Our seminar views algebraic combinatorics broadly, explicitly including algebraic enumeration and related asymptotic and bijective combinatorics, as well as algebraic combinatorics as it appears in pure algebra and in applications outside mathematics.
Our local audience consists principally of combinatorics faculty and grad students. Talks are 50 minutes with 10 minutes for questions.
| Date |
Speaker |
Title |
| January 12 | Andy Wilson (Kennesaw State University) | Coinvariants and superspace
|
| |
The ring of multivariate polynomials carries a natural action of the symmetric group. Quotienting by the ideal generated by the polynomials which are invariant under this action yields the "coinvariant algebra," an object with many beautiful algebraic and combinatorial properties. We will survey these properties and then discuss recent generalizations where the multivariate polynomials may contain anti-commuting ("superspace") variables. This talk is based on joint work with Brendon Rhoades.
|
| January 19 | François Bergeron (LACIM) | From the nabla operator to the super nabla operator |
| |
The nabla operator is certainly one of the most exploited ones in the study of: Macdonald symmetric functions and their occurrences in several areas of investigations. After recalling some of these, as well as the historical role of many related operators, we will describe a new “super” version that unifies this whole area of investigation. The talk is illustrated by explicit examples, in an effort to make it accessible to a “general” audience. The new aspects are joint work with Jim Haglund, Alessandro Iraci, and Marino Romero.
|
| January 26 | Emily Barnard (De Paul University) | Cluster combinatorics and poset topology
|
| | In this talk we use the history of cluster combinatorics to motivate the study of certain subcategories of modules called wide subcategories, coming from quiver representations. In the context of cluster combinatorics, the classical noncrossing partition lattice is isomorphic to a poset of wide subcategories for a type A linearly oriented quiver. We will play on this connection, and review some still open problems related to the W-noncrossing partition lattice. Our main result is an EL-labeling of the poset of wide subcategories which can be encoded in terms of the Kreweras complement (joint with Eric Hanson).
|
| February 2 | Nick Olson-Harris (Waterloo) | Binary tubings and Dyson-Schwinger equations
|
| | Dyson-Schwinger equations are integro-differential equations satisfied by correlation functions in quantum field theory, which play the role of the "equations of motion" of the theory. They have a recursive, tree-like structure which enables these equations and their solutions to be studied combinatorially. Marie and Yeats showed that in a special case, the solution could be expanded as a sum over connected chord diagrams; this was generalized to many more cases by Hihn and Yeats. Using Hopf algebra techniques we give new combinatorial expansions for a much larger class of Dyson-Schwinger equations and systems as sums over rooted trees equipped with a kind of recursive decomposition we call a "binary tubing". This talk is based on joint work with Paul-Hermann Balduf, Amelia Cantwell, Kurusch Ebrahimi-Fard, Lukas Nabergall, and Karen Yeats.
|
| February 9 | Elana Kalashnikov (Waterloo) | Quantum hooks and the Plücker coordinate mirror
|
| | There is a natural map from the symmetric polynomial ring in r_1 variables to the quantum cohomology ring of a type A flag variety Fl(n,r1,..rk), given by evaluating Schur polynomials in the Chern roots of the first tautological bundle. I’ll explain how for a large class of Schur polynomials, the result is a Schubert class that can be obtained by dividing the partition into a quantum-hook and smaller partitions. Surprisingly, this is the key result proving a mirror theorem for type A flag varieties. A function W is a mirror of a Fano variety X if enumerative information of X can be determined from W: for example, the Jacobi ring of W should be the quantum cohomology ring of X. Mirrors for Fano toric varieties are well-understood; and more recently Plücker coordinate mirrors have been proposed for a variety of homogeneous spaces. We use quantum hooks to prove that the Plücker coordinate mirror of the flag variety computes quantum cohomology relations. This is joint work with Linda Chen.
