Seminars and Colloquia by Series

Introduction to Kajiwara-Payne Tropicalization I

Series
Student Algebraic Geometry Seminar
Time
Friday, October 2, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1601305339104?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d
Speaker
Trevor Gunn – Georgia Tech

The goal of this talk is to present a summary of Sam Payne's 2009 paper "Analytification is the limit of all tropicalizations" (Math. Res. Lett. 16, no. 3 543–556). We will introduce Berkovich analytic spaces, tropicalization of projective varieties, and tropicalization of closed subvarieties of toric varieties, as well as the connections between these concepts. We will try to present many examples.

Note: Part I will focus on tropicalization of affine varieties and Berkovich analytic spaces, Part II will focus on tropicalization of toric varieties and discuss Sam Payne's theorem.

Noetherian operators and primary decomposition

Series
Student Algebraic Geometry Seminar
Time
Friday, September 25, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Marc Härkönen – Georgia Tech –

Please Note: Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1600608874868?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d

Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic geometry, taking a step toward the goal of numerical primary decomposition. This is joint work with Justin Chen, Robert Krone and Anton Leykin.

Tropical convex hulls of polyhedral sets

Series
Student Algebraic Geometry Seminar
Time
Friday, September 18, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Cvetelina Hill – Georgia Tech –

Please Note: Microsoft Teams Link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1600180661071?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d

Abstract: In this talk we introduce basic definitions in tropical convexity, and give an overview of some of the main results. The focus will then shift to joint work with Faye Pasley Simon and Sara Lamboglia on the interaction between tropical and ordinary convex hull. We will introduce results including the characterization of tropically convex polyhedra and give a lower bound on the degree of a fan tropical curve using only tropical techniques. The talk will conclude with some more recent results and several open questions.   

TBA by Tianyi Zhang

Series
Student Algebraic Geometry Seminar
Time
Friday, September 11, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Tianyi Zhang – Georgia Tech

Please Note: Link to meeting: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1599679148202?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d

Hyperbolic polynomial and its application to locally PSD matrices

Series
Student Algebraic Geometry Seminar
Time
Friday, September 4, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Shengding Sun – Georgia Tech –

Please Note: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1598721101106?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d

The definition of hyperbolic polynomials stems from stable polynomials, with many interesting properties related to convex geometry and optimization, including the construction of hyperbolicity cone. We will discuss some of these results and mention the application to locally PSD matrices.

Computation of invariants and Hankel index on a variety of almost minimal degree

Series
Student Algebraic Geometry Seminar
Time
Monday, March 2, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jaewoo Jung – Georgia Tech –

We have seen that Hankel index of varieties can be controlled by some invariants such as $$N_{2,p}$$ or p-base point free property. Also, we know that the Hankel index of (a linear join of) variety of minimal degree is infinity (and all invariant above are same as infinity). As next case, I will share some computations of invariants on a variety that projecting rational normal curve away from a point (which is a variety of almost minimal degree).

Hankel index of a projection of rational normal curve.

Series
Student Algebraic Geometry Seminar
Time
Monday, February 24, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jaewoo Jung – Georgia Tech –

The dual of the cone of non-negative quadratics (on a variety) is included in the dual of the cone of sums of squares. Moreover, all (points which span) extreme rays of the dual cone of non-negative quadratics is point evaluations on real points of the variety. Therefore, we are interested in extreme rays of the dual cone of sums of squares which do not come from point evaluations. The dual cone of sums of squares on a variety is called the Hankel spectrahetron and the smallest rank of extreme rays which do not come from point evaluations is called Hankel index of the variety. In this talk, I will introduce some algebraic (or geometric) properties which control the Hankel index of varieties and compute the Hankel index of rational curves obtained by projecting rational normal curve away from a point (which has almost minimal degree).

Dual spaces and Noetherian operators

Series
Student Algebraic Geometry Seminar
Time
Monday, February 17, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marc Härkönen – Georgia Tech –

Noetherian operators are a set of differential operators that encode the scheme structure of a primary ideal. We propose a framework for studying primary ideals numerically by using a combination of witness sets and Noetherian operators. We will also present a method for computing Noetherian operators using numerical data. 

The first step in the theory of Noetherian operators are the Macaulay dual spaces. Indeed, for an ideal that is primary over a maximal ideal corresponding to a rational point, the generators of the dual space are a valid set of Noetherian operators. We will start by presenting basic ideas, results and algorithms in the classical dual space theory, and then revisit some of these ideas in the context of Noetherian operators.

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