Speakers, Titles and Abstracts for BAD Math Day XVIII

Abstract: The k-equal arrangement is the collection of subspaces in Rⁿ given by equations of the form x_i1 = x_i2 = ... = x_ik, over all indices 1 ≤ i₁ < i₂ < ... < i_k ≤ n. In this talk, we describe the k-parabolic arrangement, a generalization of the k-equal arrangement for any finite real reflection group. When k = 2, these arrangements correspond to the well-defined Coxeter arrangements, including the Braid arrangement when W is of type A. In 1963, Fadell, Fox, and Neuwirth showed that the complement of the complex braid arrangement is a K(π, 1) space, and that its fundamental group is isomorphic to the pure braid group. Brieskorn (1971) generalized the last result to complexified W Coxeter arrangements by showing that the fundamental group is isomorphic to the pure Artin group of type W. Khovanov (1996) gave a real counterpart to Fadell, Fox and Neuwirth's result when W is of type A (and B) by showing that the complement of the 3-equal arrangement (over R) is a K(π, 1) space, and by giving an algebraic description of its fundamental group. We generalize Khovanov's result and obtain an algebraic description of the fundamental group of the complement of the 3-parabolic arrangement for arbitrary finite reflection group. Our description is a real analogue of Brieskorn's one. We conjecture that for W of any type, the complement of the 3-parabolic arrangement is a K(π, 1) space. This is joint work with Christopher Severs and Jacob White.

Abstract: A fibration between toric varieties, embedded in projective space, can be described by certain fibered polytopes. When the fibration has a projective space as generic fiber the polytope is called a strict Cayley polytope. It turns out that this class of polytopes encodes exceptional geometrical properties of the corresponding toric embeddings. I will present several combinatorial characterizations of Cayley polytopes, all obtained thanks to the interplay between toric geometry and convex geometry. This is partly joint work with A. Dickenstein and R. Piene.

Abstract: I will define a graph on the set of partitions of n, or alternatively on the set of monomial ideals of length n in two variables. I will then present a necessary condition on two partitions for the existence of an edge between those partitions. This is joint work with Diane Maclagan.

Abstract: The primary focus of this talk is to establish the computational complexity of satisfying certain sets of neighborhood conditions in graphs with various properties. Motivated by Ehrenfeucht−Fraïssé games for monadic second-order logic, we study the complexity of problems of the following sort: When is there a vertex-colored graph with a given set of neighborhoods? More precisely, fix a radius ρ and let N(G) be the set of isomorphism classes of ρ-neighborhoods of vertices of G, where G is a graph whose vertices are colored (not necessarily properly) by colors from a fixed finite palette. Given a set S of pointed, colored graphs, when is there a graph G with N(G) = S? Or N(G) ⊂ S? What if G is forced to be finite, or connected, or both? If the neighborhoods are unrestricted, all these problems are recursively unsolvable; this follows from the work of Bulitko but we give a simpler, independent proof in the colored case. In contrast, when the neighborhoods are cycle free, we show that all the problems are in the class P. Surprisingly, if G is required to be a regular (and thus infinite) tree, the realization problem is NP-complete (for degree 3 or higher). Finally, if time permits, we extend the former results to a slightly more general class of neighborhoods called bouquet neighborhoods.

Abstract: I will describe some combinatorial implications of interpreting the elements of a monomial ideal as differential equations, and how these facts may be used to obtain results about hypergeometric systems of differential equations. This is joint work (in progress) with Christine Berkesch.

Abstract: I will present a limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples of this special class of graphs which have an underlying variable, very useful in applications. Graphs have important applications in modern systems biology and social sciences. Edges are created between interacting genes or people who know each other. However graphs are not objects which are naturally amenable to simple statistical analyses, there is no natural average graph for instance. Being able to predict or replace a graph by hidden (statisticians call them latent) real variables has many advantages. This talk studies such a class of graphs, that sits within the larger class of interval graphs itself a subset of intersection graphs.
This is joint work with Persi Diaconis and Svante Janson.