2002-03 Colloquia talks

Abstract: In linear algebra, the question of determining the similarity classes of $n\times n$ matrices $M_n(k)$ with entries in the field $k$ is a fundamental one, for two matrices are similar exactly when they give rise to the same linear transformation of $V = k^n$ to itself. Alternatively, letting $GL_n(k)$ be the group of invertible matrices in $M_n(k),$ the similarity classes are orbits of this group acting on $M_n(k)$ by conjugation. For $k = \mathbb{C},$ the Jordan canonical form completely determines the similarity classes (orbits) in terms of two essential types of endomorphisms of $V,$ the semisimple ones (determined by diagonal matrices) and the nilpotent ones (determined by strictly upper triangular matrices). Over $k=\mathbb{R},$ however, the situation is more tricky. In this talk, we will examine some of the rich theory describing orbits when we restrict the group of matrices (e.g., to orthogonal or symplectic groups), as the special case of a Lie group or an algebraic group acting on its Lie algebra. We will see how nilpotent orbits in the case $k = \mathbb{R}$ can be related to nilpotent orbits for the case $k = \mathbb{C},$ through the intermediary of symmetric spaces and the Kostant-Sekiguchi correspondence, and we will consider what might happen if $k$ were instead an algebraically closed field of positive characteristic.

This talk is meant to give a snapshot of the world in which a pure Mathematician lives. We will begin with the question:

Question: What properties do the surface of a basketball and the surface of a football share? In what sense are they the same objects? In what sense are they different? This discussion will lead to the notion of a surface (or two dimensional space).

Next, we introduce the three basic geometries (Euclidean, Spherical, and Hyperbolic) and their various properties. Hyperbolic geometry, though the least known of the three, plays a prominent, fundamental role in our understanding of surfaces and the geometries they admit. In fact, we will see that most surfaces admit a Hyperbolic geometry.

We will finish with a discussion revolving around three dimensional spaces.

The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same

This is joint work with Alejandro Uribe from University of Michigan. We prove that, for the moduli space of flat connections on the 2-dimensional torus, the Weyl quantization and the quantization using the quantum group of SL(2,C) coincide. This is done by comparing the matrices of operators associated by the two quantizations to cosine functions. The *-product of the Weyl quantization will be discussed, and it will be shown that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus.

Can one cut a square into 100 squares, or a cube into 100 cubes? Can one cut a regular tetrahedron into regular tetrahedra? I will discuss these and other mathematical Olympiad problems, with emphasis on the methods and tricks used in the solutions.

The Liber abbaci of Leonardo Pisano is famous today as the source of the "Rabbit Problem," which gives rise to the Fibonacci numbers. But, it much more. It contains many, many more problems of a recreational nature and a good deal of the applied mathematics of its time and place. And, it is of special significance in the history of mathematics in the West.