SCU Mathematics Colloquium Series Schedule
Talks will be at 4:00pm Tuesdays in O'Connor 106. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.
Title: New Exact Solutions to Partial Differential Equations Provides a New Approximation Scheme
One of the best known equations in Physics is Schroedinger's equation. It is the equation most often used to describe quantum mechanics. The equation depends upon a given potential energy function, V(x). Until recently there have been few functions V(x), for which exact solutions to this equation were known. Using a technique called Darboux transformations, exact solutions to Schroedinger's equation when V(x) = 0 are used to obtain exact solutions when V(x) is not equal to 0 for a large family of functions V(x). This family is sufficiently large that it appears that any physically meaningful function V(x) can be approximated by one for which the solutions are now known. This provides an alternative method for approximating the solutions to Schroedinger's equation for an arbitrary function V(x). It should be noted that currently this technique only works when there is one space variable, but it extends to solve the variable wave speed equation, heat equation with source, and Helmholtz's equation.
***In this talk I will be describing some of the ideas behind what was done to transform solutions from one equation to another, that the functions V(x) for which exact solutions exist are related to solitons, the idea behind the new approximation scheme, and some of the algorithms that have been implemented to approximate real world data with the functions V(x) for which solutions are now known. Much of this work was done by undergraduates, and the talk will be understandable by anyone who knows what a partial derivative is.
Title: The evolution of Newton's Universe
After Newton developed the equations for the N-body problem, he solved the two-body problem. But when he tried to solve the three body problem -- this is a system such as the Sun-Earth-Moon -- he claimed that it gave him a headache. Newton was only the first to suffer as this problem has been giving mathematicians migraines ever since. On the other hand, we now know much more about the evolutionary aspects of the N-body problem. Some of the weird, some of the well-behaved motions which can occur, as well as the source of Newton's headache, are described here.
The list of talks from previous quarters are available via this archive link.
Last Updated: 13 October 2001