SCU Mathematics Colloquium Series Schedule

Spring 2002

Talks will be at 4:00pm Tuesdays in O'CONNOR 206. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.

**April 9**Rhonda Righter, Santa Clara UniversityTitle: The Distribution of Losses in Finite Buffer Systems

My research was motivated by the following interesting fact. Consider a queueing system with Poisson arrivals, a single server with random and arbitrarily distributed service times, and a finite buffer for holding customers. If the arrival rate equals the service rate, the expected number of losses during a busy period is 1, regardless of the buffer size. This result was originally shown with very complicated transform arguments. I will explain the simple but powerful notion of stochastic coupling, and use coupling arguments to develop a stochastic relationship among queueing systems with buffers of different sizes. This relationship allows us to examine in detail the effect of buffer size on losses and the approach is easily extended to batch arrivals and multiple servers. The result described above for mean losses is a simple consequence of the stochastic relationship.

Much of the work described here was done with Erol Pekoz of Boston University and Cathy Xia of IBM Research, and was supported by the Breetwor Fellowship of the Leavey School of Business at Santa Clara University.

**April 16**Byron Walden, Santa Clara UniversityTitle: Harmonic Measure, Brownian Motion and Conformal Invariance

Harmonic measure, essentially the solution to Dirichlet's problem using a characteristic function for boundary values, is an important conformal invariant in complex analysis.We will examine its role in the interplay between conformal mappings and Brownian motion in the plane, as well as its utility in geometric function theory, as illustrated by Øyma's proof of the Hayman-Wu theorem: the length of the image of the intersection of any line with a simply-connected domain in the plane under a Riemann mapping of the domain onto the unit disk is bounded above by 4(Pi).

**April 23**Frank Farris, Santa Clara UniversityTitle: Forbidden Symmetry: Relaxing the Crystallographic Restriction

If you look at enough swatches of wallpaper, you will see centers of 2-fold, 3-fold, 4-fold, and 6-fold rotation. Why not 5-fold centers? They cannot occur, according to the Crystallographic Restriction, a fundamental result about wallpaper patterns, which are defined to be invariant under two linearly independent translations. Even so, we offer convincing pictures that appear to show wallpaper patterns with 5-fold symmetry. What is going on? The talk is intended to be accessible to students.

**April 30**Tom Allen, Lockheed Martin Space SystemsTitle: Are your deviates random, or are they just mixed up?

When computer simulations predict the behavior of existing or contemplated equipment, random variation of many system parameters must also be simulated. The fidelity of the simulation may hinge on how well the randomness is simulated. This area of computing science has long been notorious for trapping naive programmers, and has recently become even more difficult. Today, millions of random quantities may be needed for each simulated second of behavior. Random generators that were considered adequate just two decades ago can be exhausted in minutes by today's simulations. Recently developed random generators offer both high speed and much better fidelity. Although the mathematics underlying these new generators has a futuristic "look and feel," most of it was invented long ago.

**May 7**Roger Alperin, San Jose State UniversityTitle: Geometrical Constructions: Origami, Field Theory and Algebraic Curves

I will discuss axiomatic mathematical origami, its connections with constructions in classical geometry, and the use of algebraic curves as an aid in constructions. The axiom systems are hierarchical in nature and give rise to different types of field extensions of the rationals.

**May 14**Jeffrey Sink, Santa Clara University (note the date change)Title: Why non-mathematicians care about knots

Starting in the 1800's, scientists of various fields have found knots and knot theory to be crucial to their field of study. The were used as a model for atoms before quantum theory (as knots in the ether!). Now knots and their invariants (to be explained) appear in String Theory, but not in the obvious way. The third intersection with the sciences is the structure of mitochondial DNA.

This will be a survey of the field, with a basic introduction to Knots, their invariants, and some basic notions of how/why other scientists care about them.

**May 21**Moshe Baruch, UC Santa Cruz (note the date change)Title: Unitary representations of Lie groups and the Kirillov conjecture

Abstract: We describe how irreducible unitary representations of a Lie group play a role in the study of Fourier analysis on the group. We will indicate how the Kirillov conjecture played a role in the attempt by Gelfand and Naimark around 1950 to classify the irreducible unitary representations of GL(n,C) and GL(n,R) and if time permits we will outline the proof of the conjecture.

**May 28**George Fegan, Santa Clara UniversityTitle: How to review for your Linear Algebra final

An examination of the Moore-Penrose psuedoinverse forces one to bring together many of the topics covered in a Linear Algebra course. The talk will also show how some textbooks and/or a misuse of Matlab can be more detrimental to one's consulting career than a trip to Arthur Anderson.

The talk is targeted at undergraduate students who need to cram for their linear algebra final.

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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