—  Mathematics & Statistics

The TACO Seminar | Fall 2022

Wednesdays, 3:50–4:40. Zoom or CH 217

The TACO seminar focuses on topics in Topology, Algebra, Combinatorics, and Operators.

Organizers:  Emily Peters, Carmen Rovi

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)

Some of the talks will be on zoom. If you would like to attend a zoom talk let one of the organizers know so we can add you to the list of participants.

Frobenius algebras are both (1) a reasonably straightforward type of algebra, which in addition to a unit and multiplication also have a co-unit and co-multiplication, and (2) fundamental to the classification of 2D topological quantum fields theories (TQFTs), making them part of "baby model for quantum gravity" (according to Joaquim Kock). In this talk I will define the key players and go into detail about the classification of Frobenius algebras.

September 28 Rafael González D'León - Flow polytopes as a unifying framework for some familiar combinatorial objects

Flow polytopes are a family of beautiful geometric objects which have connections to many areas in mathematics including optimization and representation theory. Computing their volumes and enumerating lattice points of some particular flow polytopes turn out to be combinatorially interesting problems that involve beautiful enumeration formulas and many familiar combinatorial objects. Baldoni and Vergne found a series of formulas for both of these purposes, which they call Lidskii formulas, that are combinatorially powerful and pleasant. A later proof of the Lidskii formulas has been achieved by Mészáros and Morales, following the ideas of Postnikov and Stanley, using polytopal subdivisions. For a smaller class of flow polytopes, these subdivisions are triangulations that coincide with a family of framed triangulations defined by Danilov, Karzanov, and Koshevoy. These triangulations have interesting hidden combinatorial structure. We will give an introduction to flow polytopes and these formulas, including some recent applications, and a series of open problems and conjectures which we are currently working on.

October 5 Thomas Brazelton - Equivariant enumerative geometry

Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these questions admit well-defined integral answers independent upon the initial parameters of the problem is Schubert's principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is an equivariant conservation result, which states that, under nice hypotheses, the sum of regular representations of the orbits of solutions to an equivariant enumerative problem is conserved. As an application, we explore the orbits of the 27-lines on an S_4-symmetric cubic.

October 26 Marie Meier - Laplacian Simplices Associated to Threshold Graphs

Polytopes associated to graphs have been studied more extensively in the past decade. One such technique is to use the Laplacian matrix of a graph to form a polytope by considering the rows of the matrix as vertices of the polytope. For a connected simple graph with n vertices, the resulting polytope is an n - 1 dimensional simplex. We relate properties of the simplex with its underlying graph for a variety of families. Ehrhart Theory provides the lens in which we view properties such as reflexivity, the integer decomposition property, and the associated h*-vector. In this talk, I will motivate the Laplacian simplex construction and provide results that lead to interesting behaviors, such as an extreme h*-vector. I also present recent results of Laplacian simplices arising from threshold graphs.

November 2 Josh Hallam - Braid Cones and the Gorenstein Property

To each poset P, we can associate a convex cone sigma_P as well as an affine toric variety U_P that sigma_P defines. sigma_P is the union of regions in the braid arrangement and thus is called a braid cone. Postnikov, Reiner, and Williams gave a characterization of when U_P is smooth. Moreover, U_P is Cohen-Macaulay for any poset P. In this talk, we consider an algebraic property that sits properly between smooth and Cohen-Macaulay, namely the Gorenstein property. We will show that U_P is Gorenstein if and only if there is a certain kind of vertex labeling on the Hasse diagram of P. When P has a maximum or minimum, we will see that the Gorenstein property of U_P is completely determined by the Moebius function of P. We will also discuss a recursive algorithm to check if U_P is Gorenstein in this case. No background knowledge on posets or varieties will be assumed. This is joint work with John Machacek.

November 9 Tianyuan Xu - 2-roots for simply laced Weyl groups

We introduce and study ``2-roots'', which are symmetrized tensor products of orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$ is the Weyl group of a simply laced Y-shaped Dynkin diagram with three branches of arbitrary finite lengths $a$, $b$ and $c$; special cases of this include types $D_n$, $E_n$ (for arbitrary $ n\geq 6$), and affine $E_ 6$, $E_7$ and $E_8$. With motivations from Kazhdan--Lusztig theory, we construct a natural codimension-$1$ submodule $M$ of the symmetric square of the reflection representation of $W$, as well as a canonical basis $\mathcal{B}$ of $M$ that consists of 2-roots. We conjecture that, with respect to $ \mathcal{B}$, every element of $W$ is represented by a column sign-coherent matrix in the sense of cluster algebras, and we prove the conjecture in the finite and affine cases. We also prove that if $W$ is not of affine type, then the module $M$ is completely reducible in characteristic zero and each of its nontrivial direct summands is spanned by a $W$-orbit of 2-roots. (This is joint work with Richard Green.)

November 16 Emily Barnard - New-biCambrian Lattices

In 2016, with N. Reading, I studied a family of counting problems related to Coxeter-Catalan combinatorics. Each counting problem considered a pair of "twin" Coxeter-Catalan objects. We showed that each counting problem has the same solution, which we called the W-biCatalan number. In this talk, we focus on the "biCatalan" object that is analogous to the Tamari lattice, called the c-biCambrian lattice. In type A, the elements of the c-biCambrian lattice consist of pairs of "compatible" pattern-avoiding permutations. In 2016, the compatibility relation of these pairs was not well-understood. Our main result is to give a simple criterion for determining when a pair of pattern-avoiding permutations (called c-sortable) are compatible which generalizes to all finite type Coxeter groups. Moreover, our result leads to the definition of a nu-biCambrian lattice. This project is joint with Emily Gunawan and Emily Meehan.

November 30 Qing Zhang - Braided Zesting and its applications

Abstract: Modular categories appear in many areas of mathematics, as well as condensed matter physics. Similar to extension theory for finite groups, there exist a number of ways to construct new modular categories out of existing ones. Classical examples include the Drinfeld center construction, (de-)equivariantization, and gauging. A more recent construction ''zesting'' has proved useful for finding realizations of modular data, producing new modular closures of super-modular categories, and classifying weakly integral modular categories. In this talk, I will present an overview of zesting constructions and some of their interesting properties.

December 7 Martha Yip - Triangulations of flow polytopes and gentle algebras

Abstract: By the work of Danilov, Karzanov and Koshevoy, the cone of nonnegative flows for a directed acyclic graph is known to admit regular unimodular triangulations induced by framings of the graph. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. These triangulations restrict to triangulations of the flow polytope for strength one flows. We establish a connection between the maximal simplices in amply-framed triangulations of a flow polytope, and tau-tilting posets for certain gentle algebras. Using this connection, we are able to prove that for certain directed acyclic graphs, the flow polytopes are Gorenstein and have unimodal h^* polynomials.