SCU Mathematics Colloquium Series Schedule

Fall 2003

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

**Sept. 25 (Thursday)**Christian Klingenberg, University of WurzburgTitle: Conservation laws with space dependent flux function

This is an introductory lecture into the theory, numerics and application of physical models which may be described by conservation laws where the flux may change in space. Applications are oil reservoirs or sedimentation processes.

**Oct. 7**Doris Schattschneider, Moravian CollegeTitle: Escher's Combinatorial Patterns

From 1938-1942,the Dutch graphic artist M.C. Escher carried out experiments in printing repeating patterns with small carved wooden squares that contained a simple motif, using a purely algorithmic scheme. He asked and answered combinatorial questions about these patterns, and opened the door to some other tantalizing questions that have recently been addressed by mathematicians and computer scientists.

**Oct. 14**Jonathan Goodman, Courant Institute, New York UniversityTitle: Numerical Solution of Stochastic Differential Equations

Stochastic differential equations (SDEs) have long been used to model the effect of noise on dynamical systems. Applications include physical systems ranging from biological molecules to large engineering systems, as well as telephone and computer networks and financial markets. We give a quick introduction to SDEs and their associated partial differential equations (PDEs). We discuss mathematical issues that arise in computer generation of approximate stochastic trajectories, including the notions of weak and strong accuracy. We suggest a third measure of accuracy that measures the behavior of paths in a statistical rather than in a pathwise sense. The main technical tools are short time asymptotic expansions of the Greens functions for the PDEs and some martingale theory.

**Oct. 21**Francis Su, Harvey Mudd CollegeTitle: Minimal Triangulations of Cubes

Among the polyhedra, cubes seem like rather simple objects, and easy to describe. Yet many aspects of the geometry of cubes are not known. For instance, what's the minimal number of tetrahedra that you can put together to build a 3-dimensional cube? (Try it! Another challenge: what is the maximal number, if your tetrahedra corners are at cube corners?) What about cubes of other dimensions? Even in "low" dimensions like 4 and 5, such questions can be challenging.

In this talk, we'll develop some intuition for cubes, and then demonstrate a new approach to obtaining lower bounds for minimal triangulations of cubes, that improve current bounds up through dimension 12. Techniques include simple geometry, and only a modest amount of linear algebra (which won't be needed to understand the pictures that we'll draw). These results are joint work with HMC undergraduate Adam Bliss.

**Oct. 28**Bob Megginson, Deputy Director, MSRITitle: Native American Mathematics

Though mathematics students often have learned quite a bit about the Indo-Mediterranean heritage of some of modern mathematics, and many have some awareness that indigenous peoples of Central and South America did mathematics prior to European contact, fewer know of the depth and variety of use of the mathematics developed in the Americas prior to 1500. This talk will explore that, with a focus on Central American mathematics but with excursions into the mathematics of some other Native North American peoples if time allows. The talk will be completely accessible to students who have had a first course in number theory, and most of it will require only an interest in number systems and other cultures.

**Nov. 4**Russ Merris, California State University, HaywardTitle: Polynomial Invariants of Graphs

Among graph theorists, the search for a short list of easily computed invariants sufficient to distinguish nonisomorphic graphs is something of a quest. Ninteenth century linear algebraists had a similar quest: To find a short list of easily computed invariants sufficient to distinguish dissimilar n-by-n matrices. The resolution in that case turned out to consist of n polynomials, of which the characteristic polynomial is one. The talk involves some largely unexplored "immanantal" graph polynomials.

**Nov. 11**Jorge Aarao, Claremont McKenna CollegeTitle: Solving two-way diffusion equations

The heat equation, relating how a distribution of temperature in space changes in time, is both well known and well understood. We know different methods for solving it, we know how to write down formulas for the solutions, we know that heat has a preferred direction of diffusion in space-time. Less well known are the so-called two-way diffusion equations, where there is more than one direction of diffusion - as if we had time running backwards in some places. I will tell you about one such equation arising from plasma physics, and will discuss a method for solving a whole class of such equations.

**Nov. 18**Byron Walden, Santa Clara UniversityTitle: The Arithmetic-Geometric Mean via Harmonic Measure

Study of the arithmetic-geometric mean touched off a blaze of mathematical progress in classical analysis, number theory and approximation theory. We will develop the basic details of the arithmetic-geometric mean iteration and its surprising connections to elliptical integrals, using a novel interpretation from harmonic measure in the plane. While the initial harmonic-measure interpretation of the agm is a bit off-the-wall, the applications to elliptical integrals follow in a much more direct fashion than the standard approach seems to allow.

**Nov. 20 (Thursday)**Edward Packel, Harvey Mudd CollegeTitle: Projectile Motion with Resistance, Experimental Mathematics, and the Lambert W Function

A new closed-form solution to the range formula for projectile motion with linear resistance will be developed with the help of symbolic computation and the increasingly applicable Lambert W function. Using a newly developed limit theorem for a class of inverse functions, the inverse range problem of finding angles that result in a given projectile range can also be solved in closed form.

Issues involving "Experimental Mathematics" and the role of computer algebra systems in doing mathematics will also be discussed.

If you have a disability and require a reasonable accommodation, please call Aaron Diaz aaron@turing.scu.edu 1-408-554-6811 or 1-800-735-2929 (TTY - California Relay).

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

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