Algebra / Number Theory / Combinatorics Seminar

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 […]

Many knot invariants are defined from features of knot projections such as arcs or crossings. Gauss diagrams provide an alternative combinatorial scheme for representing knots. In this talk we will […]

There are two different measures of how far a given Euclidean lattice is from being orthogonal -- the orthogonality defect and the average coherence. The first of these comes from […]

Consider rational polynomials in multiple variables that are linear with respect to some of the variables. In this talk we discuss the problem of finding a zero of such polynomials […]

In this talk we will consider some notions of `robustness' of graph/hypergraph properties. We will survey some existing results and will try to emphasize the following new result (joint with […]

Given a prime p, let P(t) be a non-constant monic polynomial in t over the ring of p-adic integers. Let X(n) be an n x n uniformly random (0,1)-matrix over the same ring. We compute the asymptotic distribution of the cokernel of P(X(n)) as n goes to infinity. When P(t) is square-free modulo p, this […]