Seminar for Geometry, education and visualization with applications

 

PROGRAM

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MATEMATIČKI INSTITUT SANU
Seminar geometriju, obrazovanje i vizualizaciju sa primenama

Plan rada za FEBRUAR 2024.

 

Četvrtak, 22.02.2024. u 17:15, Sala 301f, MI SANU, Kneza Mihaila 36 i On-line
Ivan Limonchenko, MI SANU
ON GEOMETRICAL METHODS IN THE THEORY OF TORIC MANIFOLDS
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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Četvrtak, 29.02.2024. u 17:15, Sala 301f, MI SANU, Kneza Mihaila 36 i On-line
Rodion Deev, Independent University of Moscow
COMPLEX SURFACES WITH MANY ALGEBRAIC STRUCTURES
Suppose two complex algebraic surfaces are biholomorphic as complex manifolds. Are they necessarily isomorphic as algebraic surfaces? In general, the answer is negative. Some complex surfaces even admit infinitely many non-isomorphic algebraic structures. We shall review the known examples of such surfaces, including the one discovered by the author in a joint work with Anna Abasheva (Columbia University), https://arxiv.org/abs/2303.10764. It is based on a gluing of a complex projective plane with a non-Kaehler Hopf surface in infinitely many ways.

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Sednice seminara odrzavaju se u zgradi Matematickog instituta SANU, Knez Mihailova 36, na trecem spratu u sali 301f.

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