SCU Mathematics Colloquium Series Schedule

Fall 2001

Talks will be at 4:00 Tuesdays, Room O'Connor 205. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.

**Oct. 2**Maria Schonbek, UC Santa CruzTitle: Decay of solutions to nonlinear fluid equations

Abstract: In this lecture I plan to discuss results concerning the decay of energies of solution to nonlinear diffusive equations arising in Fluid Dynamics. In particular I will show that solutions to equations like Navier-Stokes decay at the same rate as the solutions to heat equations, which are the underlying linear part of these equations. The norms where the decay will be considered will include energy norms like L

^{2}and L^{1}. I will also discuss the pointwise decay in space and time. When possible I will show why the methods I am introducing work for several different nonlinear equations, provided they are diffusive.

**Oct. 9**Rick Scott, Santa Clara UniversityTitle: Small covers of regular hyperbolic polyhedra

Abstract: The classical approach to understanding different types of geometries (e.g., Euclidean, hyperbolic, spherical) is to study properties of standard geometric objects like points, lines, angles, polygons, etc. In this talk, we will describe some features and examples of polygons and polyhedra in hyperbolic space. Using certain regular hyperbolic polyhedra as building blocks, we will then describe a class of hyperbolic manifolds, called "small covers", and give a geometric classification.

**Oct. 23**Maciej Zworski, UC BerkeleyTitle: Numerical linear algebra and solvability of PDEs

**Oct. 30**Nicholas Tran, Santa Clara UniversityTitle: An Easy Case of Sorting by Reversals

Abstract: Biologists have recently discovered that genome rearrangements is a common mode of molecular evolution in mitochondrial, viral, and bacterial DNA. Genome rearrangements occur when two species have the same set of genes in their genomes, but the genes are arranged in different orders.

The most common mode of rearrangement is the reversal of a chromosome portion. This observation motivates the problem of SORTING BY REVERSALS: find the minimum number of reversals required to transform one genome to another. This REVERSAL DISTANCE is believed to be a useful measure of the evolutionary path between the two species.

In this talk, we will present an efficient algorithm for sorting by reversals when the distance is half the number of breakpoints. This result answers an open question posed by Kececioglu and Sankoff in 1995.

**Nov. 6**Stan Benkoski, West Valley CollegeTitle: The Search for the Scorpion Submarine

Abstract: In 1968, the Scorpion (a nuclear powered submarine) disappeared without a trace during a westbound transit across the North Atlantic. The search for the Scorpion lasted for 5 months and one observer called it the "most difficult search operation ever undertaken and pressed to a successful completion." The mathematics used in this search includes hyperbolas, Monte Carlo, techniques, Bayes' Theorem, and Lagrange multipliers. We will demonstrate the use of mathematics in the context of this specific search.

**Nov. 13**Ernest Croot, UC BerkeleyTitle: Unit Fractions

Abstract: A unit fraction is a rational number of the form 1/n, where n is a non-zero integer. Such rationals are sometimes called Egyptian Fractions, in honor of the ancient Egyptians who first used and studied them. Since the time of the Egyptians, unit fractions have been a source of many fascinating and difficult conjectures, some of which are still unsolved. One such unsolved conjecture is the following, which is due to Erdos and Straus:

Is it true that for every integer n > 1, there exist positive integers x,y, and z such that 4/n = 1/x + 1/y + 1/z?

In this lecture I will give a survery of some of these conjectures and results, as well as mention some recent progress in the subject.

If you have a disability and require a reasonable accommodation, please call Rick Scott rscott@math.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 2001*
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