Mike Krebs (California State University Los Angeles)

Think of a graph as a communications network. Putting in edges (e.g., fiber optic cables, telephone lines) is expensive, so we wish to limit the number of edges in the graph. At the same time, we would like messages in the graph to spread as rapidly as possible. We will see that the speed of communication is closely related to the eigenvalues of the graph's adjacency matrix. Essentially, the smaller the eigenvalues are, the faster messages spread. It turns out that there is a bound, due to Serre and others, on how small the eigenvalues can be. This gives us a rough sense of what it means for graphs to represent "optimal" communications networks; we call these Ramanujan graphs. Families of k-regular Ramanujan graphs have been constructed in this manner by Sarnak and others whenever k minus one equals a power of a prime number. No one knows whether families of k-regular Ramanujan graphs exist for all k.

ML 211

Geoffrey Buhl (California State University Channel Island)

Mathematically, "Moonshine" refers to the unexpected relationship between the largest sporadic simple group, the Monster, and the modular function, j. One of the products of the study and proof of the Moonshine conjectures are new algebraic objects called vertex operator algebras. Surprisingly, these objects are exactly the so-called chiral algebras of string theory. For certain vertex operator algebras, there is an associated modular function, generalizing one aspect of the moonshine conjectures. In this talk I will describe the moonshine conjectures, give a definition of vertex operator algebras, and describe which vertex operator algebras have modularity properties.

ML 211

Francis Su (Harvey Mudd College)

In this talk, I will describe recent progress on the question of determining the smallest triangulation of a d-dimensional cube, and more generally, the smallest triangulation of a simplotope: the product of simplices. Some interesting combinatorial insights come out of the geometry.

ML 211

Alex Hoffnung (University of California Riverside)

"Groupoidification" attempts to take familiar structures from linear algebra and enhance them to obtain structures involving groupoids. This process is not entirely systematic, however. The reverse process, "degroupoidification", is systematic and combined with examples sheds light on how to achieve the former. We describe the latter process and some examples including the groupoidification of Hecke algebras.

ML 211

Dagan Karp (Harvey Mudd College)

In this talk I hope to give an introduction to Gromov-Witten theory, touching on its string-theoretic origins, applications to

enumerative geometry and through the perspective of geometric moduli. Recent theorems and conjectures may also be discussed.

ML 211

Jonathan Lubin (Brown University - emeritus)

If k is a finite field, say with p^{n} elements, then we may form the group of all formal power series u(x) ∈ k[[t]] for which u(0) = 0, u'(0) = 1, the group operation being substitution (composition). This group is often called the Nottingham group over k. It's a pro-p-group, i.e. the projective limit of finite p-groups, simple enough in definition, but in many ways, very mysterious in behavior. Camina has shown that every finite p-group can be embedded in Nottingham, and Klopsch has classified all the conjugacy classes of elements of order p. They remarked a while back that they did not know of any explicitly given elements of order even as low as p^{2}. In this talk I will apply old mathematics to give a description of how to construct all elements of the Nottingham group of p-power order, and tell a classification up to conjugacy. But a characterization of the conjugacy classes that's as satisfactory as Klopsch's seems elusive.

ML 211

Sam Nelson (Claremont McKenna College)

Quandles are a type of non-associative algebraic structure defined from the combinatorics of knot diagrams. In this talk will recall the basics of quandle theory and look at some generalizations of quandles including biquandles, racks and biracks. If time permits, we will also look at tangle functors and a connection to Hopf algebras.

ML 211

Anna E. Bargagliotti (University of Memphis)

Nonparametric statistical tests can be used to differentiate among alternatives. Each test is uniquely identified with a procedure that analyzes ranked data. Procedure results are then incorporated into a test statistic. Inconsistencies among tests occur at both the procedure level and the statistic level. In this talk, I will characterize symmetry structures of data that explain why different procedures can output different rankings when analyzing the same data. In addition, I will quantify the number of ways that two ranked data sets can be aggregated and define a strict condition data must satisfy in order to ensure consistent procedure results. Finally, I will discuss how procedure inconsistencies affect the test statistics. Using the Kruskal-Wallis test as an example, I will outline how to asymptotically find the probability with which the null is rejected.

ML 211

Ghassan Sarkis (Pomona College)

We will present the field-of-norms construction of Fontaine and Wintenberger, which associates certain totally ramified extensions of local fields with positive-characteristic fields in a way that relates the Galois group of the extension to a subgroup of automorphisms of the positive-characteristic field. Time permitting, we will discuss applications the field-of-norms theory to p-adic dynamical systems.

ML 211

Lenny Fukshansky (Claremont McKenna College)

Siegel's lemma in its simplest form is a statement about the existence of small-size solutions to a system of linear equations with integer coefficients: such results were originally motivated by their applications in transcendence. A modern version of this classical theorem guarantees the existence of a whole basis of small "size" for a vector space over a global field (that is number field, function field, or their algebraic closures). The role of size is played by a height function, an important tool from Diophantine geometry, which measures "arithmetic complexity" of points. For many applications it is also important to have a version of Siegel's lemma with some additional algebraic conditions placed on points in question. I will discuss the classical versions of Siegel's lemma, along with my recent results on existence of points of bounded height in a vector space outside of a finite union of varieties over a global field.

Millikan 211