Hiren Maharaj (CMC)

In recent joint work with Albrecht Boettcher, Lenny Fukshansky and Stephan Garcia on the Rosenbloom-Tsfasman function field lattices from the Hermitian function field we showed that these lattices are generated by their minimal vectors. We also have a lower bound on the total number of minimal vectors, and some properties of the automorphism groups of these lattices. I will talk about these results.

Mudd Science Library 126, Pomona College

Daniel Katz (Cal State Northridge)

We consider Weil sums, which are obtained by summing the values of a finite field character with a polynomial argument. These are used to count points in varieties over finite fields, as well as to compute the correlation properties of sequences that arise in information theory and cryptography. Our Weil sums involve additive characters with binomial arguments. As we vary the coefficients of the binomial, we obtain a spectrum of values, which depends on the finite field used and the exponents that appear in the binomial. Some rare choices give very elegant spectra that attain only three distinct values as the coefficients are varied through the field (two values occur only in uninteresting degenerate cases). Ten infinite families of three-valued Weil sums have been found from 1966 to 2013. Numerical evidence was found by Dobbertin, Helleseth, Kumar, and Martinsen for one further inifnite family, which they conjectured (in 2001) to exist. This talk is about the proof of their conjecture. The methods employed are diverse: algebraic number theory, graph theory, trilinear forms, character sums, and counting points on curves. This is joint work with Philippe Langevin.

Mudd Science Library 126, Pomona College

Eric Friedlander (University of Southern California)

We consider affine algebraic groups G over a field k and rational (i.e., algebraic) actions of G on k-vector spaces V. We seek to provide some understanding of such actions through examples, algebraic invariants, and geometric structures. Our first example of an algebraic group is the general linear group GL_N, but we shall primarily focus on the example of the additive group G_a over k of characteristic p > 0. The representation theory of G_a is far too complicated to fully classify, but sufficiently accessible to reveal considerable structure. Our efforts lead to finite subgroup schemes of G_a and finite dimensional sub-coalgebras of the algebraic functions k[G_a] = k[T].

MDSL 126

Karl-Dieter Crisman (Gordon College)

Those studying the mathematics of voting and choice have long used whatever tools are necessary to explain paradoxes - from dynamical systems to Voronoi diagrams. Recently, graph theory has proved to be quite useful in modeling situations where there are natural symmetry relationships between options in a given choice setup. We will see some recent results (including some due to HMC faculty) answering questions about voting for rank-orderings of candidates and committees. Then we will explore how graphs can be used to answer questions you didn't even know you had about ranking *cyclic* orders - such as how to seat people at a round table!

MDSL 126

Wai Yan Pong (Cal State Dominguez Hills)

We improve several results on algebriaic independence of arithmetic functions by a theorem of Ax's on differential Schanuel conjecture.

Mudd Science Library 126, Pomona College

Lenny Fukshansky (CMC)

Let N > 1 be an integer, and let 1 < a_1 < ... < a_N be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be represented as a linear combination of a_1,...,a_N with non-negative integer coefficients. More generally, the s-Frobenius number is defined to be the largest positive integer that has precisely s distinct representations like this, so that the classical Frobenius number can be thought of as the 0-Frobenius number. The condition that a_1,...,a_N are relatively prime implies that s-Frobenius numbers exist for every non-negative integer s. The general problem of determining the Frobenius number, given N and a_1,...,a_N, dates back to the 19-th century lectures of G. Frobenius and work of J. Sylvester, and has been studied extensively by many prominent mathematicians of the 20-th century, including P. Erdos. While this problem is now known to be NP-hard, there has been a number of successful efforts by various authors producing bounds and asymptotic estimates on the Frobenius number and its generalization. I will discuss some of these results, which are obtained by an application of techniques from Discrete Geometry.

Mudd Science Library 126, Pomona College

Nora Youngs (HMC)

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode the full combinatorial data of a neural code. We find that these objects can be expressed in a "canonical form'' that directly translates to a minimal description of the receptive field structure intrinsic to the neural code. We analyze the algebraic properties of maps between these objects induced by natural maps between codes. We also find connections to Stanley-Reisner rings, and use ideas similar to those in the theory of monomial ideals to obtain an algorithm for computing the canonical form associated to any neural code, providing the groundwork for inferring stimulus space features from neural activity alone.

Mudd Science Library 126, Pomona College

Aravind Asok (USC)

I will discuss the theory of projective modules over smooth (affine) algebras from a modern point of view. In paticular, I will explain how ideas from homotopy theory can be used to answer classical questions like: when does a projective module of rank r split as the sum of a module of rank r-1 and a free module of rank 1?

Mudd Science Library 126, Pomona College

Bogdan Petrenko (Eastern Illinois University)

See attached PDF.

Mudd Science Library 126, Pomona College

Sinai Robins (Nanyang Technological University - Singapore and Brown University)

It is natural to ask when the spherical volume defined by the intersection of a sphere at the apex of an integer polyhedral cone is a rational number. This work sets up a dictionary between the combinatorial geometry of polyhedral cones with `rational volume' and the analytic behavior of certain associated cone theta functions, which we define from scratch. We use number theoretic methods to study a new class of polyhedral functions called cone theta functions, which are closely related to classical theta functions. One of our results shows that if K is a Weyl chamber for any finite crystallographic reflection group, then its cone theta function lies in a graded ring of classical theta functions (of different weights/dimensions) and in this sense is ‘almost’ modular. It is then natural to ask whether or not the conic theta functions are themselves modular, and we prove that in general they are not. In other words, we uncover relations between the class of integer polyhedral cones that have a rational solid angle at their apex, and the class of cone theta functions that are almost modular. This is joint work with Amanda Folsom and Winfried Kohnen.

Mudd Science Library 126, Pomona College