Seminars and Colloquia by Series

Finite Automata and Transfer Matrices

Series
Algebra Student Seminar
Time
Friday, January 28, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Teams
Speaker
Trevor GunnGeorgia Tech

This talk is a primer on solving certain kinds of counting problems through regular languages, finite automata and transfer matrices. Example problems: count the number of binary strings that contain "0110", count the number of binary strings that contain 0, 1, 2,... copies of "0110," a derivation of the negative binomial distribution function.

The only requirements for this talk is a basic familiarity with directed graphs, matrices and generating functions.

Teams Link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1643050072413?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%22dc6c6c03-84d2-497a-95c0-d85af9cbcf28%22%7d

Apolarity for quadratic forms

Series
Algebra Student Seminar
Time
Friday, November 19, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jaewoo JungGeorgia Tech

Recall that, for a variety $X$ in a projective space $\mathbb{P}^d$, the $X$-rank of a point $p\in \mathbb{P}^d$ is the least number of points of $X$ whose span contains the point $p$. Studies about $X$-ranks include some well-known and important results about various tensor ranks. For example, 

  • the rank of tensors is the rank with respect to Segre varieties,
  • the rank of symmetric tensors, i.e. Waring rank, is the rank with respect to Veronese embeddings, and
  • the rank of anti-symmetric tensors is the rank with respect to Grassmannians in its Plücker embedding.  

In this talk, we focus on ranks with respect to Veronese embeddings of a projective line $\mathbb{P}^1$. i.e. symmetric tensor ranks of binary forms. We will discuss how to associate points in $\mathbb{P}^d$ with binary forms and I will introduce apolarity for binary forms which gives an effective method to study Waring ranks of binary forms. We will discuss various ranks on the Veronese embedding and some results on the ranks.

Introduction to Diophantine Approximation with Applications to Arithmetic Geometry

Series
Algebra Student Seminar
Time
Friday, November 5, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skile 005
Speaker
Ian LewisGeorgia Tech

One question addressed in the field of Diophantine approximation is precisely quantifying how many "good" approximations an algebraic number has by rational numbers. This is answered most soundly by a 1955 theorem of Klaus Roth. In this talk, I will cover this theorem, some related results and hint at how it can be used to bound the number of rational solutions to curves.

Representation of Delta-matroids and the spinor varieties

Series
Algebra Student Seminar
Time
Friday, October 29, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tong JinGeorgia Tech

Delta-matroids are natural generalizations of matroids in which we replace each difference operator by the symmetric difference operator in the basis exchange axiom. I will briefly introduce (even) Delta-matroids and their representability. I will also discuss how they are related to the spinor varieties. 

Nonnegative Quadratics over Stanley Reisner Varieties

Series
Algebra Student Seminar
Time
Friday, October 22, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kevin ShuGeorgia Tech

Nonnegative polynomials are of fundamental interest in the field of real algebraic geometry. We will discuss a model of nonnegative polynomials over an interesting class of algebraic varieties which have potential applications in optimization theory. In particular, we will discuss connections between this subject and algebraic topology and the geometry of simplicial complexes.

Characterizing multigraded regularity on products of projective spaces

Series
Algebra Student Seminar
Time
Friday, October 15, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mahrud SayrafiUniversity of Minnesota

Motivated by toric geometry, Maclagan-Smith defined the multigraded Castelnuovo-Mumford regularity for sheaves on a simplicial toric variety. While this definition reduces to the usual definition on a projective space, other descriptions of regularity in terms of the Betti numbers, local cohomology, or resolutions of truncations of the corresponding graded module proven by Eisenbud and Goto are no longer equivalent. I will discuss recent joint work with Lauren Cranton Heller and Juliette Bruce on generalizing Eisenbud-Goto's conditions to the "easiest difficult" case, namely products of projective spaces, and our hopes and dreams for how to do the same for other toric varieties.

Tropical intersection theory I

Series
Algebra Student Seminar
Time
Friday, September 24, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Trevor GunnGeorgia Tech

This is the first part of a two part introduction to tropical intersection theory. The first part will review some of the classical theory. We will mostly focus on the parts of the classical theory that have counterparts in the tropical theory but we may also cover some elements of the classical theory which do not have tropical analogues.

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The algebra of linear PDE

Series
Algebra Student Seminar
Time
Friday, September 17, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005, or ONLINE
Speaker
Marc HärkönenGeorgia Tech

Please Note: Online link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1631746973297?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d

This talk is meant to be a gentle introduction to the algebraic theory of linear PDE with constant coefficients. We will present the connection between submodules of free modules of polynomial rings and solution sets of PDEs, and establish certain results relating analytical properties of solutions with algebraic properties of polynomial modules. We will also review classical spaces of functions in distribution theory and Fourier analysis.

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