Special year - math workshop

Combinatorics of Enumerative Geometry

Sponsored by Dr. John P. Hempel and the Simons Foundation

Organizers: Paolo Aluffi, Dave Anderson, June Huh, and Leonardo Mihalcea

This workshop is dedicated to the exploration of the combinatorial dimensions of...

Abstract: I will survey new results on several enumerative problems related to the geometry, topology, and arithmetic of moduli spaces of smooth and stable curves. The enumerative problems include counting isomorphism classes of curves of genus $g$...

Abstract: Chow rings of toric varieties feature a rich combinatorial structure of independent interest. We will begin by surveying four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We will...

Abstract: There has been a lot of recent work connecting the two pipe dream formulae for Schubert polynomials (classic and bumpless). One strand is the hybrid pipe dreams of Udell and myself, giving $2^n$ different pipe dream formulae; an orthogonal...

Abstract: For a smooth projective variety, the classical Hirzebruch--Riemann--Roch (HRR) theorem asserts the isomorphism between the Chow ring and the Grothendieck K-ring of vector bundles over rational coefficients. For certain toric varieties, we...

Abstract: We present new objects called quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a...

Abstract: 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...

Abstract: 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...

Abstract: 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...

Abstract: 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.