Willamette University Department of Mathematics

2007-2008

Fall Semester

Tuesday, September 18, 4:00pm

Prof. Inga Johnson, Willamette University

*Planting Trees*

Abstract:

Your aid I want, nine trees to plant

In rows just half a score;

And let there be in each row three.

Solve this: I ask no more.

The puzzle above was published in 1821 by John Jackson in a book of problems
called *Rational Amusement for Winter Evenings*. We'll talk about the
solution to this puzzle (try to figure it out before the talk!) and the
following generalization:

Given a positive integer *p,* how can *p* points be arranged on a
plane, no four in a straight line, so that the number of straight lines with
three points on them is maximized?

We'll discuss known solutions and some accessible open problems (which there are
many) all stemming from a question about planting trees!

Thursday, October 4th, 4:00pm

Collins Room 204

Prof. Josh Laison, Willamette University

*Count on the Platonic
Solids*

Abstract:

Platonic solids are the three-dimensional analogs of regular polygons in two dimensions. They have been admired for thousands of years for their beauty and symmetry. The numbers of distinguishing features of these polyhedra, such as the number of vertices, edges, and faces, and even the number of Platonic solids, contain many interesting patterns, and they get even more interesting in higher dimensions. We'll investigate some of these patterns, and spend some time counting on the Platonic solids.

Rated: PG

Wednesday, October 10th, 4:00pm

Collins Room 204

Prof. Liz Stanhope, Lewis & Clark College

*Doughnuts sound good to
me!*

Abstract:

Suppose you made a drum that was in the shape of a doughnut. You could
then strike this drum and listen to how it sounds. Would it be possible
for doughnut-drums of different sizes to sound exactly the same? One way
to tackle this question is to go to higher dimensions. Another way is to punch
more holes in the doughnut.

We'll do both and see what the resulting doughnuts have to say.

Rated PG13

Tuesday, October 16th, 4:00pm

Collins Room 204

Prof. Gary Gislason, Willamette University

*Some Properties of the
Sum and Product of Two Uniformly Distributed Random Variables*

Abstract:

In this presentation the density and cumulative distribution functions of both V=X+Y and V=XY will be developed. The random variables, X and Y, are taken to be uniformly distributed on [0,1].

Rated PG13

Thursday, October 25th, 4:00pm

Collins Room 204

Prof. Peter Otto, Willamette University

*Is the Sacrifice Bunt a Good
Baseball Strategy?*

Abstract:

The sacrifice bunt is often used by baseball managers as a strategy to score runs by putting runners in scoring position at the "sacrifice" of losing an out. In this talk, we'll look into the value of this strategy by modeling baseball with a very useful tool in probability theory called Markov Chains.

Rated PG

Thursday, November 1st, 4:00pm

Collins Room 204

Prof. Jessica Sklar, Pacific Lutheran University

*Defeating the Robot and Unlocking
Doors: Mathematical Solutions to Computer Game Puzzles*

Abstract:

Many puzzles in computer adventure games are obviously mathematical; others are math problems in disguise. In this talk, we discuss the use of linear and abstract algebra to solve puzzles in computer games such as Myst and Timelapse.

Rated PG

Thursday, November 15th, 4:00pm

Collins Room 204

Professor Emeritus Richard Iltis, Willamette University

*An Historical View of Models for
Planetary Motion*

Abstract:

We follow the development of mathematical models for the movement of planets beginning with the Greek quest for harmony in the universe as exemplified by uniform circular motion. Scientists who contributed major insights include Ptolemy, Copernicus, Kepler, Newton, and Einstein. The models begin with spheres centered at the earth and continue to ellipses about the sun.

Tuesday, December 4th, 4:00pm

Collins Room 204

Prof. Erin McNicholas, Willamette University

Abstract:

*Mathematical Toys: the Rubik's Cube and
other permutation puzzles
*

Abstract: We'll examine the Rubik's Cube group, which is made up of all possible configurations of the cube. Using group theory to analyze this set, we'll determine some interesting properties of the cube. Which transformations of the cube commute with all others? How many moves on the cube does it take to get it into a particular configuration? What are the most common solution techniques? We'll look at a few other permutation puzzles that can be analyzed in a similar way. If you have a Rubik's cube handy, bring it! You'll have the chance to try out a few moves and master the superflip.

