I will share the results of an exploratory study in which undergraduates' conceptions of congruence of integers in the context of modular arithmetic were compared with those of advanced graduate students.

This tutorial is preparation for Professor Healey's colloquium. It presents background physics and mathematics. In the simplest setting a magnetic field is represented as the curl of a vector potential. This vector potential is determined only up to a gradient, so it would seem not to be a physical quantity. However the change of quantum mechanical phase is given by the integral of the vector potential, and there are experiments where this phase change produces measurable physical effects. This is all part of a larger mathematical framework, in which the vector potential becomes a connection on a fiber bundle, the magnetic field is the associated curvature, and the phase change is defined by parallel transport.

This talk will serve as an introduction to Paul Bressler's Friday Seminar talk. Deformation quantization was introduced by Berezin and Flato et al as a tool to construct algebras of classical and quantum observables in terms of the symplectic structure on phase space. I will give a leisurely introduction to the subject and give the Weyl formula for ${\bf R}^2n$.

Several methods have been proposed for assigning non-integer dimensions to fractal structures (such as the attractor ). I will discuss the box-counting method for assigning fractal dimension, which is conceptually very simple, but computationally quite difficult.

I will give an introduction to the basic features of cells in the nervous system as well as describe a few methods for modeling these features.

Bessel Functions are a class of functions solving a family of ODE's. They appear when solving the Laplace equation on a disk. We will show that the Bessel Functions and their "recurrence relations" are consequences of a representation of a particular Lie Algebra on a space of functions. If there is time we will examine the analogous functions and relations for such a representation of sl(2,C).

The formulas Link = Twist + Writhe and Link = Surface Link + Winding Number apply to closed ribbons. A ribbon is a closed smooth curve together with a vector field of unit length on and normal to the curve. One of the formulas will be proved. (There are applications of these formulas to, for example, fluid mechanical helicity, scroll waves, and DNA.)

I will give a brief historical introduction of the topic and present in some detail the two body problem.

I will discuss vector bundles generally and show how to construct rank 2 vector bundles in algebraic geometry.

In this talk I will introduce the reduced Maxwell-Bloch equations, give the Lax pair for the system, and then define a generalized Lax pair, which will give rise to a higher hierarchy of equations . The goal would then be, to find a set of functions that are ad-invariant. Then with the help of the Adler-Kostant-Symes theorem, we will be able to assert that the different systems resulting from the generalized Lax pair, are Hamiltonian and their flows commute.

In 1961, Kato proposed a problem involving the square root of a certain type of operator. In this talk, I will describe the problem and discuss some of the ideas used in the solution. The intent of the talk is to present some of the ideas of harmonic analysis and the types of questions that get asked. Experts in harmonic analysis are welcome to attend but should come prepared to answer questions from the audience and speaker.

We present a numerical solution of the unsteady transonic small disturbance (UTSD) and Euler equations of gas dynamics for a weak shock Mach reflection in a half-space. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of either UTSD or the Euler equations. The numerical solution of the Euler equation uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.

In this talk I will sketch a proof of why any Riemann surface of genus greater that two can be realized as the orbit space of the upper half plane by a subgroup of PSL(2,R). Following this, I will discuss how we may use this to help determine possible groups of automorphisms of compact Riemann surfaces.

Problem solving is a large part of mathematics and as the teaching of mathematics more and more emphasizes the importance of problem solving skills, more research is being devoted to what factors influence problem solving abilities. One such factor is metacognitive abilities. During this presentation, we will discuss what metacognition is, how it relates to problem solving, what research has been done regarding it, and how I plan to incorporate this into a study in the near future.