Seminars and Colloquia by Series

Introduction to differential operator rings

Series
Student Algebraic Geometry Seminar
Time
Friday, February 2, 2018 - 10:10 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Marc Härkönen – Georgia Tech –
Differential operator rings can be described as polynomial rings over differential operators. We will study two of them: first the relatively simple ring of differential operators R with rational function coefficients, and then the more complicated ring D with polynomial coefficients, or the Weyl algebra. It turns out that these rings are non-commutative because of the way differential operators act on smooth functions. Despite this, with a bit of work we can show properties similar to the regular polynomial rings, such as division, the existence of Gröbner bases, and Macaulay's theorem. As an example application, we will describe the holonomic gradient descent algorithm, and show how it can be used to efficiently solve computationally heavy problems in statistics.

Automatic Sequences and Curves Over Finite Fields

Series
Student Algebraic Geometry Seminar
Time
Friday, January 26, 2018 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Trevor Gunn – Georgia Tech –
We will first give a quick introduction to automatic sequences. We will then outine an algebro-geometric proof of Christol's theorem discovered by David Speyer. Christol's theorem states that a formal power series f(t) over GF(p) is algebraic over GF(p)(t) if and only if there is some finite state automaton such that the n-th coefficent of f(t) is obtained by feeding in the base-p representation of n into the automaton. Time permitting, we will explain how to use the Riemann-Roch theorem to obtain bounds on the number of states in the automaton in terms of the degree, height and genus of f(t).

Kushnirenko's Theorem

Series
Student Algebraic Geometry Seminar
Time
Friday, November 17, 2017 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Timothy Duff – GA Tech
Motivated by the general problem of polynomial system solving, we state and sketch a proof Kushnirenko's theorem. This is the simplest in a series of results which relate the number of solutions of a "generic" square polynomial system to an invariant of some associated convex bodies. For systems with certain structure (here, sparse coefficients), these refinements may provide less pessimistic estimates than the exponential bounds given by Bezout's theorem.

The relationship between Betti tables and clique sums on graphs

Series
Student Algebraic Geometry Seminar
Time
Friday, October 27, 2017 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Jaewoo Jung – GA Tech
For any undirected graph, the Stanley-Reisner ideal is generated by monomials correspoding to the graph's "non-edges." It is of interest in algebraic geometry to study the free resolutions and Betti-tables of these ideals (viewed as modules in the natural way.) We consider the relationship between a graph and its induced Betti-table. As a first step, we look at how operations on graphs effect on the Betti-tables. In this talk, I will provide a basic introduction, state our result about clique sums of graphs (with proof), and discuss the next things to do.

Slack Matrices for Polytopes and Polyhedra

Series
Student Algebraic Geometry Seminar
Time
Friday, October 20, 2017 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Kisun Lee – Georgia Institute of Technology
We will introduce a class of nonnegative real matrices which are called slack matrices. Slack matrices provide the distance from equality of a vertex and a facet. We go over concepts of polytopes and polyhedrons briefly, and define slack matrices using those objects. Also, we will give several necessary and sufficient conditions for slack matrices of polyhedrons. We will also restrict our conditions for slack matrices for polytopes. Finally, we introduce the polyhedral verification problem, and some combinatorial characterizations of slack matrices.

Divisor Theory on Curves

Series
Student Algebraic Geometry Seminar
Time
Friday, October 13, 2017 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Libby Taylor – GA Tech
We will give an overview of divisor theory on curves and give definitions of the Picard group and the Jacobian of a compact Riemann surface. We will use these notions to prove Plucker’s formula for the genus of a smooth projective curve. In addition, we will discuss the various ways of defining the Jacobian of a curve and why these definitions are equivalent. We will also give an extension of these notions to schemes, in which we define the Picard group of a scheme in terms of the group of invertible sheaves and in terms of sheaf cohomology.

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