Rated: PG 13

Spring Semester

Thursday, January 24th, 4:00pm

Collins Room 204

Professor Holly Swisher, Oregon State University

*Monstrous Moonshine Meets Rogers-Ramanujan
Functions*

Abstract:

Some of the most fascinating mathematics
occur when seemingly unrelated objects reveal themselves to be distinctly
intertwined. In this talk we will see a surprising connection between the
Monster, the largest finite

sporadic simple group, and Rogers-Ramanujan functions, which are connected to
Ramanujan's famous continued fraction.

One of the many things Ramanujan did in his life was to list 40 identities

involving what are now called the Rogers-Ramanujan functions on one side, and
products of functions of the form $\prod_{n=1}^\infty (1-q^{mn})$ on the other
side. It was remarked by Birch that these identities seemed too
complicated to guess, even for one with Ramanujan's incredible instinct for
formulae. Recently however, Koike devised a strategy for finding (but not
proving) these types of identities by noticing an interesting connection to the
Monster. He was able to conjecture many new Rogers-Ramanujan type
identities which we have now proved. The key to tying it all together?
Modular forms!

Rated R

Thursday, February 21st, 4:00pm

Collins Room 204

Professor Stephanie Salomone, University of Portland

*Analyzing a Multi-Scale Symphony*

Abstract:

Wavelets have applications in many fields
of science and

mathematics, including noise reduction, mammography, fingerprint analysis, and
other forms of data storage and compression. Recently, wavelets were used to
predict forgeries in works of art. Because of their ability to localize and give
a time-frequency description of a signal, we can use wavelets to decompose,
compress, and resynthesize a signal accurately and with little loss of
information.

We will discuss the symphony analogy for the wavelet transform, an accessible
heuristic that describes how we might decompose a signal using wavelets. We'll
also construct a basis for an infinite dimensional vector space, and talk about
uses of wavelets in pure mathematics.

Rated PG

Thursday, February 28st, 4:10pm

Collins Room 204

Professor Ellen Gethner, University of Colorado at Denver

*An Adventurer's Guide To The* *Treasure
Hunt For High Chromatic Thickness-Two Graphs*

Abstract:

A graph
*G* is said to have *thickness-t* if the edges
of *G* can be partitioned into *t* and no fewer planar
graphs. For example, if *G* is planar, then *G* has thickness-one. For
another example, with the help of Kuratowski's
Theorem, it is easy to see that *K5*
has thickness at least two. A
longstanding open problem is the following:

The
largest chromatic number of any thickness-two graph is known to be one of 9, 10,
11, or 12. The 9 is due to
exactly one published example of a 9-critical thickness-two graph found by Thom
Sulanke in 1973 and was reported by Martin
Gardner in 1980. The 12
is due to a straightforward argument that relies on Euler's
Formula for planar
graphs.

We introduce a catalogue of new small 9-critical thickness-two graphs, and a
construction that generates infinitely many 9-critical thickness-two graphs,
thus providing ballast to the 9,
and providing a stepping stone to the search for a 10-chromatic thickness-two
graph as well. In addition new families of thickness-two graphs will be defined,
some of which have a known asymptotically sharp upper bound for the chromatic
number.

The (re)search for the thickness-two 10-chromatic graph is a continuing treasure
hunt and this talk will highlight many of the gems that have been found so far.

http://carbon.cudenver.edu/~egethner/WillametteUniversity_EllenGethnerTalk.html

Thursday, March 6th, 4:10pm

Collins Room 204

Alex Jordan, University of Oregon

*An Introduction to the Riemann Zeta
Function*

The Riemann zeta function is very curious.
It's definition is relatively simple to write down, but a full understanding of
its properties remains elusive. The famous Riemann Hypothesis makes a
conjecture about where the function's zeros lie, and a virtual library of
published theorems use the Riemann hypothesis as an assumption. This is
one reason why a proof of the Riemann Hypothesis is considered by some to be the
biggest unsolved problem in mathematics.

We'll introduce the function and discuss well-known basic properties -
especially its connection to the Gamma function. Mathematica generated
pictures will hint at the truth of the Hypothesis, but also hint at difficulties
in proving it. There is a connection between the zeros of the function and
prime numbers that we will look at. If time allows, we'll look at one or
two examples of theorems that use the Hypothesis as an assumption.

Tuesday, March 11th, 4:10pm

Collins Room 204

Cam McLeman, University of Arizona

*Generating Functions and some Crazy
Dice*

Number theory, and mathematics in general,
is filled with interesting sequences of numbers -- square numbers, powers of
two, prime numbers, Fibonacci numbers, etc. Unfortunately, one typically
has to invest in a course (or a lifetime) in number theory to make any progress
on questions surrounding these sequences, but we'll describe a simple technique
-- "hanging your sequence on a clothesline" -- which lets you get lots
of cool number theory out of basic Taylor expansions and the factorization of
polynomials. As a particularly surprising example, we'll expose critical
information on board games that the powerful Washington dice lobby *doesn't*
want you to know...

