Wednesday, January 23 and 30, 18:30, room 306

Alexander Esterov

Geometry and algebra of lattice polytopes

We shall overview geometry of lattice polytopes and its applications, and discuss the seminar topics for this term. At the same time, I will recall some facts and notions that will be used in the subsequent talks by Anatoly Kouchnirenko (see below).
This includes: classification and statistics of lattice polytopes, Ehrhart polynomials of lattice polytopes and Hilbert series of graded algebras, multiplicities of roots and length of local algebras, shellable complexes and f-vectors of polytopes

Upcoming talks: February 6&13: Anatoly Kouchnirenko Arnold filtrations, Stanley–Reisner rings, and simplicial Newton polytopes (in Russian)

Wednesday, January 16, 18:30, room 306

Alexander Esterov

Lattice polytopes and systems of polynomial equations

The fundamental theorem of algebra states that the number of roots of the univariate polynomial (counted with multiplicities) equals the degree of the polynomial. We shall discuss several generalizations of this fact to systems of polynomial equations of several variables.
The most geometric generalization (the Kouchnirenko theorem) states that, for generic systems of equations, the number of solutions of the system equals the volume of a certain lattice polytope, associated to this system. Motivated by this application, we shall overview geometry of lattice polytopes and briefly discuss the seminar topics for this term.