SCU Mathematics Colloquium Series Schedule

Fall 2000

Talks will be at 4:00 Tuesdays, Room -- O'Connor 107. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.

**Oct. 3**Aaron Melman, St Mary's CollegeTitle: New fast algorithms for Toeplitz matrix problems

Toeplitz matrices occur in many applications, ranging from Pade approximations to image and signal processing. Common problems concerning these matrices involve computing their eigenvalues and solving linear systems that have such matrices as their coefficient matrix. Several existing methods for these problems take advantage of the symmetry properties of Toeplitz matrices, but do not exploit them to their full extent. In this talk we will show how to make better use of these properties to construct new, faster and more parallelizable methods.

**Oct. 12 (THURSDAY!)**Ellen Gethner, University of British ColumbiaTitle: A Panoply of Pontentially Pleasing Patterns

The well-known Dutch artist M.C. Escher is a familiar friend to many communities of scholars; included among them are artists, mathematicians, and computer scientists. Doris Schattschneider has recently drawn attention to some Combinatorial problems posed and partially solved by Escher.

Place an interesting asymmetric MOTIF (an abstract design) in a square. Consider the 0, 90, 180, and 270 degree rotations of the design about the center of the square. These are called the four ASPECTS of the motif. Create a decorated TILE T by selecting any four aspects of the original stamp, and placing them in a 2x2 grid formation. Tile the plane with T to create a periodic WALLPAPER PATTERN. With this set-up, how many inequivalent wallpaper patterns can be produced from the collection of 256 possible tiles? (Exercise: Why 256?) Alternatively, how many inequivalent 2x2 tiles with motifs in up to four aspects are there? Escher proved that the answer to this question is 23. His technique was to use brute force and a sketchpad!

I will talk about solutions to generalizations of Escher's problem; the harmony of techniques used are drawn from Combinatorics, Elementary Number Theory, Geometry, and Group Theory. If time permits, I will mention a Coloring Problem of a Different Colour.

Here are some relevant URLs:

http://www.combinatorics.org/Volume_4/Abstracts/v4i2r17ab.html

http://www.combinatorics.org/Volume_4/Abstracts/v4i2r23ab.html

**Oct. 17**Tim Hsu, Pomona CollegeTitle: Geometric group theory and the size of Pac-Man's world

Geometric group theory studies an infinite group G by thinking of it as the symmetry group of an object whose geometry completely captures the algebraic structure of G. In this colloquium, we'll see how to use geometric group theory to answer the following question: "To what extent can an infinite group be understood in terms of its finite quotients?" (And yes, we'll discuss what all this has to do with Pac-Man.)

As background, this talk will assume that you have seen the definitions of groups and subgroups, and perhaps a few examples. Otherwise, no knowledge of group theory will be assumed.

**Oct. 24**Michelle Marvier, Santa Clara UniversityTitle: How math is used in conservation biology

I will discuss two classes of mathematical models that are commonly used to aid the conservation of declining populations. The first class of models describes future population abundance as the result of simple Brownian motion. Such analyses are used to assess the likelihood of extinction over various time horizons and can help conservation practitioners to prioritize populations for intervention and management. The second class of models relies on matrix models, sensitivity analyses, and numerical experiments to identify particular management strategies that are most likely to result in successful recovery of endangered species.

I will apply these models to the population dynamics of Snake River salmon. Although dams are the most visible impediment to salmon survival, we find that improving survival of downstream migrants will have little impact on annual rates of population growth. Conversely, improving survival in the freshwater rearing habitat or in the estuary could greatly increase the population growth rate. These surprising and controversial findings illustrate how mathematical models can lead biologists to answers they may have otherwise ignored due to personal bias and political pressures.

**Oct. 31**Glenn Appleby, Santa Clara UniversityTitle: Invariant factors and the Littlewood-Richardson Rule

Let M and N be two n x n matrices over a discrete valuation ring R. Let \mu, \nu, and \lambda denote the list of orders (with respect to a fixed uniformizing parameter t) of the invariant factors of the matrices M, N, and MN, respectively. If there is a Littlewood-Richardson filling of the skew shape (\lambda)\(\mu) with content \nu, we show a simple and direct matrix factorization using that filling to construct matrices with the appropriate invariants. We shall also discuss the converse: Given such a pair of matrices, can one determine an actual filling of the skew shape?

**Nov. 7**Elaine MacDonald, Sonoma StateTitle: The voter model: why coalescing random walks matter

I will introduce the voter model, an interacting particle system, and give a brief history of some important results. We will focus on the duality between the voter model and systems of coalescing random walks and discuss my latest results on the asymptotics of a restricted coalescing random walk system. This talk should be accessible to anyone who has taken a course in undergraduate probability--we will develop a few needed results about random walks as we go.

**Nov. 14**Joan Hutchinson, Macalester College and MSRITitle: How to color maps (and graphs) when you have only three crayons

Many are familiar with the celebrated Four-Color Theorem, but which planar graphs (or maps) need only two or three colors? A graph can be k-colored, for k a positive integer, if one of k colors can be assigned to each vertex so that two vertices joined by an edge receive different colors. A graph is planar if it can be drawn in the plane without any crossing edges.

It is easy to determine which planar graphs and which general graphs can be 2-colored, but the detection of 3-colorable graphs and even 3-colorable planar graphs is one of the notorious NP-complete problems. We consider a few classes of planar graphs that can be 3-colored, demonstrated by Heawood's Theorem, Groetsch's Theorem, and the Art Gallery Theorem.

What about nonplanar graphs? Natural homes for these are surfaces, orientable and nonorientable, and we consider the general problem of coloring graphs on surfaces and analogues of the planar theorems for coloring some classes of graphs on surfaces with few (like 2, 3, 4, or 5) colors.

**Nov. 28**Victor Flynn, University of LiverpoolTitle: Fractions in Intervals

Suppose that I and J are two open intervals of the real line, and furthermore suppose that J is the same interval as I, but shifted by an integer. For example, I = (1/9, 1/3) and J = I + 2 = (1/9 + 2, 1/3 + 2) = (19/9, 7/3). If we consider all fractions with denominator 8 (not simplified): ..., -2/8, -1/8, 0/8, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8, 9/8, ... we see that two of them (namely 1/8 and 2/8) are in the interval I and two of them (namely 17/8 and 18/8) are in J. We could use any denominator n instead of 8, and it would still be true that the intervals I and J contain the same number of fractions with denominator n.

The same thing is true if J = -I, the negative of I, that is to say the reflection of I in the point 0, for example: I = (1/9, 1/3) and J = (-1/3, -1/9). Similarly, we can let J be -I followed by a shift by an integer, for example: I = (1/9, 1/3) and J = -I + 1 = (2/3, 8/9). In summary,

if: J = I + m or J = -I + m, for some integer m,

then: I and J contain the same number of fractions with denominator n, for every n.A natural question to ask is whether the converse is true. That is,

if: I and J contain the same number of fractions with denominator n, for every n,

must it always follow that: J = I + m or J = -I + m, for some integer m?It turns out that the converse is indeed true, and that there are at least three very different ways of proving it. I shall spend most of the talk on my own proof and another proof of Cassels, both of which use only elementary algebra. I shall also briefly sketch a proof of Montgomery, which uses ideas from analysis.

The talk is intended to be accessible to a general Mathematical audience, including students of any year.

If you have a disability and require a reasonable accommodation, please call Rick Scott, rscott@schubert.scu.edu, 1-408-554-4460 or 1-800-735-2929 (TTY - California Relay).

The list of talks from previous quarters are available via this archive link.

*Last Updated: 13 October 2000*
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