The number of inversions of a permutation is an important statistic that arises in many contexts, including as the minimum number of simple transpositions needed to express the permutation and, equivalently, as the rank function for weak Bruhat order on the symmetric group. In this talk, I’ll describe an analogous statistic on the reduced expressions […]
Given a finite quandle $X$, a set $S \subset \mathrm{Hom}(X,X)$ of quandle endomoprhisms, and an oriented knot or link $L$, we construct a quiver-valued invariant of oriented knots and links. This quiver categorifies the quandle counting invariant in the most literal sense and can be used to define many enhancements of the counting invariant. This […]
In 1846, Ernst Eduard Kummer conjectured a distribution of values of a cubic Gauss sum after computing a few values by hand. This was forgotten about for nearly 100 years until John von Neumann and Herman Goldstine attempted to verify the conjecture as a way to test the new ENIAC machine in 1953. They found […]
Combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane as a Venn diagram-like figure. A sufficient condition to do so is […]
The Bateman—Horn Conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the Green—Tao theorem, and a variety of famous conjectures, such as the Twin Prime Conjecture. In this expository talk, we start from basic principles and provide a heuristic argument in favor of the conjecture. […]
Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a […]
Moment generating functions (Laplace transforms) are a means for transforming probability problems into problems involving polynomials. Here I will concentrate on the binomial distribution, and use the mgf to link this distributions probabilities directly to the binomial theorem. The mgf is also a key ingredient in Chernoff bounds, which give upper bounds on the tail […]
In this talk we use the unit-graphs and the special unit-digraphs on matrix rings to show that every n x n nonzero matrix over F_q can be written as a sum of two SL_n-matrices when n>1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove […]
We consider sums in which an additive character of a finite field F is applied to a binomial whose individual terms (monomials) become permutations of F when regarded as functions. These Weil sums characterize the nonlinearity of power permutations of interest in cryptography. They also tell us about the correlation of linear recursive sequences over finite fields that are used […]
The classical Sperner - KKM (Knaster - Kuratowski - Mazurkiewicz) lemma has many applications in combinatorics, algorithms, game theory and mathematical economics. In this talk we consider generalizations of this lemma as well as Gale's colored KKM lemma and Shapley's KKMS theorem. It is shown that spaces and covers can be much more general and […]
We begin with a review of the Bateman—Horn conjecture, which sheds light on the intimate relationship between polynomials and prime numbers. In this expository talk, we survey a host of applications of the conjecture. For example, Landau’s conjecture, the twin prime conjecture, and the Green—Tao theorem are all consequences of the Bateman—Horn conjecture. Moreover, the […]
In this talk we will survey recent work on Niebzydowski Tribrackets and Niebrydowski Algebras, algebraic structures related to region colorings the planar complements of knots and trivalent spatial graphs.
