Askold Khovanskii: Tropicalizations of subvarieties of algebraic tori

There will be two preparatory lectures:

Smooth toric varieties, A. Esterov, 13.09
I will introduce smooth toric varieties https://en.wikipedia.org/wiki/Toric_variety, trying to keep things as simple as possible: 2nd year students are welcome.

Intersection theory on complex tori, V. Kiritchenko, 20.09
In the 19th century, Chasles, Schubert and others obtained many spectacular results in enumerative geometry by heuristic methods (calculus of conditions or Schubert calculus). In the 20th century, their results were justified using intersection theory. In particular, De Concini and Procesi developed the concept of the ring of conditions for (possibly) non-compact homogeneous varieties. I will define the ring of conditions and describe its applications to some classical problems of enumerative geometry. The main examples will be complex tori and Grassmannians.

Tropicalizations of subvarieties of algebraic tori, A. Khovanskii, 27.09
In the first lecture we will recall a description of cohomology rings of toric varities in terms of polytopes, in particular, the relation between the Dehn-Sommerville equations for simple convex polytopes and Poincare duality for smooth toric varieties.