Dr. Sabir Gusein-Zade (Moscow State University), Physically Inspired Mathematics

Title:  Power structures over the Grothendieck ring of varieties and Macdonald type equations.

Abstract:  One has a simple formula due to I. Macdonald for the generating series of the Euler characteristics of the symmetric powers of a topological space $X$: \sum_{n=0}^\infty \chi(S^nX)t^n= (1-t)^{-\chi(X)}$. Notice the the power series $(1-t)^{-1)$ does not depend on the space $X$. A Macdonald type equation is a formula for the generating series of the values of an invariant for symmetric powers of spaces or varieties (or for their analogues) which gives this series as a series not depending on the space in the power equal to the value of the invariant for the space itself. If the invariant takes values in a ring different from the ring of integers, an expression of this sort requires an interpretation. It is given by the so called power structure over the ring. The most important example is a geometric power structure over the Grothendieck ring of complex quasiprojective varieties. This power structure can be used to formulate (and sometimes also to prove) equations for generating series of classes of spaces of some sort. One example is the generating series of classes in the Grothendieck ring of complex quasiprojective varieties of Hilbert schemes of zero-dimensional subschemes in a non-singular quasiprojective variety. Another example is Macdonald type equations for generalized orbifold Euler characteristic (with values in a modification of the Grothendieck ring of complex quasiprojective varieties) and for their higher order versions.  The talk is mostly based on joint works with I. Luengo and A. Melle-Hernandez.