Public key quantum money is a replacement for paper money which has cryptographic guarantees against counterfeiting. We propose a new idea for public key quantum money. In the abstract sense, […]
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 […]
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 […]
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 […]
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 […]
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 […]
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 […]
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 in 2012, is the vector of coefficients of the Ehrhart polynomial with respect to the binomial coefficient basis . Similarly to h and h^* -vectors, […]
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 challenging task, even for simple examples. In this talk, we will discuss our efforts to tackle this problem for a specific class of varieties, […]
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 […]
