Representations, Algebraic Topology, and Combinatorics On Wednesdays

(image by Ira Young)

Spring 2024 Schedule

All talks are Wednesdays at 10am in 203 Condon, onless otherwise noted.

April 3: No meeting.

April 10: Dan Dugger, "A (sort of) combinatorial look at algebraic K-theory"

Abstract: This will be a very gentle introduction to some ideas from algebraic K-theory, aimed towards algebraists and combinatorialists. Algebraic K-theory is a machine that takes input that is essentially algebro-combinatorial and outputs a topological space, whose homotopy groups end up having interesting behavior. I want to talk about some conjectures that currently interest me, but I won't have anything earth-shattering to say about them. Honestly, this talk is kind of a fishing expedition, to see if there are any potential connections to algebra and combinatorics that no one has yet thought about. But I think (hope) I can tell a story that is very interesting in its own right and really should be more widely known.

April 17: Nick Proudfoot, "Categorical valuations for polytopes and matroids"

Abstract: Valulations of matroids are very useful and very mysterious. After taking some time to explain this concept, I will categorify it, with the aim of making it both more useful and less mysterious.

April 24: No seminar.

May 1: Nick Addington, "On the integral cohomology of hypersurfaces, and some Torelli problems"

Abstract: Torelli's theorem states that a smooth complex projective curve X is determined by the polarized Hodge structure on H^1(X,Z). One can ask whether similar results hold for other classes of varieties -- hypersurfaces in P^n, K3 surfaces, hyperkahlers, Calabi-Yau threefolds, either up to isomorphism, or birational isomorphism, or derived equivalence. I'll discuss the Torelli problem for Calabi-Yau threefolds, and an example of Aspinwall, Morrison, and Szendroi that was supposed to break things in one way, but we've found that it breaks them another way.

Our main tool is a description of the integral cohomology of the Fermat quintic threefold that seems to be new: the real points determine a class in H^3, and we show that the orbit of this class under the automorphism group spans H^3.

This is joint work with Ben Tighe.

May 8: No seminar.

May 15: Nicolle Gonzalez (UC Berkeley), "Calibrated Representations of the Double Dyck Path Algebra"

Abstract: The double Dyck path algebra, Bqt, and its polynomial representation arose originally in the proof of the shuffle theorem of Carlsson and Mellit and shortly thereafter in the work of Carlsson-Gorsky-Mellit where the polynomial representation was realized geometrically on the K-theory of parabolic flag Hilbert schemes. In this talk we study the representation theory of Bqt, in particular those representations that are semisimple with respect to a family of commuting operators, and give a classification result by realizing this class of Bqt-representations combinatorially. This construction allows us to endow the subcategory of such representations with a 'generically' monoidal structure and prove that the r-fold tensor product of the polynomial representation arises geometrically on the equivariant K-theory of parabolic Gieseker moduli spaces. This is joint work with Eugene Gorsky and Jose Simental.

May 22: Ben Tighe, "Applications of the du Bois Complex to Symplectic Singularities"

Abstract: The du Bois complex is a generalization of the holomorphic de Rham complex in the presence of singularities and is a formal consequence of Deligne's mixed Hodge structure. While complicated to describe, the du Bois complex has been extremely useful in detecting the complexity of a singularity. The goal of this talk is to convince you that this object is very useful in studying symplectic singularities. For these singularities, the du Bois complex admits many symmetries you might expect to hold on a compact hyperkahler manifold. I will discuss how these symmetries are useful in answering some conjectures related to local vanishing and the existence of symplectic resolutions.

May 29:

June 5:


Spring 2023 Schedule

All talks are Wednesdays at noon in 104 University Hall, onless otherwise noted.

April 5: No meeting.

April 12:

April 19: Nick Proudfoot, "Interesting filtrations of boring rings".

Abstract: The Varchenko-Gelfand ring of a real hyperplane arrangement is a boring ring with an interesting filtration. I'll describe the associated graded and the Rees algebra, and interpret these things as cohomology rings of certain manifolds (such as the configuration space of points in R^3). No background about filtered rings will be assumed.
This talk will involve the work of my former students Daniel Moseley and Jayden Wang, as well as our future RTG postdoc Galen Dorpalen-Barry.

April 26: Ben Elias, "When does Poincare duality work for invariant subrings?"

Abstract: It is a famous fact that symmetric polynomials in n variables themselves form a polynomial ring with n generators. More generally, let G be a group acting homogeneously on a polynomial ring R. When is the subring R^G of invariant polynomials itself a polynomial ring with the same number of generators? A theorem of Shephard-Todd gives a precise answer: when G is finite and generated by complex reflections, a so-called "complex reflection group." They also classified these groups: there is one infinite family G(m,d,n), and 34 sporadic groups.

