Representations, Algebraic Topology, and Combinatorics On Wednesdays

(image by Ira Young)

Winter 2022 Schedule

All talks are at 3pm in 260 Tykeson unless otherwise noted.

January 12: Ben Young, "Double-Dimer condensation and the PT/DT correspondence".

Abstract: I'll discuss joint work with Helen Jenne and Gautam Webb in which we prove, combinatorially, a relationship between the dimer and "tripartite double-dimer" models on the hexagon lattice, with a wide range of specified boundary conditions at infinity. This result was motivated by a geometric conjecture of Pandharipande-Thomas, which we prove (as a corollary of our work and of a lot of other geometric work over the past decade).

January 19: John Machacek, "Shelling the m=1 amplituhedron"

Abstract: We start from the totally nonnegative Grassmannian and a related space called the amplituhedron. We then make use of a known cell decomposition to study a certain class of amplituhedra with poset topology. The poset arising generalizes the Boolean algebra and shares properties with the Boolean algebra including shellability and Spernicity. Based on joint work with Steven Karp.

January 26: Anna Cepek, "The combinatorics of configuration spaces of R^n".

Abstract: We approach manifold topology by examining configurations of finite subsets of manifolds within the homotopy-theoretic context of infinity-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of Euclidean space in terms of the category Theta_n.

February 2: Sarah Brauner (U Minnesota), "A type B analog of the Whitehouse representations".

Abstract: The Eulerian idempotents of the symmetric group generate a family of representations---the Eulerian representations---that have connections to configuration spaces, equivariant cohomology, and Solomon's descent algebra. These representations are defined in terms of S_n, but can be "lifted" to representations of S_{n+1} called the Whitehouse representations. I will describe this story in detail and present recent work generalizing it to the hyperoctahedral group (e.g. Type B). In this setting, configuration spaces will be replaced by certain orbit configuration spaces and Solomon's descent algebra is replaced by the Mantaci-Reutenauer algebra. All of the above will be defined in the talk.

February 9: George Nasr, "Ehrhart Theory of Paving and Panhandle Matroids"

Abstract: The Ehrhart polynomial for a polytope P records the number of lattice points in integral dilations of P. Recently, a lot of work has been done for the special case where P is the base polytope for a matroid. In this talk, I'll discuss some recent work I've done with collaborators to find a formula for the Ehrhart polynomial of a large class of matroids known as paving matroids, and how it led us to define an important subclass of lattice-path matroids, which we call Panhandle matroids.

February 16: Nick Proudfoot, "The algebraic geometry of KLS-polynomials"

Abstract: I'll give a general introduction to Kazhdan-Lusztig-Stanley polynomials, focusing on three cases of interest: classical KL polynomials, g-polynomials of polytopes, and KL polynomials of matroids. Then I will explain a unified formalism for interpreting all of these polynomials as Poincare polynomials.

February 25, 4pm (Note special day and time): Nir Gadish (University of Michigan), "The inclusion-exclusion principle in topology"

Abstract: Inclusion-exclusion is a combinatorial principle whereby we count a set by counting each of its components, but then must repeatedly correct for overcounting overlaps. The result is an alternating sum, or more generally, a weighted sum that counts a union from the counts of its constituents. Now think of the sets as topological spaces, and instead of counting their elements we want to understand their topology. It turns out that inclusion-exclusion still applies, when interpreted correctly, with coefficients in the weighted sums replaced by certain spaces. I will explain the sense in which this topological inclusion-exclusion applies, and give some examples of calculations it produces.

March 2: Nir Gadish (University of Michigan), "From configurations on graphs to top weight cohomology of the moduli spaces M_{2,n}".

Abstract: Configuration spaces of points on a graph are hard to understand topologically. Their one-point compactifications, on the other hand, are proper homotopy invariants and their homology admits a concrete algebraic description. Via the machinery of tropical geometry, such compactified configuration spaces detect cohomology of moduli spaces of curves with marked points. I will discuss joint work with C. Bibby, M. Chan and C. Yun, in which we combine this approach with some representation theory magic to compute the top weight rational cohomology of the moduli space M_{2,n} of genus 2 curves with n marked points, along with their action of the symmetric group, for all n < 11.

March 9: No talk this week.

Spring 2022 Schedule

Date TBA: Ben Elias, "Reduced expressions and braid relations for double cosets"

Abstract: Coxeter groups are ubiquitous in representation theory and combinatorics. The theory of expressions and reduced expressions for elements of a Coxeter group W has a generalization to double cosets inside W. In joint work with Hankyung Ko, we found all the braid relations between reduced expressions. One appearance of these ideas in geometry is as follows. For all parabolic subgroups P of a semisimple lie group, consider the P-equivariant cohomology (resp. K-theory) of a point. There are pushforward and pullback maps between these equivariant cohomology rings. Consider the category whose objects are parabolic subgroups P, and whose morphisms are generated by pushforward and pullback. We present this category by generators and relations.

Date May 25: Christin Bibby (LSU), "Matroid schemes and geometric posets"

Abstract: The intersection data of an arrangement of hyperplanes is described by a geometric lattice, or equivalently a simple matroid. There is a rich interplay between this combinatorial structure and the topology of the arrangement complement. In this talk, we will similarly characterize the combinatorial structure underlying certain arrangements of subvarieties by defining a class of geometric posets and a generalization of matroids called matroid schemes. Notions such as the independence complex and the Tutte polynomial extend to this setting, and we will touch on the topological implications of this framework.

Date June 1: Shiyue Li (Brown University), "K-rings of wonderful varieties and matroids"

Abstract: The wonderful model of a realizable matroid and its Chow ring have played key roles in solving many long-standing open questions in combinatorics and algebraic geometry. I will be sharing with you some discoveries on K-rings of wonderful varieties associated with realizable matroids. We also compute the Euler characteristic of every line bundle on wonderful varieties, and give a purely combinatorial formula. This in turn gives a new valuative invariant for an arbitrary matroid. As an application, we study the K-ring and compute the Euler characteristic of every line bundle of $\overline{\mathcal{M}}_{0, n}$ --- the Deligne-Mumford-Knudsen compactification of the moduli space of rational stable curves with $n$ distinct marked points. Joint work with Matt Larson, Sam Payne and Nicholas Proudfoot.