# MATH COLLOQUIUM ABSTRACTS ARCHIVE '10 - '11

## Fall 2010

### September

9/16 Prof. Sheldon Joyner, University of Western Ontario

1. The Riemann zeta function and the complex iterated integrals
It has long been known that the values of the Riemann zeta function at positive integers can be paired with those at negative integers via the functional equation satisfied by this function. But enlarging the realm of these positive zeta values to include multiple versions thereof, simple algebraic relations can be seen to arise among them, the so-called stuffle and shuffle relations. The latter arise from iterated integral expressions for the zeta values. In this talk, I will introduce these ideas, due to Zagier, Hoffman and many others, and discuss extensions of these iterated integral expressions to functions of a complex variable.

2. p-adic L-functions and the Bernoulli numbers
For each prime p, there exists an analogue of the real numbers, namely the field of p-adic numbers. As in the real world, in this p-adic domain, L-functions can be defined as a tool to study properties of field extensions. This is done by p-adically interpolating a family of very special rational numbers, (the so-called Bernoulli numbers), by exploiting their congruence properties. In this talk, I will explain some of these ideas, which go back to work of Kummer, Hensel, Tate, Iwasawa, Kubota, Leopoldt, Mazur and others.

9/23 Prof. Colin Starr, Willamette University

Eigenvalues of Graphs and the Inverse Inertia Problem
Every graph has an associated adjacency matrix, which has eigenvalues. We can include edge weights in our idea of "adjacency matrix," which means that a given graph can have many matrices associated to it, and thus many sets of eigenvalues are possible. The Inverse Eigenvalue Problem is to determine, for a given graph and a given set of eigenvalues, whether there is a matrix with those eigenvalues that represents the graph. The Inverse Inertia Problem is similar but only considers the signs of those eigenvalues. We will investigate the Inverse Inertia Problem and examine some recent results.

#### 9/30 Andrew Hoskins

Stealing Cars and Securing Free Public Transportation:
Everyday Applications of Mathematics
Cryptosystems are sets of algorithms for encoding and decoding messages and are in use constantly throughout the world. Car lock remotes from several leading manufactures use a cryptosystem called HiTag2, for example. As with any cryptosystem, HiTag2 can be modeled as a system of multivariate equations. This system of equations can then be attacked directly via algebraic methods or indirectly by converting the equations into an equivalent Boolean satisfiability problem and using logic-based methods. In this talk, I will discuss how cryptosystems work, how they are broken, and my summer research designing a hybrid attack which leverages logic-based techniques to feed extra information into a well known algebraic technique.

### October

10/14 Dr. Scott Beaver, Western Oregon University

The Wonderful (But Not-So-Mysterious) World of Fourier
We'll take a journey into an alternate description of reality. The Fourier Transform provides an accounting of a signal, whether a time-series or an image, which offers information about the frequencies of sinusoids present in the signal. We'll show how this amazing tool can be used to get rid of most of the noise in a noisy sound wave, and consider some image processing applications as well.

10/28 Mark Walsh, Oregon State University

Topology, Curvature and the Theorem of Gauss-Bonnet
Topology is a branch of geometry concerned with the aspects of a shape which are preserved under continuous deformation. Thus, while a round ball and a cube are topologically equivalent (as one could imagine a continuous deformation from one to the other), neither have the topology of a bagel. One of the dominant themes in modern geometry is understanding the relationship between the topology of a shape and another geometric property: its curvature. A round sphere, for example, is easily shown to have positive curvature. A torus (the surface of a bagel) in its usual form, has positive curvature around the outer circle and negative curvature near the inner circle. It is a fascinating consequence of the classical theorem of Gauss-Bonnet, that no matter how much we continuously deform its shape, it is not possible to make the curvature of the torus positive at every point. In this talk, I will discuss some of the main ideas behind this theorem as well as some of its topological consequences.

### November

11/11 Prof. Bill Kronholm, Whittier College

Euler Characteristic
The birth of topology can be traced back to the development of the Euler characteristic. In this talk, we will explore the Euler characteristic and its role in relation to platonic solids, surfaces, Pick's Theorem, and map colorings.

