Will Murray (California State University Long Beach)

Question: Given a field, what interesting finite multiplicative groups does it contain?

Answer: None. In algebra class we prove that any finite group in a field is cyclic.

However, division rings (noncommutative fields) are much more interesting. At the very least, we know that the quaternions contain the finite quaternion group {1,-1,i,-i,j,-j,k,-k}, which is not cyclic. Lots of interesting geometry gives us a classification of all finite subgroups of the quaternions. They correspond to subgroups of SU(2) and SO(3), which correspond in turn to rotation groups of the Platonic solids.

MDSL 126

Jesse Elliott (California State University, Channel Islands)

The polynomial (1+X)(2+X) is reducible as a polynomial over the integers but is irreducible as a formal power series over the integers. On the other hand, the polynomial 6+X is irreducible as a polynomial over the integers but is reducible as a formal power series over the integers. How does 6+X factor as a product of formal power series? More generally, how does one represent a given polynomial over the integers as a product of irreducible formal power series? We will provide a complete answer to this question. The solution involves the ring of p-adic integers for primes p.

MDSL 126

Mark Huber (CMC)

Suppose that I have a stack of books on my desk. To find a book, I start at the top and work my way down to the desired book. After finding the book, I can move the book one position towards the top of the stack so that it will be slightly easier to find next time. Under the model that all the books are chosen independently, and book i is chosen with probability p_i, this gives rise to a Markov chain called the move ahead 1 chain. Books with relatively large p_i will tend to move to the top of the stack, while books with low p_i will tend to stay in the bottom. For this reason, this is called a self-organizing list, and provides a simple model for database organization. After many steps in this chain, the state will be in a long run distribution on permutations of the objects in the stack. In order to estimate quantities such as the expected time needed to access an object, it is necessary to be able to draw random samples from this distribution over permutations. In this talk I'll discuss a new method for doing so and prove that it runs in linear time as long as the p_i (when ordered) decrease sufficiently quickly.

Mudd Science Library 126, Pomona College

Stephan Ramon Garcia (Pomona College)

The theory of *supercharacters*, which generalizes classical character theory, was recently developed in an axiomatic fashion by P. Diaconis and I.M. Isaacs, based upon earlier work of C. Andre. When this machinery is applied to abelian groups, a wide variety of applications emerge. In particular, we develop a broad generalization of the discrete Fourier transform along with several combinatorial tools. This perspective illuminates several classes of exponential sums (e.g., Gauss, Kloosterman, and Ramanujan sums) that are of interest in number theory. We also consider certain exponential sums that produce visually striking patterns of great complexity and subtlety.

Mudd Science Library 126, Pomona College

David Krumm (CMC)

A major open problem in the field of arithmetic dynamics is a uniform boundedness conjecture formulated by Morton and Silverman in 1994. Sadly, despite many efforts, not even the simplest case of this conjecture has been proved yet. By work of Poonen this basic case has been reduced to an extremely simple statement about the possible periods of a rational number under iteration of a quadratic polynomial. In this talk we will discuss a new approach to proving Poonen’s conjecture by showing that the theory of quadratic dynamical systems is a rare case in which global behavior can be determined by purely local considerations.

Mudd Science Library 126, Pomona College

Hiren Maharaj (CMC)

In recent joint work with Albrecht Boettcher, Lenny Fukshansky and Stephan Garcia, we defined a class of lattices constructed from Abelian groups. I will

talk about aspects of this work and also about a relationship between this work and a construction of lattices from algebraic curves over finite fields due

to Tsfasman and Vladut.

Mudd Science Library 126, Pomona College

Lenny Fukshansky (CMC)

Well-rounded lattices are vital in extremal lattice theory, since the classical optimization problems can usually be reduced to them. On the other hand, many of the important constructions of Euclidean lattices with good properties come from diferent algebraic settings. This prompts a natural question: which of the lattices coming from algebraic constructions are well-rounded? We consider three such well known algebraic constructions: ideal lattices from number fields, cyclic lattices from quotient polynomial rings, and function field lattices from curves over finite fields. In each of these cases, we provide a partial answer to the above question, as well as discuss some generalizations and directions for future research.

Mudd Science Library 126, Pomona College

Sam Miner (UCLA)

There are dozens of different "Catalan structures", which are combinatorial objects enumerated by the Catalan numbers. Many of these have been classically studied in probability to obtain often delicate results on the "shape" of these random structures. We investigate two classes of permutations without forbidden 3-patterns, which were introduced by Knuth in his studies of sorting. These are known Catalan structures which have been extensively generalized and studied in the past two decades. We prove some rather detailed results about the shapes of random permutations in these two classes. All the proofs come from bijective and asymptotic combinatorics - we will sketch the reasoning behind these proofs, and also mention potential generalizations to permutations with forbidden sets of k-patterns. Joint work with Igor Pak.

Mudd Science Library 126, Pomona College

Sam Nelson (CMC)

Finite type invariants, also known as Vasiliev invariants, are integer-valued knot invariants satisfying a certain skein relation. Many of the coefficients of the Jones and Alexander polynomials, for example, are known to be Vassiliev invariants, and the set of all Vassiliev invariants dtermines a powerful invariant known as the Kontsevich integral. We adapt a scheme for computing finite type invariants due to Goussarov, Polyak and Viro to enhance the biquandle counting invariant. The simplest nontrivial case has connections to the concept of parity in virtual knot theory. This is joint work with Pomona student Selma Paketci.

Mudd Science Library 126, Pomona College

There will be a short organizational meeting preceding the talk at 12:00 noon.

Sam Nelson (Claremont McKenna College)

Biracks are algebraic structures with axioms derived from the framed Reidemeister moves. Associated to a finite birack is an integer-valued invariant of knots and links known as the birack counting invariant. An enhancement of the counting invariant is a stronger and more sensitive invariant which determines the counting invariant, e.g. a polynomial invariant which evaluates to the counting invariant at 1. In this talk we will see some new enhancements of the birack counting invariant under development in Claremont.

Mudd Science Library 126 at Pomona College