Willamette University                                                                                                    Department of Mathematics

Mathematics Colloquium


2007-2008 Abstracts

2006-2007 Abstracts

Fall Semester

Wednesday, September 10th, 4:00pm

Eaton 209

Prof. Alan Taylor, Union College

The Mathematics of Voting 


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.


Thursday, September 18th, 4:05pm

Collins 204

Prof. Cam McLeman, Willamette University

The Arithmetic Derivative 


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.

Thursday, September 25th, 4:05pm

Collins 204

Prof. Alex Jordan, Willamette University

Transcendence of e and π


The numbers e and π can't be made to "fit" into a polynomial equation.  Unlike say i, which is a root of x2+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.

Wednesday, October 1st, 4:00pm

Eaton 209

Prof. Roger Nelson, Lewis & Clark College

Presidential Primaries: Is Democracy Possible?


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.

Thursday, October 9th, 4:05pm

Collins 204

Jared Nishikawa & Chelsea McLennan, Willamette University

Expected Length of Minimal Spanning Trees on Graphs With Random Edge Weights


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.

Thursday, October 16th, 4:05pm

Collins 204

Prof. Aaron Leeman, University of Oregon

How to tell if your n-dimensional doughnut has a k-dimensional hole


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.

Thursday, October 30th, 4:05pm

Collins 204

Prof. Josh Laison, Willamette University

Obstacle Numbers of Graphs


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.


Thursday, November 6th, 4:05pm

Collins 204

Prof. Mark Janeba, Willamette University

Roots, Cobwebs, and some Chaos: A Tate of Numerical Analysis


   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.


Thursday, November 13th, 4:05pm

Collins 204

Prof. John Caughman, Portland State University

The Friendship Theorem: Graphs Heart Matrices



Spring Semester