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HSE University
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Faculty of Mathematics
Faculty of Mathematics
Contacts

119048Moscow, Usacheva str., 6

phone/fax: +7 (495) 624-26-16
phone: +7 (495) 916-89-05

e-mail: math@hse.ru

Administration
Alexandra Skripchenko
Sergei Lando
Scientific Supervisor – Sergei Lando
Vasily Gorbounov
Deputy Dean for Research – Vasily Gorbounov
Alexander Kolesnikov
Deputy Dean for Academic Progress – Alexander Kolesnikov
Svetlana Balaeva
Deputy Dean – Svetlana Balaeva
Pavel Nikolaevich Pyatov
Deputy Dean for Admissions – Pavel Nikolaevich Pyatov

Convex Geometry and Intersection Theory

Organizers: Alexander Esterov esterov@gmail.com, Valentina Kiritchenko vkiritchenko@yahoo.ca

Announcement: Alexander Esterov "Spherical characteristic classes", Oct. 17, 2013, room 302

Abstract: "Spherical characteristic classes" are defined for (not necessarily compact or regular) subvarieties in a spherical homogeneous space,
and take values in the ring of conditions of the ambient space (as defined by de Concini and Procesi).

A special case that will be of primary interest for me is the "tropical characteristic class" of a subvariety of a complex torus.
I will explain how to prove its existence constructively,
how to deduce known facts about Newton polytopes from basic properties of tropical characteristic classes
(for instance, how to prove Kouchnirenko-Bernstein-Khovanskii theorem in two lines),
how tropical characteristic classes are related to Schwarz-MacPherson classes,
and how they lead to a new approach to Mikhalkin-type tropical correspondence theorems in enumerative geometry.

Program:

  • Patchworking (1st module): Introduction to real algebraic geometry, Harnack's curve inequality, Viro's patchworking, moment maps, ℝ+-toric varieties, proof of the patchworking theorem.

  • Toric varieties (2nd module): polytopes, fans and toric varieties; convex-algebro-geometric dictionary; Kouchnirenko theorem; mixed volumes; Bernstein theorem; Dehn–Sommerville equations.

  • Tropical geometry (3rd-4th module): tropical algebra, amoebas, tropical Bezout theorem, introduction to enumerative geometry, Mikhalkin's tropical correspondence theorem, sparse resultants and A-discriminants, secondary polytope, proof of Mikhalkin's theorem for 1 and 2 nodes, outline for the general case.

 

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