Seminars and Colloquia by Series

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 JungGeorgia 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 JungGeorgia 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 HärkönenGeorgia 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.

Higher connectivity of the Bergman fan

Series
Student Algebraic Geometry Seminar
Time
Monday, November 18, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Kisun LeeGeorgia Tech

The Bergman fan is a tropical linear space with trivial valuations describing a matroid combinatorially as it corresponds to a matroid. In this talk, based on a plenty of examples, we study the definition of the Bergman fan and their subdivisions. The talk will be closed with the recent result of the Maclagan-Yu's paper (https://arxiv.org/abs/1908.05988) that the fine subdivision of the Bergman fan of any matroid is r-1 connected where r is the rank of the matroid.

Cayley-Bacharach Relations and Sums of Squares

Series
Student Algebraic Geometry Seminar
Time
Monday, November 11, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Kevin ShuGeorgia Tech (grad student)

This talk is based on a paper by Grigoriy Blekherman. In most cases, nonnegative polynomials differ from positive polynomials. We will discuss precisely what equations cause these differences, and relate them to the well known Cayley-Bacharach theorem for low degree polynomials.

Nonnegative symmetric polynomials and sums of squares with many variables

Series
Student Algebraic Geometry Seminar
Time
Monday, October 28, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Jose Gabriel Acevedo HabeychGeorgia Tech

Please Note: By using the representation theory of the symmetric group we try to compare, with respect to two different bases of the vector space of symmetric forms, the cones of symmetric nonnegative forms and symmetric sums of squares of a fixed even degree when the number of variables goes to infinity.

Tropical convex hulls of convex sets

Series
Student Algebraic Geometry Seminar
Time
Monday, October 21, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Cvetelina HillGeorgia Tech

This talk is based on work in progress with Sara Lamboglia and Faye Simon. We study the tropical convex hull of convex sets and of tropical curves. Basic definitions of tropical convexity and tropical curves will be presented, followed by an overview of our results on the interaction between tropical and classical convexity. Lastly, we study a tropical analogue of an inequality bounding the degree of a projective variety in classical algebraic geometry; we show a tropical proof of this result for a special class of tropical curves. 

 

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