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].

Mudd Science Library 126, Pomona College

David Krumm (CMC)

Let $K$ be a number field and let $f \in K[x]$ be a polynomial. For any nonnegative integer $n$, let $f^n$ denote the $n$-fold composition of $f$ with itself. If $\tilde K$ is a field containing $K$, we say that an element $\alpha \in \tilde K$ is periodic for $f$ if there exists a positive integer $n$ such that $f^n(\alpha)=\alpha$. In that case, the least such $n$ is called the period of $\alpha$. It is clear that if $f$ has a point of period $n$ in $K$, then it has a point of period $n$ in any extension of $K$; in particular, for every finite place $v$ of $K$, $f$ has a point of period $n$ in the completion $K_v$. In this talk we will discuss whether the converse holds: if $f$ has a point of period $n$ in every nonarchimedean completion of $K$, must it then have a point of period $n$ in $K$?

Mudd Science Library 126, Pomona College

Mark Huber (CMC)

In an induction, a problem is reduced recursively to a base case. In perfect simulation, there is no base case. Instead, a problem is randomly reduced to one of two problems, one of which is the original problem! The Fundamental Theorem of Perfect Simulation says that as long as the chance that the reduction does not require the original problem is greater than zero, then such a procedure terminates with probability 1 in finite time. In this talk, I'll explore and prove this fascinating theorem, and consider multiple ways in which it can be used to draw samples from one of the cornerstone models of statistical physics, the Ising model.

Mudd Science Library 126, Pomona College

Sam Nelson (CMC)

Knotted surfaces can be represented with diagrams known as ch-diagrams where two diagrams represent ambient isotopic knotted surfaces iff they are related by a sequence of Yoshikawa moves. Recently Kauffman introduced a generalization of knotted surfaces by adding virtual crossings to ch-diagrams. In this talk we will define invariants of knotted and virtual knotted surfaces using algebraic objects known as ribbon biquandles.

Mudd Science Library 126, Pomona College

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)

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

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