Spring 2022

Title: A concrete construction of the cographic hyperplane arrangement

Abstract: Geometrically carrying a trove of information about the underlying simple graph, the graphic hyperplane arrangement H_G yields an interesting mathematical object to study a simple graph G. For example, one proves that the normal vectors of H_G are linearly independent if and only if they induce forests on G and the regions of H_G are in a one-to-one correspondence to the acyclic orientations of G. With the graphic hyperplane arrangement we can associate the graphic matroid whose bases are spanning forests of G.

What information do we obtain if we apply duality, i.e., if we start with the dual of the graphic matroid, called the cographic matroid whose bases are complements of spanning forests of G?

In this talk we are going to start answering the above question by constructing the normal vectors of the cographic hyperplane arrangement associated with the cographic matroid for simple, connected and bridgeless graphs.

Prerequisites: Basic understanding of matroids, hyperplane arrangements, graphs and cellular chain complexes (the last is more optional than required).

Title: Wasserstein Distance to Independence Models

Abstract: An independence model for discrete random variables is a variety in a probability simplex.

Given any data distribution, we seek to minimize the Wasserstein distance to the model.

That distance comes from a polyhedral norm whose unit ball is dual to an alcoved polytope.

The solution to our optimization problem is a piecewise algebraic function of the data.

In this talk we discuss the combinatorial and algebraic structure of this function.

Title: Ehrhart Theory of Paving and Panhandle Matroids

Abstract: Ehrhart theory is a topic in geometric combinatorics which involves the enumeration of lattice points in integral dilates of polytopes. We show that the base polytope P_M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off subpolytopes. The pieces removed are base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams, whose Ehrhart polynomials we can calculate explicitly. Consequently, we can write down the Ehrhart polynomial of P_M. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain gangs and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters. (This is joint work with D. Hanely, J. Martin, D. McGinnis, D. Miyata, G. Nasr, and, M. Yin).

Prerequisites: definitions of polytope, matroids, Ehrhart polynomials (but the talk will attempt to explain most concepts).

Title: Rational Ehrhart Theory

Abstract: The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define y-rational Gorenstein polytopes, which extend the classical notion to the rational setting. This is joint work with Matthias Beck and Sophia Elia.

Prerequisites: TBA

Title: Combinatorics of singularities in vexillary Schubert varieties

Abstract: Schubert varieties in Grassmannians and flag varieties parametrize linear subspaces subject to incidence conditions. In general, these are singular varieties, and the nature of their singularities has deep connections with representation theory. I’ll describe joint work with Takeshi Ikeda, Minyoung Jeon, and Ryotaro Kawago in which we give combinatorial formulas for the Hilbert-Samuel multiplicity of points on a special class of Schubert varieties, the vexillary ones.

Prerequisites: I won’t assume any special knowledge of singularities or of Schubert varieties; it will help to be familiar with the idea using a manifold to parametrize mathematical objects.

Title: Coxeter groups, Artin groups, and the word problem

Abstract: Coxeter groups are a general family of groups that contain the isometry groups of the Platonic solids and the symmetry groups of regular Euclidean tilings. These groups are ubiquitous and well-understood. They are also closely linked to their less understood braided versions known as Artin groups. In this talk, I will introduce the connection of Artin groups to Coxeter groups via hyperplane arrangements. Then I will give a survey of some recent results towards a solution to the word problem for Artin groups using their connection to the geometry and combinatorics Coxeter groups as well as some interesting combinatorial problems that arise along the way.

Prerequisites: I won't assume any knowledge of Coxeter groups or Artin groups, but familiarity with groups will be helpful.

Title: Signed Poset Polytopes

Abstract: Posets can be viewed as subsets of the type-A root system that satisfy certain properties. Geometric objects arising from posets, such as order cones, order polytopes, and chain polytopes, have been widely studied. In 1993, Vic Reiner introduced signed posets, which are subsets of the type-B root system that satisfy the same properties. In this talk, we will explore the analogue of order and chain polytopes in this setting, focusing on the Ehrhart theory of these objects.

Prerequisites: Familiarity with posets and the definition of polytopes. We will attempt to define terms.

Title: Permutations, patterns and permutons

Abstract: Consider a large random permutation. What does it look like? We will answer this question for different classical models of random permutations, such as uniform permutations, pattern-avoiding permutations, Mallows permutations and many others. An appropriate framework to describe the asymptotic behaviour of these pemutons is to use a notion of limiting continuous objects for permutations, called permutons. We will investigate some more sophisticated examples of these objects, such as the skew Brownian permuton. Permutons lead to some nice connections between probability theory and combinatorics, and during the talk, we will investigate some of them. In the last part of the talk, we will also present a new and complementary recent local limit approach for the study of large random permutations. Indeed, permutons are appropriate to describe the ”global shape” of permutations but not the ”finer details''. These are on the contrary encoded by local limits.

Prerequisites: Basic probability

Title: Complex dynamics, Gleason polynomials, and irreducibility.

Abstract: Complex dynamics began with the study of iterating polynomials (with complex coefficients). I’ll introduce the Mandelbrot set and Gleason polynomials, and explain how both are examples of parameter spaces of dynamical systems. I’ll then discuss the notion of irreducibility, as applied to these parameter spaces.