Title: Symbolic Generic Initial Systems and Matroid Configurations

Abstract: We survey dissertation work of my academic sister Sarah Mayes-Tang (2013 Ph.D.). As time allows, we aim towards two objectives. First, in terms of combinatorial algebraic geometry we weave a narrative from linear star configurations in projective spaces to matroid configurations therein, the latter being a recent development investigated by the quartet of Geramita -- Harbourne -- Migliore -- Nagel. Second, we pitch a prospectus for further work in follow-up to both Sarah's work and the matroid configuration investigation. 

Title: Betti Number Maximization and Bounds

Abstract: Think about all ideals with your favorite Hilbert function. What are the largest possible Betti numbers for those ideals? Is there an ideal whose Betti numbers attain maximums? Now, think about your favorite Hilbert polynomial. Let's ask the same questions. These questions have been answered by Bigatti and Hulett (1993) and Caviglia and Murai (2010). We will talk about other, related, families of ideals that we could ask these questions about.

Title: Hyperplane Arrangements and Compactifying the Milnor Fiber

Abstract: Milnor fibers are invariants that arise in the study of hypersurface singularities. A major open conjecture predicts that for hyperplane arrangements, the Betti numbers of the Milnor fiber depend only on the combinatorics of the arrangement. I will discuss how tropical geometry can be used to study related invariants, the virtual Hodge numbers of a hyperplane arrangement's Milnor fiber. This talk is based on joint work with Max Kutler.

Title: Asymptotic Syzygies for Products of Projective Space

Abstract: I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.

Title: Generalized minimum distance functions for schemes and linear codes

Abstract In coding theory, a linear code is a subspace of a finite dimensional vector space. To a linear code one can associate a set of points in projective space. If these points are distinct, then the Hamming distance of the code can be computed based on the geometry of the points. 

Motivated by these considerations, we introduce commutative algebraic generalizations for the notion of Hamming distance, which are better suited for working with non-reduced schemes. We describe the properties of these generalized minimum distance functions, as well as bounding them in the spirit of the classical singleton bound and by means of a Cayley Bacharach-type conjecture.

This talk is based on joint work with Susan Cooper, Stefan Tohaneanu, Maria Vaz Pinto and Rafael Villarreal.

Title: Quantitative Properties of Ideals arising from Hierarchical Models

Abstract: We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented.  One goal is to study the invariance properties of families of ideals arising from hierarchical models with varying parameters. We will present classes of examples where we have information about an equivariant Hilbert series. This is joint work with Uwe Nagel.

Title: Elements of an infinite product of finite fields coming from geometry

Abstract: The ring A is defined to be the quotient of the product the prime finite fields, modulo those elements with only finitely many non-zero coordinates. In this talk, I will describe some arithmetically interesting elements of A coming from algebra and geometry, and I will explain how these elements are analogous to integrals of algebraic differential forms.

Tropical curves, graph complexes, and the cohomology of M_g

Joint with Søren Galatius and Sam Payne. The cohomology ring H^*(M_g,Q)
of the moduli space of curves of genus g is not fully understood, even
for g small. For example, in the 1980s Harer-Zagier showed that the
Euler characteristic (up to sign) grows super-exponentially with g
---yet most of this cohomology is not explicitly known. I will explain
how we obtained new results on the rational cohomology of moduli
spaces of curves of genus g, via Kontsevich's graph complexes and the
moduli space of tropical curves.

Title: Tropical geometry and amoebas in matrix groups

Abstract: We start with the basic and remarkable notions of amoeba and tropical variety of a subvariety Y in the algebraic torus $(\mathbb{C} \setminus \{0\})^n$. We will demonstrate how these notions lead us to finding a minimal compactification of Y (usually referred to as "tropical compactification"). In the course of this we will introduce the notion of a toric variety as well. Next, I will discuss recent results about extending these notions from the algebraic torus to other matrix groups such as $GL(n, \mathbb{C})$. Some interesting linear algebra, such as singular value decomposition and Smith normal form, pops up. For the most part, I assume only basic background from algebra and geometry and the talk should be understandable to a general math crowd. There will be a nonzero number of pictures!

Title: Equivalence of Classical and Quantum Codes

Abstract: In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture.