Speakers, Titles and Abstracts for BAD Math Day 22

Anton Dochtermann
Stanford University
Syzygies and Subdivisions
Abstract: The study of minimal free resolutions of ideals is a classical topic in algebra and geometry, but providing explicit descriptions of these complexes for even very simple ideals is a challenging task. In the case of monomial ideals, one approach is through the use of `cellular resolutions', where the generators, relations, and higher dimensional syzygies of an ideal are encoded in the facial structure of a CW complex. We will describe some methods of constructing cellular resolutions from subdivisions of dilated simplices, incorporating methods from discrete geometry/topology. We obtain new minimal cellular resolutions of certain classes of monomial ideals, including Borel fixed ideals and initial ideals of certain blowup algebras. These provide an interesting connection between the algebraic properties of the complex and the combinatorial properties of the subdivision, and also generalize and unify some constructions from the literature including the well-known Eliahou-Kervaire resolution of a stable ideal as well as the recently studied resolutions of so-called cointerval ideals.

This includes joint work with Alex Engstrom, Michael Joswig, Fatemeh Mohammadi, and Raman Sanyal.

Daniel Erman
Stanford University
Boij-Söderberg Theory
Abstract: Boij-Söderberg theory describes the structure of two types of objects: free resolutions over the polynomial ring, and the cohomology of coherent sheaves on projective space. Two convex cones that satisfy a twisted duality lie at the heart of the theory. Many open questions remain about the convex geometry of these two cones. I will outline the basics of Boij-Söderberg theory, and then I will discuss some of these open questions.: TBA

Steve Klee
University of California, Davis
Centrally Symmetric Manifolds with Few Vertices
Abstract: A simplicial complex is \textit{centrally symmetric} if it admits a free involution. We will present a construction of a centrally symmetric triangulation of the product of spheres S^i times S^{d-i-2} on 2d vertices for all pairs of nonnegative integers i and d with 0 <= i <= d-2. The constructed complex admits a vertex-transitive action by a group of order 4d. The existence of such complexes was conjectured by Lutz and Sparla in the case that i = d-2-i. The crux of this construction is a definition of a certain manifold with boundary, B(i,d), whose combinatorial structure is easy to analyze, and whose boundary complex gives the desired triangulation of S^i times S^{d-i-2}.

Jeffrey C. Lagarias
University of Michigan/MSRI
Packing Space with Regular Tetrahedra
Abstract: The problem of determining the densest packing of space by congruent regular tetrahedra has a long history. It starts with Aristotle's assertion that regular tetrahedra fill space, and continues through its appearance in Hilbert's 18-th Problem. This talk describes its history and many recent results obtained on this problem, including contributions from physicists, chemists and materials scientists. The current record for packing density is held by my former graduate student Elizabeth Chen, jointly with Michael Engel and Sharon Glotzer.

Dan Romik
University of California, Davis
Alternating Sign Matrices: Some Open, Recently Closed and Not-so-recently Closed Problems
Abstract: An alternating sign matrix is a square matrix of 0's, (+1)'s and (-1)'s such that in every row and every column the non-zero entries sum to 1 and appear with alternating signs. Alternating sign matrices have fascinated algebraic combinatorialists and statistical physicists since their discovery by Mills, Robbins and Rumsey in the early 1980's, and even today are the focus of significant research. In this talk I will give a brief introduction to their study and recount some of the beautiful results proved about them -- covering various aspects such as enumeration results of Zeilberger and others and the beautiful recent proof of the Razumov-Stroganov conjecture by Cantini and Sportiello -- and some equally beautiful conjectures still waiting to be proved.

Kelli Talaska
University of California, Berkeley
Trek Separation in Gaussian Graphical Models
Abstract: This talk will explore a connection between combinatorics and algebraic statistics. In particular, we will look at Gaussian graphical models, whose covariance matrices can be given in terms of certain path families called treks. Inspired by classical results in algebraic combinatorics, we develop a graph-theoretic criterion for determining the rank of a submatrix of the covariance matrix. (No prior knowledge in statistics will be assumed.) This is based on joint work with Seth Sullivant and Jan Draisma.