For some groups (e.g. the symmetric group) Demazure proved that there is a nondegenerate trace map from R to R^G. Being a complex reflection group is a prerequisite. When G is a Weyl group, this corresponds to integration along the flag variety, which is nondegenerate by Poincare duality. Moreover, Demazure proved that the trace map is a composition of smaller operators known as Demazure operators. When G is a Coxeter group this still happens, but there is no geometry to supply an explanation. No attempt has been made to classify the complex reflection groups admitting nondegenerate trace maps.

In recent work with Ben Young and Daniel Juteau, we prove that G(m,m,n) admits a nondegenerate trace map, and explore Demazure operators. This particular case is motivated by quantum geometric Satake at a root of unity.

May 3: Hiatus.

May 10: Hiatus

May 17: Hiatus

May 24: Dan Dugger, "Bredon equivariant cohomology of configuration spaces"

Abstract: I will give an introduction to the singular cohomology of ordered configuration spaces of points in a Euclidean space, which has several interesting connections to combinatorics. Following this, I will also give an introduction to Bredon cohomology for topological spaces with group actions, and how this relates to classical equivariant cohomology. Finally, I will explain some recent work on Bredon cohomology of configuration spaces and raise some questions about possible connections to combinatorics.

May 31: John Machacek, "q-Rational and q-Real Binomial Coefficients"

Abstract: We consider q-binomial coefficients built from the q-rational and q-real numbers defined by Morier-Genoud and Ovsienko in terms of continued fractions. We establish versions of both the q-Pascal identity and the q-binomial theorem in this setting. These results are then used to find more identities satisfied by the q-analogues of Morier-Genoud and Ovsienko including a Chu--Vandermonde identity. We will start with a review of q-integers and Gassian binomial coefficients before moving on the q-rational numbers. This is joint work with Nick Ovenhouse.

Winter 2023 Schedule

All talks are at 10am in 210 University, unless otherwise noted.

February 9: Ben Elias, "Reduced expressions for double cosets in Coxeter groups".

Abstract: There is a rich and well-developed theory of reduced expressions for elements of Coxeter groups. For example, one has Matsumoto's theorem, which states that any two reduced expressions for a given element are related by a sequence of braid relations. Reduced expressions also play a role in many representation-theoretic and geometric constructions. We highlight two of these: the construction of Bott-Samelson varieties which resolve singularities in Schubert varieties, and the study of Demazure operators, acting on the cohomology rings of flag varieties.

Meanwhile, consider two parabolic subgroups W_J and W_K inside a Weyl group W. To a double coset [w] inside W_J \ W / W_K, one can also associate a Schubert variety and a Demazure operator, this time related to the geometry of partial flag varieties. Geordie Williamson in his thesis introduced a theory of expressions and reduced expressions for double cosets, in order to construct Bott-Samelson varieties in this context.

In recent work with Hankyung Ko, we develop the theory of reduced expressions for double cosets. We provide several equivalent definitions for a reduced expression, easier to work with than the original definition of Williamson. We prove a relationship between the Bruhat order on double cosets and the appropriate analogue of subexpressions. Most importantly, we introduce the double coset braid relations, and prove the double coset Matsumoto theorem: that any two reduced expressions for [w] are related by the double coset braid relations. For example, in type A_2, the ordinary braid relation sts=tst actually follows from a sequence of four (smaller and more elementary) double coset braid relations. We describe the implications to the study of Demazure operators (and don't describe the applications to the study of Bott-Samelson varieties).

March 15: Dev Sinha, "Rational measurement of groups". [4pm, Room TBA]

Abstract: How can you tell if some (power of a) word in a finitely presented group is an n-fold commutator? How do cochains 'see' the fundamental group?

We give explicit answers to these questions, in work in progress with N. Gadish, A. Ozbek and B. Walter. In particular, we take an abstract isomorphism between the dual of the zeroth Harrison homology of rational cochains of a space with the Malcev completion of its fundamental group, established by U. Buijs, Y. Felix, A Murillo and D. Tanre, and show it is realized through Hopf invariants. We thus complete a program initiated by B. Walter and myself to explicitly construct homotopy periods on nilpotent spaces. When the space is a two-complex, the resulting 'letter linking' invariants for words are defined for all finitely presented groups, a result which has not been achieved for other explicit funcationals, namely those arising from Magnus expansion and Fox calculus. We outline a planned application to Milnor's link invariants, showing how different choices of Seifert surfaces can lead to either Cochran's approach or diagrammatic formulae.

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.