Seminars and Colloquia by Series

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.

Polynomial Decompositions in Machine Learning

Series
Algebra Seminar
Time
Monday, April 22, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe KileelPrinceton University

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.

Projective geometry of Wachspress coordinates

Series
Algebra Seminar
Time
Tuesday, April 16, 2019 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kathlén KohnICERM and University of Oslo
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.

Prime tropical ideals

Series
Algebra Seminar
Time
Monday, April 15, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kalina MinchevaYale University

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.

Limits of split rank two bundles on P^n

Series
Algebra Seminar
Time
Monday, April 8, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mengyuan ZhangUniversity of California, Berkeley

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.

Combinatorics of line arrangements on tropical cubic surfaces

Series
Algebra Seminar
Time
Monday, April 1, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Maria Angelica CuetoOhio State University

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.

Gattaca

Series
Algebra Seminar
Time
Saturday, March 30, 2019 - 14:00 for 8 hours (full day)
Location
Atlanta
Speaker
Georgia Tech Tropical Arithmetic and Combinatorial Algebraic-geometryGeorgia Institute of Technology

This is a two day conference (March 30-31) to be held at Georgia Tech on algebraic geometry and related areas. We will have talks by Sam Payne, Eric Larson, Angelica Cueto, Rohini Ramadas, and Jennifer Balakrishnan. See https://sites.google.com/view/gattaca/home for more information.

Cohen-Macaulayness of invariant rings is determined by inertia groups

Series
Algebra Seminar
Time
Monday, March 25, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ben Blum-SmithNYU

If a finite group $G$ acts on a Cohen-Macaulay ring $A$, and the order of $G$ is a unit in $A$, then the invariant ring $A^G$ is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On the other hand, if the order of $G$ is not a unit in $A$ then the Cohen-Macaulayness of $A^G$ is a delicate question that has attracted research attention over the last several decades, with answers in several special cases but little general theory. In this talk we show that the statement that $A^G$ is Cohen-Macaulay is equivalent to a statement quantified over the inertia groups for the action of G$ on $A$ acting on strict henselizations of appropriate localizations of $A$. In a case of long-standing interest—a permutation group acting on a polynomial ring—we show how this can be applied to find an obstruction to Cohen-Macaulayness that allows us to completely characterize the permutation groups whose invariant ring is Cohen-Macaulay regardless of the ground field. This is joint work with Sophie Marques.

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