Title: Stress and the Stanley-Reisner Ring
Abstract: I will discuss some connections between classical stress on bar and joint frameworks, a generalization of stress to simplicial complexes, the Stanley-Reisner ring, and a consequent interpretation of the g-theorem for simplicial polytopes.
Title: On Artin's Conjecture for diagonal forms
Abstract: Emil Artin conjectured that any form of degree d in more than d^2 variables with coefficients in a p-adic field K has a nontrivial zero in K. Terjanian provided a counterexample to the conjecture, and many more have been found afterwards. However, in the case of diagonal forms, the result is known to hold for K=\mathbb{Q}_p. The conjecture for diagonal forms over arbitrary p-adic fields remains unproved. We investigate partial results.
Title: Bouquet algebra of toric ideals
Abstract: To any integer matrix A one can associate a toric ideal I_A, whose sets of generators are basic objects in discrete linear optimization, statistics, and graph/hypergraph sampling algorithms. The basic algebraic problem is that of implicitization: given the matrix A, find a set of generators with some given property (minimal, Groebner, Graver, etc.). Then there is a related problem of complexity: how complicated can these generators be? In general, it is known that Graver bases are much more complicated than minimal generators. But there are some classical families of toric ideals where these sets actually agree, providing very nice results on complexity and sharp degree bounds.
This talk is about combinatorial signatures of generating sets of I_A. For the special case when A is a 0/1 matrix, bicolored hypergraphs give the answer. It turns out that such hypergraphs give an intuition for constructing basic building blocks for the general case too. Namely, we introduce the bouquet graph and bouquet ideal of the toric ideal I_A, whose structure determines the Graver basis. This, in turn, leads to a complete characterization of toric ideas for which the following sets are equal: the Graver basis, the universal Groebner basis, any reduced Groebner basis and any minimal generating set. This generalizes many of the classical examples.
Title: Crossing numbers for tropical curves
Abstract: In tropical geometry, curves have both an intrinic side, as metric graphs and an embedded representation in terms of so-called balanced polyhedral complexes. I will discuss the relationship between these two representations. Since most curves can't be embedded in the plane, it is often useful to relax the embedding condition by allowing transverse crossings. A tropical crossing number for a metric graph is defined to be the fewest number of crossings in a planar immersion, and I will give some results on this crossing number.
Title: Tropical plane quartics
Abstract: I will begin with a brief introduction to tropical geometry, and explain how algebraic curves give rise to tropical curves. I will then show that every tropical plane quartic admits 7 families of bitangent lines. This is analogous to the remarkable fact in classical geometry that a smooth plane quartic has exactly 28 bitangent lines. While the proof is purely combinatorial, I will discuss recent developments which suggest that the classical and tropical results are closely related. This is joint work with Matt Baker, Ralph Morrison, Nathan Pflueger, and Qingchun Ren.
Title: A result by Davenport and Lewis on additive equations
Abstract: We will present a result by Davenport and Lewis which states that an additive form with coefficients in $\mathbb{Q}_p$ of degree d in s>d^2 variables has a non-trivial p-adic solution.
Title: Chip-Firing and Tropical Independence, II
Abstract: We continue discussing the basic theory of divisors on graphs, with a primary focus on concrete examples. If time permits, we will describe how these tools are used to provide new proofs of some well-known theorems in algebraic geometry.
Title: Chip-Firing and Tropical Independence
Abstract: We will discuss the basic theory of divisors on graphs, with a primary focus on concrete examples. If time permits, we will describe how these tools are used to provide new proofs of some well-known theorems in algebraic geometry.
Title: Constructing ideals with large projective dimension
Abstract: Projective dimension is a homological measure of the complexity of algebraic objects. Motivated by computational considerations, M. Stillman asked for upper bounds on the projective dimension of homogeneous ideals in polynomial rings, based solely on invariants of the ideal, not of the ambient ring. In this talk, we discuss several constructions that shed some light on what ideal invariants can or cannot be used to bound projective dimension and we give lower bounds on any possible answer to Stillman's question. The talks is based on joint work with Huneke-Mantero-McCullough and Beder-McCullough-Nunez-Snapp-Stone.