|
| February 16 | Stephen Gillen (Waterloo) | Geometry of gradient flows for analytic combinatorics
|
| |
Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of series coefficients of multivariate rational functions in an exponent direction r by analyzing the singular set V of a multivariate rational function. The poly-torus of integration T that arises from the multivariate Cauchy Integral Formula (it would be a circle in one complex variable) is deformed away from the origin into cycles around critical points of a “height function" h on V. The deformation can sometimes flow to infinity at finite height in the presence of a critical point at infinity (CPAI): a sequence of points on V approaching a point at infinity, and such that the log-normals to V converge projectively to the direction of r. The CPAI is called heighted if the height function also converges to a finite value. In this talk we discuss under what conditions we know that all CPAI are heighted, and in which directions CPAI can occur, by compactifying in a toric variety. In smooth cases under generically satisfied conditions, CPAI must always be heighted. Non-generic cases are also studied under other conditions.
|
| February 23 | Reading week (no seminar) |
|
| March 2 | Social hour |
|
| March 9 | Joel Lewis (George Washington University) | Bargain hunting in a Coxeter group
|
| |
Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost-minimization problem over the factorizations of a permutation into transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t be the distance between the integers transposed by t in the combinatorial representation of the group (à la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula. This work is joint with Bridget Tenner.
|
| March 16 | Kartik Singh (Waterloo) | Taking limits in Go-diagrams
|
| |
Most decompositions of the Grassmannian are described as subsets of the Grassmannian by setting certain Plücker coordinates to zero, demanding certain other Plücker coordinates to be non-zero, and leaving the remaining Plücker coordinates unspecified. In the case of the Deodhar decomposition, these coordinates are determined by the location of stones in the corresponding Go-diagram. We shall be interested in answering the question as to when one Deodhar component lies in closure of another by looking at their corresponding Go-diagrams. We will define restricted paths on a graph determined by the Go-diagram and show how they can be used to solve the above problem. This work is joint with Kevin Purbhoo and Olya Mandelshtam.
|
| March 23 | Lucas Gagnon (York University) | Quasisymmetric varieties, excedances, and bases for the Temperley--Lieb algebra
|
| |
This talk is about finding a quasisymmetric variety (QSV): a subset of permutations which (i) is a basis for the Temperley--Lieb algebra TLn(2), and (ii) has a vanishing ideal (as points in n-space) that behaves similarly to the ideal generated by quasisymmetric polynomials. While this problem is primarily motivated by classical (co-)invariant theory and generalizations thereof, the course of our investigation uncovered a number of remarkable combinatorial properties related to our QSV, and I will survey these as well. Of particular interest is a new equivalence relation on permutations defined using their excedance sets. This relation has many nice properties: each equivalence class is naturally indexed by a noncrossing partition and also forms an interval in the (strong) Bruhat order. This allows us to define a quotient Bruhat order and gives a simple method for constructing many new bases of TLn(2), generalizing known results of Williams--Gobet and Zinno. Surprisingly, the combinatorics of this equivalence relation turn out to be key in solving the QSV problem: collecting the maximal element of each excedance class produces a QSV, and the ensuing noncrossing partition combinatorics are essential to prove this fact. Based on joint work with Nantel Bergeron; arXiv:2302.10814.
|
| March 30 | Freddy Cachazo (Perimeter) | Arrangements of Pseudolines, Tropical Grassmannians, and Generalized Scattering Amplitudes.
|
| | For each arrangement of (pseudo)lines on the projective plane, it is possible to construct a differential form that captures its combinatorial structure. The forms have simple poles whenever triangles shrink to a point in the arrangement, and share the same residue when two arrangements are connected via a "triangle flip". In this talk I will explain the construction and give evidence for the conjecture that integrating such differential forms, with the appropriate measure, computes generalized scalar scattering amplitudes. These amplitudes are defined as sums over arrangements of metric trees or generalized Feynman diagrams. While generic arrangements of metric trees span the Dressian, it is also conjectured that the "physical" ones define cones in the tropical Grassmannian.
|
| April 6 | |
|
| | |