Algebra / Number Theory / Combinatorics Seminar

An elliptic curve $ E: y^2 + a_1 \, x \, y + a_3 \, y = x^3 + a_2 \, x^2 + a_1 \, x + a_6 $ is […]

The existence of a set $A\subset \N_0$ of positive upper Banach density such that $A-A:=\{m-n:m, n\in A, m>n\}$ does not contain a set of the form $S-S$ with $S$ a […]

An arithmetical structure on a finite, connected graph G without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there […]

This talk discusses a puzzle called “Spinning Switches,” based on a problem popularized by Martin Gardner in his February 1979 column of “Mathematical Games". This puzzle can be generalized to […]

The slice rank polynomial method, motivated by groundbreaking work of Croot, Lev and Pach and refined by Tao, has opened the door to the resolution of many problems in extremal […]

Given a lattice L, an extension of L is a lattice M of strictly greater rank so that L is equal to the intersection of the subspace spanned by L […]

Markov chains have become widely-used to generate random political districting plans. These random districting plans can be used to form a baseline for comparison, and any proposed districting plans that differ significantly from this baseline can be flagged as potentially gerrymandered. However, very little is rigorously known about these Markov chains - Are they irreducible? […]

"A Tale of Two Cities" is a novel told in three books/parts. Here we describe three projects related both to published work and ongoing pieces: PROJECT 1: In the world of combinatorics, parking functions are combinatorial objects arising from the situation of parking cars under a parking strategy. In this part of the talk, we […]

Given a degree d polynomial f(x) in Q, consider the subset S_f  of Q consisting of rational numbers t for which the translated polynomial f(x) - t factors completely in […]

The Mahler measure of a polynomial is the modulus of its leading term multiplied by the moduli of all roots outside the unit circle.  The Mahler measure of an algebraic […]

Young diagrams are all possible arrangements of n boxes into rows and columns, with the number of boxes in each subsequent row weakly decreasing. For a partition λ of n, […]

Let  L be a full-rank lattice in R^n and write L+ for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L+. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative […]