Algebra / Number Theory / Combinatorics Seminar

The orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2n)$ is rich in representation theory: while the finite dimensional $\mathfrak{osp}(1|2n)$-module category is semisimple, the study of infinite dimensional representations of $\mathfrak{osp}(1|2n)$ is wide open. In this talk, we will define the orthosymplectic Lie superalgebras, realize $\mathfrak{osp}(1|2n)$ as differential operators on complex polynomials, and describe the space of polynomials in commuting […]

Given a homogeneous multilinear polynomial F(x) in n variables with integer coefficients, we obtain some sufficient conditions for it to represent all the integers. Further, we derive effective results, establishing bounds on the size of a solution x to the equation F(x) = b, where b is any integer. For a special class of polynomials […]

In this talk we will survey recent developments in the use of ternary algebraic structures known as Niebrzydowski Tribrackets in defining invariants of knots, with some perhaps surprising applications.

In this talk we introduce a new modification of the Jacobi-Perron algorithm in the three dimensional case. This algorithm is periodic for the case of totally-real conjugate cubic vectors. To the best of our knowledge this is the first Jacobi-Perron type algorithm for which the cubic periodicity is proven. This provides an answer in the […]

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially […]

The Kauffman bracket skein algebra was originally defined as a generalization of the Jones polynomial for knots and links on a surface and is one of the few quantum invariants […]

A Belyi map $\beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$.  Replacing $\mathbb{P}^1$ with an elliptic curve $E: \ y^2 = x^3 + A \, x + B$, there is a similar definition of a Belyi […]