Abstract: I'll discuss (orbifold) symmetric products of projective spaces Sym^d(P^r), focusing on the structure of orbits of the natural torus action. These orbits vary in nice toric moduli spaces -- for example, components of the moduli space of 1-dimensional orbits are naturally identified with certain toric compactifications of M_{0,n}. I will then discuss how one can use these orbits to calculate Gromov-Witten invariants of Sym^d(P^r). If I have time, I'll…

Abstract: Fix the coordinate ring R of a complex non-singular affine algebraic variety V, and a radical ideal I in R. The set of polynomial functions on V vanishing to order at least N at all points of the zero locus of I in V has the structure of an ideal -- the N-th symbolic power I^{(N)} of I. In particular, while I^N always lies in I^{(N)}, the latter may…

Abstract: Let G be a connected reductive algebraic group. Let us consider a product X of flag varieties and the diagonal action of G, which extends to a Hamiltonian action on the cotangent bundle T*X. Hence we get a moment map. Steinberg considered a conormal variety and deduced a map from the Weyl group W of G to the nilpotent coadjoint orbit in the Lie algebra. In type A, this…

Abstract: The (affine cone over the) Grassmannian is a prototypical example of a variety with "cluster structure"; that is, its coordinate ring is a cluster algebra. Scott (2006) gave a combinatorial description of this cluster algebra in terms of Postnikov's plabic graphs. It has been conjectured essentially since Scott's result that Schubert varieties also have a cluster structure with a description in terms of plabic graphs. I will discuss recent…

Abstract: Connected sums were defined for local Gorenstein algebras by Ananthnarayan Avramov-Moore (A-A-M) in a 2012 paper. In the graded Artinian case, this construction is related to a topological construction that pastes two manifolds together along a common submanifold. In this case, the A-A-M construction can be described using algebraic versions of the Thom class of the normal bundle of a submanifold. We discuss this description here, as well as…

Abstract: Traditionally, Macaulay Duality furnishes a useful canonical bijective correspondence between Artinian quotients of a polynomial ring over a base field and modules of linear functionals on forms. In joint work in progress with Jan Kleppe of Oslo, this duality is generalized over an arbitrary Noetherian base ring, thus providing a suitable framework for studying a family of Artinian quotients by investigating its dual family.

Abstract: Chapoton triangles are polynomials in two variables defined by Coxeter-Catalan objects. These polynomials are related by some remarkable identities that only depend on the rank of the associated (finite) Coxeter system. The multidimensional Catalan numbers enumerate the number of standard Young tableaux of a rectangular shape. It also counts the number the vertices of a polytope known as the Grassmann associahedron. Using the structure of this polytope, I will…

In this talk, we'll share an explicit description of the corresponding "affine evacuation'' map on tabloids, and we show that the number of tabloids fixed by this map is equal to the evaluation of a certain Green's polynomial at q = -1. Along the way, we discover a combinatorial interpretation of the evaluation of the Kostka-Foulkes polynomials at q=-1. These findings are based on joint work with Mike Chmutov, Dongkwan…

Abstract: Puzzles in Schubert calculus were originally developed by A. Knutson and T. Tao as combinatorial objects for computing the expansion of the product of two Grassmannian Schubert classes. I will describe how self-dual puzzles in turn allow us to compute the restriction of a Grassmannian Schubert class to the symplectic Grassmannian in equivariant cohomology. The proof uses the machinery of quantum integrable systems. Time permitting, I will also discuss…