"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 Q. For example, if f is linear or quadratic then S_f is always infinite, but if degree of f is at least 3, then interesting […]
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 number b, M(b) is the Mahler measure of its minimal polynomial. By a result of Kronecker, an algebraic number b satisfies M(b)=1 if and only […]
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, a standard Young tableau S of shape λ is built from the Young diagram of shape λ by filling it with the numbers 1 to […]
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 […]
Given votes for candidates, what is the best way to determine the winner of the election, if some of the votes have been corrupted or miscounted? As we saw in […]
Discrete Calculus studies discrete structures, such as sequences and graphs, using techniques similar to those used in Calculus for continuous functions. The basic idea of generating functions is to associate […]
