Lenny Fukshansky

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, […]

In late March, students, staff, and faculty were invited to help collaboratively build a large-scale geometric sculpture on the campus of Harvey Mudd College, demonstrating a relationship between truncated octahedra and rhombic dodecahedra, which are two examples of space-filling polyhedra. I’ll talk about the process of designing and building the sculpture, some geometry and combinatorics […]

The famous primitive element theorem states that every number field K is of the form Q(a) for some element a in K, called a primitive element. In fact, it is clear from the proof of this theorem that not only there are infinitely many such primitive elements in K, but in fact most elements in […]