Thursday, March 20th, 4:10pm

Collins Room 204

Meike Niederhausen, University of Portland

How does a Mathematician get a Nobel
Prize?

or

An Introduction to Mathematical Finance and Option Pricing

Abstract:

In this talk we will introduce the basics of mathematical finance. In particular
we will discuss how to price a European option, which is a

certain type of financial derivative. Options allow investors to hedge losses in
the stock market by capping how much they will be able to buy or sell a stock
for in the future. We will start out with using the discrete time binomial tree
to model stock prices and derive a formula

for the fair price of an option, which will then be generalized to a continuous
time model which uses Brownian motion to model stock prices. This
will lead us to the widely celebrated Black-Scholes formula, for which Fischer
Black and Robert Merton received the Nobel Prize in Economics in 1997.

PG: 45% G, 35% PG, 10% PG13, and 10% R

Tuesday, April 1st, 4:00pm

Collins Room 204

Prof. Tim Chartier, Davidson College

*Improving on your Mistakes: solving
linear systems iteratively
*Abstract:

Learning to solve a linear system (matrix system) Ax = b using Gaussian elimination is a part of many undergraduates' education. Complex mathematical models common in modern science lead to linear systems containing millions or even billions of unknowns. For such systems Gaussian elimination is crippled due to its inefficiency. This talk will discuss how iterative methods attempt to solve Ax = b efficiently and quickly. The first step in an iterative process is to guess at the solution. This guess does not need to be accurate. Then the method uses a series of iterations that generate a sequence of "guesses" that converge quickly to a true solution. After establishing a framework for iterative methods, I will look closely at multigrid methods, which are designed to solve linear systems resulting from partial differential equations.

Wednesday, April 9th, 4:10pm

Collins Room 204

Professor Shereen Khoja, Pacific University

Part-of-Speech Tagging of the Arabic Language

Abstract:

Computational Linguistics is an
interdisciplinary field that researches applications that allow humans and
computers to interact using human language. In this talk, I will give a brief
history of the field, then describe to you my research on Arabic part-of-speech
tagging. Part-of-speech tagging is the process of automatically assigning
grammatical tags to raw text. It is used in the linguistic analysis of texts, in
machine translation, and in speech systems. The talk will also include a
description of the statistical underpinnings of my tagger, which include Hidden
Markov Models (HMMs) and the Viterbi algorithm.

Thursday, April 10th, 4:10pm

Collins Room 204

Andrea Walker and Kyle Evans-Lee, Willamette University

*Parrondo's Paradox Using Markov Chains*

*
*Abstract:

Given two games, each with a higher probability of losing than winning, it is possible to construct a new game, composed of the two original games, with a higher probability of winning than losing. This phenomena is known as Parrondo's Paradox. A Markov Chain is a discrete time chain with the memoryless property. That is, a discrete time chain such that given the present state, future states are independent of the past states. We give the most basic example of Parrondo's Paradox, a coin tossing game, and a basic example of Markov Chains, a random walk around campus. Then, we show how, given two losing games, one can force Parrondo's Paradox to occur. Perhaps this paradox is not so paradoxical after all.

Thursday, April 17th, 11:40am

Collins Room 408

Prof. Eugene Luks, University of Oregon

*The 15 Puzzle, 15000-Page Proofs, and
Parallel Computation*

*
*Abstract:

How do you write a program to solve your 15 puzzle? How about your Rubik's Revenge? Even more basic is testing whether such puzzles are solvable; this is an instance of Permutation-Group-Membership (PGM), which is fundamental to computational investigations in group theory. Standard algorithms for PGM run in time O(n

Thursday, April 24th, 4:10pm

Collins Room 204

Tatiana Mac, Willamette University

*Tangles*

Abstract:

Tangles have served as a building block for mathematicians to
study knots, and for biologists to understand DNA recombination. A specific
type of tangle, rational tangles, are created by twisting two pieces of string
in a specific manner. We will investigate some basic properties of these
tangles which will lead us to a simplified proof of Conway's Classification
Theorem (1970) by J. R. Goldman and L. H. Kauffman (1997), which states that
any two tangles associated with the same rational number are equivalent.