#### 11/18 Prof. Michael Hitchman, Linfield College

Please (stop trying to) tile this region!
Can a given region be tiled by a particular set of tiles? Tiling questions have long been interesting to ponder for all sorts of people, from toddlers to architects to physicists, and a rigorous investigation of such questions has often been considered "recreational mathematics". In the last 20 years some group-theoretic techniques have been brought to bear on some tiling questions that could not have been solved with more traditional techniques. This talk will begin with a discussion of the growing array of tools for proving the non-existence of a tiling of a region by a set of tiles. We will then use some of these new tools to solve a tiling question in the integer lattice that was investigated in an undergraduate research project at The College of Idaho.

## Spring 2011

### February

#### 2/3 Lance Thornton

The Computer Animation Pipeline for Feature Films

### March

#### 3/9 Mat Titus

A Tale of Two Transformations
The utility of complex analysis was invoked long before mathematicians would openly admit to using it. Today, complex numbers have proven especially useful in the domain of physics and in this colloquium, we study one of the various connections between the two fields of math and physics: the equivalence of Mobius and Lorentz transformations. Mobius transformations are angle-preserving maps of the complex plane to itself. Lorentz transformations are the cornerstone of Einstein's theory of special relativity. The first transforms two-dimensional space, the second, four-dimensional space-time. That a correspondence between the two exists at all is surprising, let alone an equivalence! Please join us in exploring and exploiting this connection in aggregate detail.

#### 3/31 Prof. Yevgeniy Kovchegov, Oregon State University

Introduction to Quantum Computation
In this expository talk, we will discuss the foundations of quantum computation, qubits, quantum gates and circuits, the Grover's search algorithm, and quantum Fourier transform.

### April

#### 4/7 Colloquium and Fair

This week's colloquium will be a preview of math and CS courses offered in the fall. Come hear professors talk about their courses as you mull over your fall schedule. Students have reported that these previews were instrumental, not only in setting their upcoming semester's schedule, but in their long-term course planning as well.

4/13 Greg Hanselman

Orbiting Robots, Sensor Networks, and Million-Dimensional Vector Spaces: Topology's Up-and-Coming Applications in Research and Industry
The last 20 years have seen surprising and powerful new applications of algebraic topology in the fields of motion planning , sensor networks, and data analysis. We'll first look at homology over the real line and examine how it can be used to gain a better understanding of high-dimensional shapes encountered in scientific research. Rips, Cech, and Witness complexes, which will all be defined, feature prominently. Next we'll take up some elegant homological approaches to pursuit-evasion and power conservation problems in sensor networks which are of great important in industry and defense. Time permitting, we'll formally define topological complexity and see how work in this subject has advanced since its introduction in computer science roughly 30 years ago. The talk should be fun and accessible to all!

4/21 James Bernhard, University of Puget Sound

Markov Chains: Where Probability Theory Meets Topology
Markov Chains are usually thought of as belonging to the field of probability theory. However, they can be also viewed geometrically, and from this point of view, some of their fundamental convergence properties can be seen to follow from well-known theorems in topology. In this talk, we will be briefly define and introduce Markov Chains from a traditional point of view, for those not already familiar with them. We will then show how this material can be viewed geometrically, which will allow us to use topology to establish some convergence properties of Markov Chains. While some background in probability theory of topology would be helpful for understanding this talk fully, anyone familiar with some linear algebra will be able to follow the talk well enough to appreciate its main message: the interconnectedness of mathematics.

4/28 Lee Gibson, University of Louisville

Rates of Escape for Random Walks on Infinite Graphs
A random walk on an infinite graph has a tendency to wander farther and farther away from the vertex at which it began. How fast does the displacement from the starting point increase as the number of steps grows? What role does the way that the vertices of the graph are connected to one another play in the answer to this question? We will discuss some basics, motivation, and the dichotomy between walks whose speed tends to zero and those with positive speed. Some interesting and non-intuitive results can sometimes emerge!

### May

5/2 Taylor McKenzie

Bayesian Networks in Disease Modeling
Bayesian networks are widely used in modeling systems, specifically in predicting the course of disease epidemics. In this talk, we will be using Bayesian networks to examine an SIR model that includes media influenced self-isolation. We will specifically look at how the choice of healthy individuals to limit their exposure to infected individuals affects the progression and duration of an epidemic.

The Netflix Grand Prize
In October 2006, Netflix announced to the world a \$1,000,000 prize for a team achieving 10 percent more accurate predictions of user ratings of movies relative to their existing algorithm. The competition lasted 33 months, and spurred numerous advances in the science of recommender systems and machine learning. I will describe some of the techniques and methods used by the winning team.

Willamette University