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 […]
Biquandle brackets are skein invariants of biquandle-colored knots, with skein coefficients that are functions of the colors at a crossing. Biquandle power brackets take this idea a step further with state component values that also depend on biquandle colors. This is joint work with Neslihan Gügümcü (IYTE).
A classic problem in graph theory is to find the chromatic number of a given graph: that is, to find the smallest number of colors needed to assign every vertex a color such that whenever two vertices are adjacent, they receive different colors. This problem has been studied for many families of graphs, including cube-like […]
Beilinson gave a resolution of the diagonal for complex projective space, which gives a strong, full exceptional collection of line bundles as a generating set for the derived category of coherent sheaves. Bayer-Popescu-Sturmfels generalized Beilinson's resolution of the diagonal by giving a cellular resolution of the diagonal for a proper subclass of smooth toric varieties […]
For a lattice L in the plane, we define the affiliated deep hole lattice H(L) to be spanned by a shortest vector of L and the furthest removed vector from the lattice contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices, […]
We explore the application of spectral graph theory to the problem of characterizing linguistically-significant classes of tree structures. We focus on various classes of syntactically-defined tree graphs, and show that the spectral properties of different matrix representations of these classes of trees provide insight into the linguistic properties that characterize these classes. More generally, our […]
We study variants of the Frobenius coin-exchange problem: Given n positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combination of the […]
The Ehrhart polynomial of a lattice polytope P counts the number of integer points in the nth integral dilate of P. The f^* -vector of P, introduced by Felix Breuer […]
The Cox ring of a projective variety is the ring of all its meromorphic functions, together with a grading of geometric origin. Determining whether this ring is finitely generated is […]
A simple question about chicken nuggets connects everything from analysis and combinatorics to probability theory and computer-aided design. With tools from complex, harmonic, and functional analysis, probability theory, algebraic combinatorics, and spline theory, we answer many asymptotic questions about factorization lengths in numerical semigroups. Our results yield uncannily accurate predictions, along with unexpected results about […]
The Kauffman bracket skein algebra of a surface is at once related to quantum topology and to hyperbolic geometry. In this talk, we consider a generalization of the skein algebra due to Roger and Yang for surfaces with punctures. In joint work with Han-Bom Moon, we show that the generalized skein algebra is a quantization […]
The original Bost-Connes system was constructed in 1990 and is a QSM system with deep connections to the field of rationals. In particular, its partition function is the Riemann-zeta function and its ground states evaluated on certain arithmetic objects yield generators of the maximal Abelian extension of the rationals. In this talk we describe the […]
