quasi)toric manifolds have become key players in toric topology and found various applications in bordism theory, mirror symmetry, polytope theory and other areas of research.

In this talk, I`m going to introduce some of the basic constructions and fundamental results concerned with geometry, topology, and combinatorics of (quasi)toric manifolds. We will also discuss in brief the theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. Then I will show how to apply this theory in order to obtain a topological version of the classical Bernstein-Kushnirenko-Khovanskii theorem and a Pukhlikov-Khovanskii type description for cohomology rings of a wide class of smooth orientable closed manifolds with a compact torus action, which we called generalized quasitoric manifolds.

 The talk is based in part on the joint works with Askold Khovanskii (University of Toronto) and Leonid Monin (EPFL)

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