Speaker
Description
Monomial combinatorics describes the geometric and combinatorial properties of supports of polynomials belonging to polynomial ideals. A classic example is the theory of Gröbner bases, which allows us to find sets of polynomials in an ideal with minimal leading terms, with respect to any admissible orderings. Other tasks, the simplest in terms of formulation, but by no means simple algorithmically, is the search for monomials or binomials in ideals. The first task is equivalent to determining the intersection of an ideal with a lattice of monomials. The second problem is closely related to the torical geometry. I will describe various methods and algorithms of monomial combinatorics that allow, for example, to build monomial bases in factoralgebras, as well as alternative methods to Buchberger's algorithm for constructing Gröbner bases of polynomial ideals based on interesting combinatorial structures such as involutive divisions. We will also discuss some applications of monomial combinatorics to the tropical varieties and give tropical interpretation of the universal Gröbner basis.
