Monday, April 10, 2023
04/10/2023 - 4:00pm
We will discuss applications of a discrepancy trick, which sometimes can be used to prove quantitative disjointness results. As an illustration, consider the following example. Fix an imaginary quadratic number field and consider a "diagonal" measure on a product of the unit tangent bundles of two modular curves given by the push-forward under two multivariate monomials of the uniform measure on two copies of the class group. If the monomials are independent, assuming GRH for the Dedekind zeta function, this diagonal measure is quantitatively equidistributed in the product in terms of the size of the discriminant of the quadratic field.
This version of the result implies (conditional) bounds on the size of torsion subgroups of the class group, which was proven previously by Ellenberg and Venkatesh. The main dynamical theorem has several applications and, for example, can be used to prove for most dimensions quantitatively the equidistribution of the triples consisting of a rational subspace, the shape of the integer lattice in this subspace, and the shape of the integer lattice in the orthogonal complement.
This is joint work with Menny Aka, Manfred Einsiedler, Philippe Michel, and Andreas Wieser.
04/10/2023 - 4:30pm
In analogy with the (generalized) Springer correspondence relating perverse sheaves on a nilpotent cone to representations of the Weyl group, we consider perverse sheaves on the symmetric product of n copies of the plane C2, constructible with respect to the natural stratification by collision of points. This category is semisimple when the coefficients have characteristic zero, but with positive characteristic coefficients it can be very complicated. We show that this category is equivalent to modules over a convolution algebra given by K-theory of sheaves on the symmetric group, equivariant for the action of Young subgroups on the left and right. Up to Morita equivalence, this algebra has a Schur algebra as a quotient. I will also explain how this algebra arises using the K-theory of Hilbert schemes and a theorem of Bridgeland, King, and Reid. Joint work with Carl Mautner.