## Seminars and Colloquia by Series

### 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

Let X be an elliptic surface with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.

### Fall recess

Series
Algebra Seminar
Time
Tuesday, October 15, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Speaker
No seminar.

### Heights and moments of abelian varieties

Series
Algebra Seminar
Time
Wednesday, October 2, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Farbod ShokriehUnviersity of Washington

We give a formula relating various notions of heights of abelian varieties. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener, and it extends the Faltings-Silverman formula for elliptic curves. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a key role.   Based on joint works with Robin de Jong (Leiden).

### Certifying solutions to a square analytic system

Series
Algebra Seminar
Time
Tuesday, October 1, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kisun LeeGeorgia Tech

In this talk, we discuss about methods for proving existence and uniqueness of a root of a square analytic system in a given region. For a regular root, Krawczyk method and Smale's $\alpha$-theory are used. On the other hand, when a system has a multiple root, there is a separation bound isolating the multiple root from other roots. We define a simple multiple root, a multiple root whose deflation process is terminated by one iteration, and establish its separation bound. We give a general framework to certify a root of a system using these concepts.

### Positively Hyperbolic Varieties

Series
Algebra Seminar
Time
Tuesday, September 3, 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.

### Prym–Brill–Noether loci of special curves

Series
Algebra Seminar
Time
Tuesday, August 27, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Steven Creech & Derek WuGeorgia Tech

Prym varieties are a class of abelian varieties that arise from double covers of tropical or algebraic curves. The talk will revolve around the Prym--Brill--Noether locus, a subvariety determined by divisors of a given rank. Using a connection to Young tableaux, we determine the dimensions of these loci for certain tropical curves, with applications to algebraic geometry. Furthermore, these loci are always pure dimensional and path connected. Finally, we compute the first homologies of the Prym--Brill--Noether loci under certain conditions.