## Seminars and Colloquia by Series

### Oriented Matroids and Combinatorial Neural Codes

Series
Algebra Seminar
Time
Monday, February 17, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zvi RosenFlorida Atlantic University

A combinatorial neural code is convex if it arises as the intersection pattern of convex open subsets of Euclidean space. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to feasibility and complexity. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.

### The foundation of a matroid

Series
Algebra Seminar
Time
Monday, February 3, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matt BakerGeorgia Tech

Originally introduced independently by Hassler Whitney and Takeo Nakasawa, matroids are a combinatorial way of axiomatizing the notion of linear independence in vector spaces. If $K$ is a field and $n$ is a positive integer, any linear subspace of $K^n$ gives rise to a matroid; such matroid are called representable over $K$. Given a matroid $M$, one can ask over which fields $M$ is representable. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid $M$, which governs the representations of $M$ over all partial fields. Unfortunately, most matroids (asymptotically 100%, in fact) are not representable over any partial field, and in this case, the universal partial field gives no information.

Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of $M$ is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. Pastures form a natural class of field-like objects within Lorscheid's theory of ordered blueprints, and they have desirable categorical properties (e.g., existence of products and coproducts) that make them a natural context in which to study algebraic invariants of matroids. The foundation of a matroid $M$ represents the functor taking a pasture $F$ to the set of rescaling equivalence classes of $F$-representations of $M$; in particular, $M$ is representable over a pasture $F$ if and only if there is a homomorphism from the foundation of $M$ to $F$. (In layman's terms, what we're trying to do is recast as much as possible of the theory of matroids and their representations in functorial Grothendieck-style'' algebraic geometry, with the goal of gaining new conceptual insights into various phenomena which were previously understood only through lengthy case-by-case analyses and ad hoc computations.)

As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for ternary matroids (matroids representable over the field of three elements). The proof of this classification theorem relies crucially on Tutte's celebrated Homotopy Theorem.

### Brill–Noether theory of Prym varieties

Series
Algebra Seminar
Time
Monday, January 13, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yoav LenGeorgia Tech

The talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian varieties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concerning Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci.

### Free resolutions of function classes via order complexes

Series
Algebra Seminar
Time
Tuesday, November 19, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin ChenGeorgia Institute of Technology

Function classes are collections of Boolean functions on a finite set. Recently, a method of studying function classes via commutative algebra, by associating a squarefree monomial ideal to a function class, was introduced by Yang. I will describe this connection, as well as some free resolutions and Betti numbers for these ideals for an interesting collection of function classes, corresponding to intersection-closed posets. This is joint work with Chris Eur, Greg Yang, and Mengyuan Zhang.

### Positively Hyperbolic Varieties

Series
Algebra Seminar
Time
Tuesday, November 12, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josephine YuGeorgia Tech

A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points.  Stable polynomials have close relations to matroids and hyperbolic programming.  We will discuss a generalization of stability to algebraic varieties of codimension larger than one.  They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.

### Tropical curves of hyperelliptic type

Series
Algebra Seminar
Time
Tuesday, November 5, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel CoreyUniversity of Wisconsin

We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. Using the tropical Torelli theorem (due to Caporaso and Viviani), this characterization may be phrased in terms of 3-edge connectiviations. We show that being of hyperelliptic type is independent of the edge lengths and is preserved when passing to genus ≥2 connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.

### Tropical covers with an abelian group action

Series
Algebra Seminar
Time
Tuesday, October 29, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitry ZakharovCentral Michigan University

Given a graph X and a group G, a G-cover of X is a morphism of graphs X’ --> X together with an invariant G-action on X’ that acts freely and transitively on the fibers. G-covers are classified by their monodromy representations, and if G is a finite abelian group, then the set of G-covers of X is in natural bijection with the first simplicial cohomology group H1(X,G).

In tropical geometry, we are naturally led to consider more general objects: morphisms of graphs X’ --> X admitting an invariant G-action on X’, such that the induced action on the fibers is transitive, but not necessarily free. A natural question is to classify all such covers of a given graph X. I will show that when G is a finite abelian group, a G-cover of a graph X is naturally determined by two data: a stratification S of X by subgroups of G, and an element of a cohomology group H1(X,S) generalizing the simplicial cohomology group H1(X,G). This classification can be viewed as a tropical version of geometric class field theory, and as an abelianization of Bass--Serre theory.

I will discuss the realizability problem for tropical abelian covers, and the relationship between cyclic covers of a tropical curve C and the corresponding torsion subgroup of Jac(C). The realizability problem for cyclic covers of prime degree turns out to be related to the classical nowhere-zero flow problem in graph theory.

Joint work with Yoav Len and Martin Ulirsch.

### The Mori Dream Space property for blow-ups of projective spaces at points and lines

Series
Algebra Seminar
Time
Tuesday, October 22, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zhuang HeNortheastern University

Mori Dream Spaces are generalizations of toric varieties and, as the name suggests, Mori's minimal model program can be run for every divisor. It is known that for n5, the blow-up of Pn at r very general points is a Mori Dream Space iff rn+3. In this talk we proceed to blow up points as well as lines, by considering the blow-up X of P3 at 6 points in very general position and all the 15 lines through the 6 points. We find that the unique anticanonical section of X is a Jacobian K3 Kummer surface S of Picard number 17. We prove that there exists an infinite-order pseudo-automorphism of X, whose restriction to S is one of the 192 infinite-order automorphisms constructed by Keum.  A consequence is that there are infinitely many extremal effective divisors on X; in particular, X is not a Mori Dream Space. We show an application to the blow-up of Pn (n3) at (n+3) points and certain lines.  We relate this pseudo-automorphism to the structure of the birational automorphism group of P3. This is a joint work with Lei Yang.

### The moduli space of matroids

Series
Algebra Seminar
Time
Wednesday, October 16, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Oliver LorscheidInstituto Nacional de Matematica Pura e Aplicada (IMPA)

Matroids are combinatorial gadgets that reflect properties of linear algebra in situations where this latter theory is not available. This analogy prescribes that the moduli space of matroids should be a Grassmannian over a suitable base object, which cannot be a field or a ring; in consequence usual algebraic geometry does not provide a suitable framework. In joint work with Matt Baker, we use algebraic geometry over F1, the so-called field with one element, to construct such moduli spaces. As an application, we streamline various results of matroid theory and find simplified proofs of classical theorems, such as the fact that a matroid is regular if and only if it is binary and orientable.

We will dedicate the first half of this talk to an introduction of matroids and their generalizations. Then we will outline how to use F1-geometry to construct the moduli space of matroids. In a last part, we will explain why this theory is so useful to simplify classical results in matroid theory.

### Mordell-Weil rank jumps and the Hilbert property

Series
Algebra Seminar
Time
Wednesday, October 16, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker