Schedule of Graduate Student Colloquium: 2008- 09

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 hyper-reals have been appearing in many areas of math, in part because they offer conceptual simplification and shorter proofs. See Terence Tao's blog http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ 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 and topology with the hyper-reals. Then, again, I might not! [this will be, more or less, a repeat of the talk I gave 363 days ago]

If M is a domain in the plane, the frequencies of a drumhead shaped like M are equal to the eigenvalues of the Laplace operator on M with Dirichlet boundary condition. So, mathematically, one may consider the question: with a perfect ear capable of distinguishing all frequencies produced by a drumhead, is it possible to determine the shape of the drumhead? We will discover the answer to this problem which took over fifty years to solve! This talk will also provide some history and introduction to spectral geometry and discuss currently active areas of spectral geometry research.

A finite projective plane is a combinatorial structure with a geometric flavor. While the definition---which is where the talk will begin---is simple, finite projective planes continue to be somewhat mysterious. (This blend of simplicity and mystery is common in mathematics. For example, "prime number" has a simple definition, but what could be more mysterious?) We will consider an interesting and surprising number-theoretic connection that does give some information about finite projective planes, yet leaves much more to be done.

From vector calculus you know the problem of finding a potential function for a conservative vector field. However, there is also a vector field on the punctured 2-plane which does not have a potential function. Looking at this example motivates the introduction of the winding number of a path in the plane around a point and more generally the fundamental group of the punctured plane. These are powerful tools which enable us to prove important theorems. As an example, I will outline the proof of the fundamental theorem of algebra, i.e., every nonconstant polynomial has a complex root.

"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.

Starting in the XIX century we shall try to explain some of the facts that have made the KdV equation so FAMOUS.