Edray Goins (Purdue University)

A Belyi map $\beta: P^1(C) \to P^1(C)$ is a rational function with at most three critical values; we may assume these values are ${0, 1, \infty }$. A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: $\beta^{-1} ([0,1]) \subseteq P^1(C) \simeq S^2(R)$. Replacing $P^1$ with an elliptic curve $E$, there is a similar definition of a Belyi map $\beta: E(C) \to P^1(C)$. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: $\beta^{-1} ([0,1]) \subseteq E(C) \simeq T^2(R)$. In this talk, we discuss the problems of (1) constructing examples of Belyi maps for elliptic curves and (2) drawing Dessins d'Enfants on the torus. This work is part of PRiME (Purdue Research in Mathematics Experience) with Leonardo Azopardo, Sofia Lyrintzis, Bronz McDaniels, Maxim Millan, Yesid Sanchez Arias, Danny Sweeney, and Sarah Thomaz with assistance by Hongshan Li and Avi Steiner.

Millikan 2099, Pomona College

Mark Huber (CMC)

Suppose I have a non-fair coin with unknown probability $p$ of heads that I can flip as many times as I'd like. A Bernoulli factory is a way of flipping the coin to construct a single coin flip that has chance $f(p)$ of heads for some function $f(p)$. For example, to get a $p^2(1-p)$-coin, I flip the original coin 3 times, and count it as heads if the sequence is HHT, and tails otherwise. My goal in this talk is to build a fast Bernoulli factory for the innocent looking function $f(p) = Cp$, where $C$ is a given constant. It might look innocent, but it turns out that for a fixed number of coin flips, you cannot build a Bernoulli factory that gives a $Cp$ coin for $C > 1$.

So we're going to use a random number of flips to build our $Cp$-coin. I'll show that when $Cp$ is small, this new method uses (to first order) $C$ flips, which for reasons I'll talk about is probably optimal.

Millikan 2099, Pomona College

Alissa Crans (Loyola Marymount University)

Analogous to the case for groups, the collection of quandle homomorphisms, Hom(Q, X), has no natural quandle structure. However, if X is an abelian quandle, then the hom set does become a quandle with the obvious pointwise operation. We will consider examples and investigate properties of this hom quandle.

Millikan 2099, Pomona College

Lenny Fukshansky (CMC)

Given a quadratic equation in N variables over a fixed number field K, there exists an algorithm to determine whether it has a non-trivial solution over K and to find such a solution. This matter becomes considerably more complicated for a system of quadratic equations: existence of a general such algorithm would contradict Matijasevich's negative answer to Hilbert's 10th problem. The problem however is more tractable if we allow searching in extensions of K. I will discuss an approach to this problem, which involves height functions, the common tools of Diophantine geometry. Our investigation extends previous results on small-height zeros of quadratic forms, including Cassels' theorem and its various generalizations and contributes to the literature of so-called "absolute" Diophantine results with respect to height.

Millikan 2099, Pomona College

Organizational meeting in the same room preceding the talk at 12:00 noon.

Lily Silverstein (CGU)

Some machine learning problems are naturally modeled by probability distributions (or other data) defined over a finite group. In this case the generalized Fourier transform, based on irreducible group representations, is a useful tool for designing efficient algorithms. In the case of the symmetric group, we can use combinatorial objects like Young diagrams to define an analogue to bandlimiting. Finally, I will talk about how certain probabilistic inferences can be performed directly in the Fourier domain, by considering the combinatorial decomposition of tensor products of representations. This talk is expository and based mainly on work done by Jonathan Huang and Risi Kondor.

Mudd Science Library 126, Pomona College

Mike Orrison (HMC)

Group frames are special spanning sets of vectors in representations of finite groups that sometimes behave remarkably like orthogonal bases. In this talk, I’ll explain why group frames are beginning to play a prominent role in the work I have been doing with Anna Bargagliotti on linear tests of uniformity for probability distributions defined on finite sets with symmetry. Along the way, I’ll try to convince you that group frames should be a part of the next linear algebra course you teach!

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].

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