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 come across as more puzzling or even mysterious. Explanatory proofs, in the sense of Steiner, transform what might initially seem mysterious or even magical into […]
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 where the connection to hyperbolic geometry is fairly well-established. Explicating this connection to hyperbolic geometry requires an understanding of the non-commutative structure of the skein algebra, […]
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 […]
The complete homogeneous symmetric (CHS) polynomials can be used to define a family of norms on Hermitian matrices. These 'CHS norms' are peculiar in the sense that they depend only on the eigenvalues of a matrix and not its singular values (as opposed to the Ky-Fan and Schatten norms). We will first give a general overview behind […]
It is widely believed that Weierstrass ignored Eisenstein's theory of elliptic functions and developed an alternative treatment, which is now standard, because of a convergence issue. In particular, the Eisenstein series of weight two does not converge absolutely while Eisenstein's theory assigned a value to this series. It is now well-known that the quantity which […]
We consider the problem of comparing the number of discrete points that belong to a set with the measure (or volume) of the set, under circumstances where we expect these two numbers to be approximately equal. We start with a locally compact, abelian, topological group G. We assume that G has a countably infinite, torsion […]
Hassett spaces in genus 0 are moduli spaces of weighted pointed stable rational curves; they are important in the minimal model program and enumerative geometry. We compute the Chow ring of heavy/light Hassett spaces. The computation involves intersection theory on the toric variety corresponding to a graphic matroid, and rests upon the work of Cavalieri-Hampe-Markwig-Ranganathan. […]
We will discuss the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its […]
In this talk we discuss some problems related to finding large induced subgraphs of a given graph G which satisfy some degree-constraints (for example, all degrees are odd, or all degrees are j mod k, etc). We survey some classical results, present some interesting and challenging problems, and sketch solutions to some of them. This […]
