Given an isotropic integral quadratic form which assumes a value t, we investigate the distribution of integer points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at […]

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 […]