January 31, Wednesday, 15:30, room 211

Anastasiya Tyurina

Constructing polynomial system with many positive solutions using tropical geometry (by Boulos El  Hilany)

The number of positive real solutions of a system of two polynomial equations in two unknowns with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. Tropical geometry is a powerful tool to construct polynomial systems with many positive solutions. The classical combinatorial patchworking method arises when the tropical hypersurfaces intersect transversally. Using this method the author constructs  a system that has at most 6 positive solutions. He also shows that this bound is sharp. Moreover, using non-transversal intersections of tropical curves, he constructs a system that has  7 positive solutions.

January 24, Wednesday, 15:30, room 211

Sofia Aleshina

On some problems of sum-product type

Let A be a finite non-empty set of elements of a ring. Consider the set A+A of pairwise sums and the set A*A of pairwise products of the elements in A. Obviously, the cardinality of such sets is at least |A|. The Erdős–Szemerédi hypothesis says that max(|A+A|,|A*A|) ≫ C|A|^(2-ɛ), where ɛ and C are positive constants. That is, either A+A or A*A is quite large. In this talk, I will say a few words about the Erdős–Szemerédi theorem and other results over Z and Z/pZ. Later I consider a generalization of the problem of sum-product type. Namely, I consider the set of values f(x,y) that a homogeneous polynomial f in two variables takes if x and y belong to A. I obtain a lower bound on the cardinality of this set.

January 17, Wednesday, 15:30, room 211

Note the change of the speaker and topic!

Valentina Kiritchenko

Introduction to Newton-Okounkov convex bodies

We discuss the main ingredients of the construction of Newton-Okounkov convex bodies for graded semigroups and algebras. As application we consider a generalization of Kouchnirenko’s theorem to the non-toric setting. We also explain the relation between the theory of Newton-Okounkov convex bodies and fundamental algebro geometric concepts such as the degree of a projective variety, algebra of global sections of a line bundles and Hilbert function  (all necessary definitions will be given in the talk).

The talk is intended as an overview of topics from the first block. The participants who are interested in taking a topic from this block are especially encouraged to attend.

January 10, Wednesday, 15:30, room 211

Dmitry Korshunov