Maciej Dołęga

Title: Higher-order Macdonald positivity conjecture, Hilbert schemes 
       and Springer correspondence 

Abstract: 
The main purpose of this talk is to introduce a certain conjecture with its origin 
in algebraic combinatorics to the community of algebraic geometers. We believe that 
it might be possible to prove our conjecture using standard techniques from the 
algebraic geometry and we would like to discuss the possible interaction between 
algebraic combinatorics and algebraic geometry in the settings of this problem. 
A typical question in the algebraic combinatorics of the symmetric functions 
is the following: given a symmetric function, is it Schur-positive or not? Given a 
finite family of Macdonald polynomials we transform it into a new symmetric function 
using the notion of cumulants, and we conjecture that it is Schur-positive. 
We discuss the "trivial" cases of this problem which can be interpreted as the 
Frobenius series of some graded modules using the Springer correspondence and the 
geometry of Hilbert schemes of $n$ points in the plane. We also present some partial 
results of our conjecture. The talk will be elementary and assume no prior knowledge 
in the discussed fields.