Abstract: Polynomials are fundamental objects in many branches in Mathematics. Several problems in combinatorics and number theory have been solved in an interesting way using polynomials. For example, Cauchy-Davenport theorem and Snevily’s Conjecture can be proved easily by this method. In this talk, we mainly focus on Combinatorial Nullstellensatz and Chevalley-Warning theorems, with some applications in combinatorial number theory. The Combinatorial Nullstellensatz is a theorem about the roots of a polynomial. Chevalley-Warning theorem deals with the number of solutions of system of polynomial equations. As a consequence of Chevalley-Warning, finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin in 1935.

Abstract: I will talk about the definition of higher Toda brackets in triangulated categories and possibly about how these are determined by the triple Toda brackets.