Fall 2016 Schedule

October 5

Speaker: David Wen

Title: Ghosts, Ladders and Permutations

Abstract: Amidakuji is a lottery game that randomly assigns n persons to n objects. It is a game more well known in China, Korea and Japan, as a way to make decision in a seemingly random way. The game consists of verticle paths from n starting points to n objects and rungs are randomly placed between two paths. Each player traces along a path and crossing over rungs to the end of their route to their unique assigned object. Mathematically speaking the game is a function assigning n objects to n persons. Since each object is being uniquely assigned to each person, this function in fact a permutation. This talk will discuss some of the mathematical aspects of Amidakuji and reprove some very well known theorems about permutations in this context.

October 12

Speaker: Nadir Hajouji

Title: Descent: From Grothendieck to Fermat

Abstract: In the mid-20th century, Weil developed, and Grothendieck generalized, a technique called descent. Using descent, one is able to answer the following question: given a non algebraically closed field k, and a variety X defined over the algebraic closure of k, can we find a variety Y/k whose base extension to the algebraic closure is isomorphic to X? After reviewing the basics of group cohomology and twisting, I will show how the descent principle allows us to easily conclude that a certain cohomology group vanishes, which in turn allows us to easily prove a famous theorem in Galois theory, which we can then use to parametrize all solutions to a certain class of diophantine equations.

October 19

No Seminar

October 26

Speaker: Naomi Burkhart

Title: An Introduction to Leavitt Path Algebras

Abstract: A Leavitt path algebra is a type of algebra which is built from an arbitrary directed graph, and which has been intensely studied over the past couple of decades. I will do a short introduction to these algebras (including definition, motivation, and examples) and then discuss some results regarding their prime spectrum, which is currently the focus of my Master's thesis.

November 2

Speaker: David Nguyen

Title: Formal power series and applications to combinatorics

Abstract: The famous French mathematician Laplace discovered the remarkable correspondence between set theoretic operations and operations on formal power series and put it to great use to solve a variety of combinatorial problems. This talk is an introduction to formal power series and how they can be used to solve seemingly complicated combinatorial problems. It is an invitation to the fascinating and beautiful world of generating functions.

November 9

Speaker: Zach Blumenstein

Title: Geometry of Syzygies

Abstract: The syzygies of a graded module M over a graded ring are the relations among a minimal set of generators of M—that is, the kernel of a map of a free module F onto M. We discuss syzygies and their applications to geometric topics, including configurations of finitely many points in the plane and the interpolation problem. We also include a brief introduction to local cohomology, as part of a sketch of the solution to the interpolation problem.

November 16

Speaker: Cole Hawkins

Title: Baby Boij-Soderberg Theory

Abstract: The multiplicity of a module is an invariant (and number) that relates to the Hilbert series. Under Boij-Soderberg theory we study the ranks of free modules and twists that appear in the graded minimal free resolution of a module. By organizing that information nicely and appealing to a slick geometric argument we can bound the multiplicity of Cohen-Macaulay modules.

November 23

Speaker: Nadir Hajouji

Title: Introduction to the Arithmetic of Elliptic Curves

Abstract: In a nutshell, an elliptic curve is the solution set to a polynomial of the form y^2 = x^3+ax^2+bx+c. I will introduce elliptic curves first in an algebraic context, then show how one can study them analytically as Riemann surfaces, and finally use the insight from the Riemann surface story to show how one can find all of the rational points on an elliptic curve.

November 30

Speaker: Ebrahim Ebrahim

Title: The Category of Sets

Abstract: Sets are often thought of as lacking structure, but the category of sets is actually rather rich as a category. We attempt to pin down the category of sets up to equivalence by reinterpreting the axioms of set theory in categorical language. Membership is stripped of its role as the primitive statement, with morphisms and universal properties taking its place.