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 […]
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 […]
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 this talk we will survey recent work in this area.
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 points X(k) of the variety, for any field k. Now suppose we have two different fields k and k'. How does the behavior of X(k) […]
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 inside of certain space called a ``quintic threefold." This answered longstanding problems in enumerative geometry from antiquity. In particular, the number of rational degree d=1 […]
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 dynamical phenomena are known). In the first part of this talk, I will provide the necessary background and relevant history to state an effective equidistribution […]
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 order is odd? Now let R be a random subring of Zn. What is the probability that Zn/R is cyclic? We will see how these questions fit […]
I will give a brief introduction to well-rounded lattices and to their utility in wireless communications and post-quantum security. We will see how the lattice theta series naturally arises in these contexts and discuss its connections to well-rounded lattices. The talk is based on joint work with Laia Amoros, Amaro Barreal, Taoufiq Damir, Oliver Gnilke, […]
