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, […]
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 […]
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 […]
Quiver structures are naturally associated to subsets of the endomorphism sets of quandles and other knot-coloring structures, providing a natural form of categorification of homset invariants and their enhancements. In […]
A projective algebraic variety X is the zero locus of a collection of homogeneous polynomials, in projective space. When the polynomials have integer coefficients, we can think of the k-valued […]
For a finite field F_q, a subset of F_q^N is a Kakeya set if it contains a line in every direction (i.e., a coset of every one-dimensional linear subspace). The finite field Kakeya problem is to determine the minimal size K(N,q) of a Kakeya set in F_q^N. This problem was posed by Wolff in 1999 as […]
Though mirror symmetry requires much technical background, it gained traction in the mathematical community when physicists Candelas-de la Ossa-Green-Parkes discovered enumerative invariants counting the number of rational degree d curves […]
There is a rich connection between homogeneous dynamics and number theory. Often in such applications it is desirable for dynamical results to be effective (i.e. the rates of convergence for […]
How many sublattices of Zn have index at most X? If we choose such a lattice L at random, what is the probability that Zn/L is cyclic? What is the probability that its […]
