Algebra / Number Theory / Combinatorics Seminar

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 Adva Mond and Kaarel Haenni): The binomial random digraph $D_{n,p}$ typically contains the minimum between the minimum out- and in-degrees many edge-disjoint Hamilton cycles, given […]

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 […]

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 Florida in 2000, where a difference of 537 votes determined the president of the United States, the electoral college system does not seem to be […]

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 a function with a sequence so that the coefficients of the power series expansion of the function represent the terms of the sequence. They provide […]

It is elementary and well known that a nonzero polynomial in one variable of degree d with coefficients in a field F has at most d zeros in F. It is meaningful to ask similar questions for systems of several polynomials in several variables of a fixed degree, provided the base field F is finite. […]

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann Hypothesis) concerning the bias of cubic Gauss sums. This explains […]

In a remarkable series of papers Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers, and independently Olga Kharlampovich and Alexei Myasnikov did the same using equivalent structures they called regular NTQ groups. It was later proved by Chloé Perin that if H is an […]

Public key quantum money is a replacement for paper money which has cryptographic guarantees against counterfeiting. We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We show that the proposal is secure against black […]

A numerical semigroup is a subset S of the natural numbers that is closed under addition.  One of the primary attributes of interest in commutative algebra are the relations (or trades) between the generators of S; any particular choice of minimal trades is called a minimal presentation of S (this is equivalent to choosing a […]