SCU Mathematics Colloquium Series Schedule

Spring 2004

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

**Apr. 6th**Charles Hamaker, St. Mary's CollegeTitle: A Protocol for Scalable, Peer-to-Peer Resource Sharing on Networks

A peer-to-peer network distributes the holdiing of resources equally among its members. CHORD, a protocol for such a network, provides a mechanism that permits stable network performance as members and resources join and leave in such a way that resources and transactions are uniformly distributed across the membership.

**Apr. 13th**Keith Devlin, StanfordTitle: How much mathematics can be for all?

In my book The Math Gene, I presented an evolutionary argument to show that the basic capacity for mathematical thinking is present in everyone as part of our genetic inheritance. But how much mathematics comes in this way? Is there a point beyond which most people will simply never "get it"? I believe there is sufficient evidence to indicate that the answer is yes. Moreover, among those parts of mathematics that can be mastered only by a few are some topics taught in the middle school. This talk is a sequel to Devlin's book The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. (Basic Books, 2000)

**Apr. 20th**Maria Schonbeck, UC Santa CruzTitle: Fluid Equations and their Asymptotic Behavior

Phenomena arising in fluid dynamics often evolve driven by some form of diffusion. In this lecture we will consider different types of systems of nonlinear partial differential equations which describe such models. Due to the dissipative mechanism of the systems the corresponding solutions can be shown to decay at different speeds in appropriately chosen norms or energies spaces. The models I will discuss are represented by the Navier-Stokes, the Magneto-Hydrodynamics, the Convection Diffusion and the Quasi-Geostrophic equations. Attention will be focused on the long time behavior of the solutions to these systems. We will discuss methods that will give upper and lower bounds on the rates of decay. In particular we want to explore common features regardless of the nonlinearity of the corresponding system. In this direction we will show cases were the solutions and their underlying linear counterpart decay at the same rate which as such will be optimal.

**Apr. 27th**Arek Goetz, SFSUTitle: The World of Microscopic Geometric Tiles in the Dynamics of Piecewise Isometries

In this talk, accessible to beginning undergraduate students, we will illustrate a class of dynamical systems that give rise to intriguing microscopic geometric structures. Dynamical Systems is an an active area of mathematics in which objects of interest the behavior of orbits. Given a space X and a function acting on this space, f, an orbit is a sequence, {x, f(x), f(f(x))}, etc. While the talk will be augmented by a multimedia presentation and the computer is central in the visualization aspect, the talk will be rigorous in nature and it will include an illustration of number theoretic and geometric tools. The archive of multimedia used is located at: math.sfsu.edu/goetz/

**May 11th**Stephanie Basha, SCUTitle: Systems of Orthogonal Polynomials Arising from the Modular j-function

Let $S_p (x) \in F_p[x] $ be the polynomial whose zeros are the $j$-invariants of supersingular elliptic curves over $\overline{F}_p$. Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier, we define an inner product ${\left\langle , \right\rangle_{\psi}}$ on $R[x]$ for every $\psi(x) \in Q[x]$. Suppose a system of orthogonal polynomials $\{P_{n,\psi}(x) \}_{n=0}^{\infty}$ with respect to ${\left\langle , \right\rangle_{\psi}}$ exists. We prove that if $n$ is sufficiently large and $\psi(x)\P{n}(x)$ is $p$-integral, then $S_p (x) \mid \psi(x) P_{n,\psi}(x)$ over $F_p[x]$

**May 18th**David Nash, SCUTitle: Cayley Graphs and Expanders

Expanders are certain families of graphs that have many important applications to computer science. A hard open question is whether Cayley graphs of the symmetric groups can contain an expanding family. We consider the special case of Cayley graphs generated by reversals (a reversal is an involution that reverses the order of an entire subinterval of 1234...n).

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).

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