Seminars and Colloquia by Series

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 Lee – Georgia 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 Shu – Georgia 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 Habeych – Georgia 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 Hill – Georgia 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. 

 

The foundation of a matroid

Series
Student Algebraic Geometry Seminar
Time
Monday, October 7, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles
Speaker
Tianyi Zhang – GA Tech –

Foundation is a powerful tool to understand the representability of matroids. The foundation of a matroid is a pasture which is an algebraic structure genrealize the field. I will briefly introduce matroids, algebraic structures (especially pastures) and matroid representability. I will also give some examples on how foundation works in representation of matroids.

The essential variety and degrees of minimal problems

Series
Student Algebraic Geometry Seminar
Time
Monday, September 30, 2019 - 13:15 for 1 hour (actually 50 minutes)
Location
Skiles
Speaker
Tim Duff – GA Tech

It is a fundamental problem in computer vision to describe the geometric relations between two or more cameras that view the same scene -- state of the art methods for 3D reconstruction incorporate these geometric relations in a nontrivial way. At the center of the action is the essential variety: an irreducible subvariety of P^8 of dimension 5 and degree 10 whose homogeneous ideal is minimal generated by 10 cubic equations. Taking a linear slice of complementary dimension corresponds to solving the "minimal problem" of 5 point relative pose estimation. Viewed algebraically, this problem has 20 solutions for generic data: these solutions are elements of the special Euclidean group SE(3) which double cover a generic slice of the essential variety. The structure of these 20 solutions is governed by a somewhat mysterious Galois group (ongoing work with Regan et. al.)

We may ask: what other minimal problems are out there? I'll give an overview of work with Kohn, Pajdla, and Leykin on this question. We have computed the degrees of many minimal problems via computer algebra and numerical methods. I am inviting algebraic geometers at large to attack these problems with "pen and paper" methods: there is still a wide class of problems to be considered, and the more tools we have, the better.

The Jacobian Conjecture

Series
Student Algebraic Geometry Seminar
Time
Monday, September 23, 2019 - 13:15 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Stephen McKean – Georgia Tech –

The Jacobian Conjecture is a famous open problem in commutative algebra and algebraic geometry. Suppose you have a polynomial function $f:\mathbb{C}^n\to\mathbb{C}^n$. The Jacobian Conjecture asserts that if the Jacobian of $f$ is a non-zero constant, then $f$ has a polynomial inverse. Because the conjecture is so easy to state, there have been many claimed proofs that turned out to be false. We will discuss some of these incorrect proofs, as well as several correct theorems relating to the Jacobian Conjecture.

Gram spectrahedra

Series
Student Algebraic Geometry Seminar
Time
Monday, September 16, 2019 - 13:15 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Jaewoo Jung – Georgia Tech –

The structure of sums-of-squares representations of (nonnegative homogeneous) polynomials is one interesting subject in real algebraic geometry. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. In this talk, I will introduce Gram spectrahedron, connection to toric variety, a new result that if a variety $X$ is arithmetically Cohen-Macaulay and a linearly normal variety of almost minimal degree (i.e. $\deg(X)=\text{codim}(X)+2$), then every sum of squares on $X$ is a sum of $\dim(X)+2$ squares.

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