There are two major research trends in the theory of symmetric
functions arising from Dyck paths. One is the theory of Catalan
symmetric functions and its geometric realization conceived by
Chen-Haiman, following the works of Broer and Shimozono...
Suppose X is the affine cone of a projective variety. The
Hilbert series of the coordinate ring C[X] is the character of an
algebraic torus. More generally, one considers a reductive
algebraic group G
While mirror symmetry for flag varieties and Grassmannians has
been extensively studied, Schubert varieties in the Grassmannian
are singular, and hence standard mirror symmetry statements are not
well-defined. Nevertheless, in joint work with...
Springer fibers are subvarieties of the flag variety
parameterized by partitions. They are central objects of study in
geometric representation theory. Given a partition λ, one of the
key conclusions of Springer theory is that the top
dimensional...
Chow quotients of projective varieties by affine torus actions
provide alternative constructions of interesting geometric objects.
For example, the moduli space of stable genus 0 curves with n
marked points arises as the Chow quotient of the...
We will discuss invariants of lattice polytopes and their
subdivisions arising from Ehrhart and Hodge theory and introduce
their matroid theoretic analogues which are enriched versions of
the characteristic and Tutte polynomials.
I will discuss some regular subdivisions of the permutahedron,
one for each Coxeter element in the symmetric group. These
subdivisions are "Bruhat interval" subdivisions, meaning that each
face is the convex hull of the permutations in a Bruhat...
In his 2018 paper "Some Schubert shenanigans" Richard Stanley
asked for the asymptotic behavior of the maximal principle
specialization of a Schubert polynomial. Motivated by this, still
open, question we explore the generalization to Grothendieck...
The general rule for the interactions between tropical geometry
and moduli spaces of course is the following: everything you may
wish is going to work like a charm in genus zero, and break down
horribly in higher genus. This is the case for the...
The spectral norm on the group of Hamiltonian diffeomorphisms of
a symplectic manifold is defined via a homological min-max process
on the filtered Floer homology. Based on the spectral norm one
defines the spectral capacity of domains which is...
