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

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