Schedule of Graduate Student Colloquium: 2009-10

I will explain how to solve a system of simultaneous polynomial equations like    x²+ x y + 1 = 0     y² + xy +3 = 0
using resultants. This example can be easily done, others are a bit trickier.

First, we shall discuss the dynamic of point vortices in some simple cases. Then we shall consider the self-induced motion of a vortex filament (all this without using the equations). Finally, in a more formal manner we shall study the motion of almost parallel vortex filaments.

"Free-to-move" boundaries occur naturally in fluids and materials. The surface of the ocean is a familiar example of a free boundary between two fluids, water and air. Soft materials like plastics, resins for making DVDs, and liquid crystals are also characterized by tiny structures which are free to move and interact with processing flows. The complex motion of these boundaries is the source of fascinating phenomena of both scientific and technological importance. In this talk, I will illustrate with several examples the crucial role that Mathematics plays in the development and analysis of both models and computational methods to better understand these "complex fluids" and eventually to be able to predict their properties.

The properties of a material depend on its atomic structure. A complete understanding of the material at this level is not possible nowadays, due to the large number of degrees of freedom in the system. As a result, a number of methodologies have been developed in order to describe the macroscopic properties of a solid, while including microscopic information at some level. I will discuss how Mathematics is playing a fundamental role in the development and the analysis of so-called multiscale approaches that are consistent and rigorously derived from first principles. I will focus most of the discussion on the quantum-mechanical approach.

You may already know the answer to the question "Which odd primes can be represented as the sum of two square?" We will explore generalizations of the question, using Mathematica to generate data. This talk is completely elementary (The connection to RH is tenuous, but see http://aimath.org/RH150/rhdayschedule.html )

The hyper-reals are an ordered field containing the real numbers as well as infinitesimals and infinitely large numbers. They have languished for 50 years, spurned by most professionals. The situation recalls the slow acceptance of other extensions of the concept of number. For example, as late as the 1880's, Kronecker disputed the existence of irrational numbers. Recently hyperreals have been appearing in many areas of math, in part because they offer conceptual simplification and shorter proofs. See Terence Tao's blog. We will construct the hyper-reals, and it will become evident that, just from this simple definition, one can deduce most things one wants to know. At the end I might say a few words about doing geometry with the hyperreals. In particular how non-standard hyperbolic and projective structures on manifolds over the hyper-rreals can be used to give insight into standard structures. This talk is a repeat.