### Fall recess

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

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

- Series
- Algebra Seminar
- Time
- Wednesday, October 2, 2019 - 10:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Farbod Shokrieh – Unviersity of Washington – farbod@uw.edu

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).

- Series
- Algebra Seminar
- Time
- Tuesday, October 1, 2019 - 13:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Kisun Lee – Georgia Tech – kisunlee@gatech.edu

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.

- Series
- Algebra Seminar
- Time
- Tuesday, September 3, 2019 - 13:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Josephine Yu – Georgia Tech – jyu@math.gatech.edu

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.

- Series
- Algebra Seminar
- Time
- Tuesday, August 27, 2019 - 13:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Steven Creech & Derek Wu – Georgia Tech – screech6@gatech.edu

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.

- Series
- Algebra Seminar
- Time
- Monday, April 22, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Joe Kileel – Princeton University – jkileel@math.princeton.edu

This talk will be about polynomial decompositions that are relevant in machine learning. I will start with the well-known low-rank symmetric tensor decomposition, and present a simple new algorithm with local convergence guarantees, which seems to handily outperform the state-of-the-art in experiments. Next I will consider a particular generalization of symmetric tensor decomposition, and apply this to estimate subspace arrangements from very many, very noisy samples (a regime in which current subspace clustering algorithms break down). Finally I will switch gears and discuss representability of polynomials by deep neural networks with polynomial activations. The various polynomial decompositions in this talk motivate questions in commutative algebra, computational algebraic geometry and optimization. The first part of this talk is joint with Emmanuel Abbe, Tamir Bendory, Joao Pereira and Amit Singer, while the latter part is joint with Matthew Trager.

- Series
- Algebra Seminar
- Time
- Tuesday, April 16, 2019 - 12:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Kathlén Kohn – ICERM and University of Oslo – kathlen_kohn@brown.edu

Wachspress defined barycentric coordinates on polygons in 1975. Warren generalized his construction to higher dimensional polytopes in 1996. In contrast to the classical case of simplices, barycentric coordinates on other polytopes are not unique. So the coordinates defined by Warren are now commonly known as Wachspress coordinates. They are used in a variety of applications, such as geometric modeling.

We connect the constructions by Warren and Wachspress by proving the conjecture that there is a unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. This is the adjoint polynomial introduced by Warren. Our formulation is the natural generalization of Wachspress' original idea.

The algebraic geometry of the map defined by the Wachspress coordinates was studied in the case of polygons by Irving and Schenk in 2014. We extend their results to higher dimensional polytopes. In particular, we show that the image of this Wachspress map is the projection from the image of the adjoint. For three-dimensional polytopes, we show that their adjoints are adjoints of K3- or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.

- Series
- Algebra Seminar
- Time
- Monday, April 15, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Kalina Mincheva – Yale University – kalina.mincheva@yale.edu

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a prime (tropical) ideal is either empty or consists of a single point. This is joint work with D. Joó.

- Series
- Algebra Seminar
- Time
- Monday, April 8, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Mengyuan Zhang – University of California, Berkeley – myzhang@berkeley.edu

In this talk we discuss the following problem due to Peskine and Kollar: Let E be a flat family of rank two bundles on P^n parametrized by a smooth variety T. If E_t is isomorphic to O(a)+O(b) for general t in T, does it mean E_0 is isomorphic to O(a)+O(b) for a special point 0 in T? We construct counter-examples in over P^1 and P^2, and discuss the problem in P^3 and higher P^n.

- Series
- Algebra Seminar
- Time
- Monday, April 1, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Maria Angelica Cueto – Ohio State University – cueto.5@osu.edu

The classical statement that there are 27 lines on every smooth cubic surface in $\mathbb{P}^3$ fails to hold under tropicalization: a tropical cubic surface in $\mathbb{TP}^3$ often contains infinitely many tropical lines. This pathology can be corrected by reembedding the cubic surface in $\mathbb{P}^{44}$ via the anticanonical bundle.

Under this tropicalization, the 27 classical lines become an arrangement of metric trees in the boundary of the tropical cubic surface in $\mathbb{TP}^{44}$. A remarkable fact is that this arrangement completely determines the combinatorial structure of the corresponding tropical cubic surface. In this talk, we will describe their metric and topological type as we move along the moduli space of tropical cubic surfaces. Time permitting, we will discuss the matroid that emerges from their tropical convex hull.

This is joint work with Anand Deopurkar.

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