Willamette University Department of Mathematics

2008-2009

Fall Semester

We will give a quick survey illustrating the kinds of mathematical questions
and answers that arise from real-world voting systems. Many of these results
assert that certain election-theoretic desiderata are impossible to attain.
Examples (cryptically stated) include: “simple” description of the
US federal system, an equally good alternative to majority rule, a fair method
of apportionment on which to base the electoral college, and voting systems
for three or more alternatives in which honesty is the best policy. Along the
way we’ll see that sometimes (i) having a vote is just like not having a
vote, (ii) a candidate can lose to an opponent that everyone likes less, and
(iii) gaining a vote can lead to a loss. This talk will be accessible to a
general audience.

One typically doesn't think of number theory and calculus as having much to do with each other. It comes as somewhat of a shock, therefore, that there is a way of defining the "derivative" of an integer which leads to number-theoretically interesting results. In this talk, we'll see how this rather exotic notion of derivative relates to many of number theory's biggest open questions -- twin primes, the Goldbach conjecture, and the prime number theorem. This talk will be accessible to a general audience.

The numbers e and π can't be made to "fit" into
a polynomial equation. Unlike say i, which is a root of x^{2}+1,
there is no integer polynomial of ***any***
degree having either e or π as roots. Numbers like this are called
transcendental. Most students of math know these things, because they have
heard it from reputable sources (like math professors). But it's uncommon
to find a math student who really knows why. We will see an elementary
proof drawing a little each from calculus and number theory. For perspective,
we'll start with a historical summary of transcendental numbers.

American presidential primaries are example of multicandidate elections in which plurality usually determines the winner. Is this the "best" way? How should a society determine a collective judgment from individual preferences? While plurality is a common procedure, it has serious flaws. Are there alternative procedures which are in some sense more "fair"? How do we determine the "fairness" of an election procedure? With no more mathematics than arithmetic, we'll examine some alternative procedures, and some "fairness" criteria.

Every connected graph has spanning trees, and if the graph has
edge weights, we can calculate the length of the minimal spanning tree.
What happens when those weights are assigned randomly? If we consider the
sum of the edge weights in the minimal spanning tree as a random variable, we
can find the expected value (a term in probability that describes the average
value if we repeat the experiment over and over). A previously proven
theorem has been the backbone of our efforts, and we can now quantitatively
describe the expected value of the length of the minimal spanning tree for
several families of graphs. This talk will be accessible to a general
audience, although a foundational understanding of graph theory and probability
would be helpful.

Mathematicians are often interested in classification. Simply
put, sorting

objects into like piles. If those objects are topological spaces, which

for the purposes of this talk can be thought of as subsets of Euclidean

space, one method for sorting is the notion of "homotopy equivalence."

I'll define what this means and give several examples. For instance, we'll

see that two-dimensional Euclidean space with a point removed belongs in

the same pile with the circle. Once we know what "homotopy
equivalence" is

we'll discuss some ways to associate groups to topological spaces

(homology and the fundamental group) in an attempt to distinguish spaces

which are not homotopy equivalent.

This talk will be about work done this past summer with two students in the Willamette Valley REU-RET project. An obstacle representation of a graph G is a drawing of G in the plane with straight line edges, together with a set of polygons called obstacles, such that an edge exists in G if and only if it does not intersect an obstacle. The obstacle number of G is the smallest number of obstacles in any obstacle representation of G. Previous research about obstacle number seemed to suggest that most, if not all, graphs had obstacle number 1. In this talk we'll show that there exist graphs with arbitrarily large obstacle number. On the other hand, most of the graphs we know still have obstacle number 1, and there are a large number of questions still open.

How do we solve equations when algebra isn’t up to the task? For instance, how do we solve x·ln(x+1) = 3? You probably know an approach or two: Graph both sides, use Maple, or use a “solver” on a calculator. We’ll talk about efficient ways to get approximate solutions (to whatever degree of accuracy desired), get some insight into the algorithms used by Maple and calculator “solvers,” and see the pretty “cobweb” diagrams associated with iterated root finding. As a bonus, we’ll see how the method gives gives rise to a study of chaotic behavior, an area of much interest in the mathematical community since the 1980’s.

Spring Semester