Combinatorics and Number Theory Seminar

I will describe a certain two-parameter family of automorphisms of the p-adic affine plane. Each of these automorphisms admits a chaotic attractor, with non-integral Hausdorff dimension, on which it is topologically conjugate to a full two-sided shift map. The attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction; this measure is a p-adic analogue of the SRB measures in smooth dynamics.

When D is a positive non-square integer, D^1/2 has a periodic regular continued fraction expansion. This expansion gives a means to calculate units of the ring of integers in the field generated by the square root over the rationals. Using continued fractions with ``partial quotients" which are polynomials, the units that one can find in a similar way in the function field of a finite genus hyperelliptic curve correspond to rational points of the Jacobian variety of the curve. I'll sketch the basics of these continued fractions, mainly by examples, and report on a new family of genus two curves over ℚ with torsion elements of order 11 that my former student K. Daowsud and I determined.

The concept of pseudo-distance-regularity around a vertex of a graph was introduced by Fiol, Garriga and Yebra in 1996. This concept is a natural generalization of the standard distance-regularity around a vertex. Given a vertex x of a connected graph G, let T(x) denote the Terwilliger algebra of G with respect to x. In this talk, we give an algebraic characterization of pseudo-distance-regularity around x via the corresponding trivial module of T(x).

Imagine a system in which particles of energy are randomly added to sites in a finite network. When the number of particles at a site reaches a certain threshold, the site becomes unstable and fires, sending one particle to each of its immediate neighbors in the network. This model of dispersion has interesting connections to a variety of areas of mathematics: combinatorics (domino tilings, Stanley's h-vector conjecture, spanning trees, necklaces, Young's lattice of partitions, hyperplane arrangements, and graph orientations), Coxeter systems, algebraic geometry and commutative algebra, pattern formation, and statistical physics. My goal is to advertise the subject of chip-firing by illustrating several of these connections.

TBA.

Many decision problems can be modeled as constraint satisfaction problems (CSPs). In this talk, I will discuss a class of CSPs whose computational complexities can be classified using algebraic and model theoretic techniques, giving examples with graphs and permutations.