Willamette University Department of

The Frobenius Problem, also called the postage stamp problem, has motivated two summers of math research with Willamette undergraduates. Come hear about the Summer Mathematics Undergraduate Research Program at Willamette and learn about some fun new math!

What do Evariste Galois, Marius Sophus Lie and Bobby Fischer have in common? Each used a novel approach to problem solving which involves imagining a solution has been found and exploiting its properties. Some consequences of this perspective will be explored with examples from differential equations. In certain cases, these properties have significant physical meaning, mathematically formulated as conservation laws. A couple of well-known examples will be presented.

Is this the graph of g(x)=((x^2)-1)/(x-1) near x=1? Your computer and/or calculator thinks it is!

The distortion in the figure is in fact found with all graphing devices that plot points to produce a graph, carrying a fixed number of digits in computation. This talk will explain precisely how roundoff error gives rise to the distorted graph. In studying this graph, as it is given by different electronic graphers, we will not only learn about mechanisms of roundoff error, but we will also see how to determine from the graphs quite a bit about the internal workings of our graphing devices.

The problem of determining the group $G$ of biholomorphic automorphisms which can act on a compact Riemann surface $X$ of genus $g\geq 2$ is classical with results dating back to the late nineteenth century. Recently, due to the great advances in computer algebra systems and computer power there has been a major resurgence in this area. The main goal of this talk is to outline the general method taken to solve the problem of determining automorphism groups, present the recent major results and discuss applications to other areas of mathematics including algebraic geometry, graph theory, Galois theory and cryptography. Time permitting, we shall discuss the problem of classifying infinite families of surfaces whose automorphism groups share certain properties. This talk is mainly expository and should be understandable to those with a reasonable grasp of finite group theory.

Like a handwritten signature, a digital signature is meant to uniquely identify the signer. A cryptographic digital signature algorithm takes as input a secret key and a message, and outputs a signature on the message. In the case of digital signature, the signer is the person who holds the secret key. Suppose that Joe, Betty, Marsha, Alan and Carol are co-members of a budget committee. The committee wishes to digitally sign messages and so would like to have a "group key" where any signature generated with the group key is attributed to the entire group. However, the members don't trust each other. They don't want any one person to know the key lest that person cheat and sign messages on behalf of the committee without the approval of everyone. In this talk we will see how cryptographic secret sharing can solve this problem and identify any cheaters.

As the building blocks of natural numbers, primes have been the subject of intense study since the time of the ancient Greeks. Still, many questions about primes remain unanswered. Examining the number of primes up to some number x lead to the famous Riemann Hypothesis. After 150 years of scrutiny, this hypothesis has yet to be proved or disproved. This talk will examine the link between the distribution of primes, the Riemann Hypothesis, and the study of Random Matrices. Current research at the interface of Random Matrix Theory and Graph Theory will be discussed, and we will see how a chance meeting over tea led to one of the most amazing discoveries of Random Matrix Theory.

Competitive Graph Coloring: How to be Stingy with the Crayons while Coloring Maps with Your Niece

Suppose you are playing the following game with your niece. You
start with a map of the contiguous states in the US. You alternate coloring
uncolored states so that two states that share a nontrivial border may not
receive the same color. You know how to do this efficiently, but your niece
is more interested in making the final product as colorful as possible. How
many colors must be available so that no matter what choices are made, you
will get the entire map colored? We will see that this game extends
naturally to graphs and will consider bounds for the associated parameter
with different classes of graphs.

In mathematics, much attention is given to organizing knowledge so 
that it is readily accessible to all those who wish to build upon its 
foundation. A primary goal for such organization is classifying the objects 
studied. In the 19th century topologists classified 2-dimensional manifolds and 
in recent years there has been a general consensus that the 3-dimensional 
manifolds have been classified.  More recently algebraists have classified the 
finite simple groups.    In this talk I will describe the problem of classifying 
some familiar and some not so familiar number systems.

Determining the number of twin primes is one of the most
easily-understood open questions in mathematics, and one of the hardest to
answer. Nevertheless, there are many facts about twin primes that we can prove,
and there are many interesting conjectures to explore. We shall examine several
of these, and then focus on one fairly new method of determining how many twin
primes there are. We shall show that the sum of the reciprocals of the twin
primes converges, and then see how much we can say about the value of this sum,
called Brun's Constant.

Graphs are frequently used to model networks of connections, such as components of an electric circuit connected by wires, or genetic similarities among species. Intersection graphs model spatial relationships between geometric figures such as polygons or polyhedra. So the study of intersection graphs brings together ideas from the fields of graph theory and discrete geometry in mathematics, and also algorithms in computer science. This is an exciting new field of mathematics, where the interesting problems outnumber the researchers working on them. In this talk I'll highlight some beautiful theorems and open questions about intersection graphs

Mathematics has well-documented relationships with a large collection of other disciplines (physics, economics, biology, etc.), but there are less mainstream connections that can yield equally fruitful results. Today's talk focuses on one such connection. We'll discuss the intersection of linguistics and mathematics, looking primarily at their applications to each other, but addressing along the way methods of detecting cheaters, eliminating spam, and naming new mathematical objects. Finally, we'll address a new methodology for unification called Rooter which could help you get your degree and runs in an amazing O(log log n) time.

In 1933, three Harvard junior-fellows tied together some recurring themes in mathematics, into what Gian Carlo Rota called one of the most important ideas of our day. Do you find independence everywhere you look too?
Now we find that matroids are everywhere. Vector spaces are matroids. We can define matroids on a graph.
Matroids are useful in situations that are modeled by both graphs and matrices. Bicircular
matroids model generalized network flow problems whose algorithms are more efficient than those
available for general linear programming codes.

We introduce graphs in order to define two matroids on a graph, the well-known cycle matroid and the
lesser-known bicircular matroid. In the classical matroid associated with a graph, a set of edges
is independent in the matroid if it contains no cycles in the graph, and the circuits of the matroid are the single cycles of $G$. In the {\it bicircular matroid} of a graph, two cycles form a circuit.
More specifically, the circuits are the subgraphs which are subdivisions of one of the following graphs:
(i) two loops on the same vertex, (ii) two loops joined by an edge, (iii) three edges joining the same pair of vertices.

What questions can we ask about matroids and what might we count? We will discuss some recent results for
bicircular matroids.

A Model of a Least-time Path for a Sailing Ship.

In this talk a model is developed for the path of a ship sailing from
point A(-a,b) to point B(a,b) under the action of wind alone. The wind is
assumed parallel to a straight shoreline, with intensity proportional to
distance from the shore. The relevant Euler-Lagrange differential
equation is solved, yielding an explicit solution for the least-time path.

Expected Value of Minimum Spanning Trees of Graphs with Random Edge Lengths

In this talk, I will introduce the theory of graphs with random edge lengths and explain a recent result that gives the exact formula for the expected length of minimal spanning trees on these graphs

Mission: Impossible: Angle Trisection

Abstract:

It is well-known that it is not possible to construct a trisector
of an arbitrary angle using only compass and straightedge. What tools will
suffice? We will examine Archimedes' classic construction and discuss some
other constructions. Compasses and straightedges will be provided.

In this talk, we will actually see that mathematics can
be employed when playing Tic-Tac-Toe. Everybody knows that the first
player in 3x3 Tic-Tac-Toe should never lose, as long as she plays
intelligently. After all, if a CHICKEN in New York City never loses
the game, it must be easy. We will discuss various strategies in
Tic-Tac-Toe, both the standard game and the game played on "larger
boards." We will discuss different properties of these various
Tic-Tac-Toe games. and Tic-Tac-Toe-type games.