[PAST EVENT] Mathematics Colloquium/CSUMS Lecture

April 13, 2012
2pm - 3pm
Jones Hall, Room 301
200 Ukrop Way
Williamsburg, VA 23185Map this location
Title: Bridge Disasters Lead to Beautiful Mathematics

Abstract: Abstract: The Tacoma Narrows Bridge collapsed into the Puget Sound on November 7, 1940, after four months of oscillations that earned this suspension bridge the name of "Galloping Gertie." Videos of this disaster have now become a favorite of nonlinear analysts, such as myself, who study nonlinear eigenvalue problems. A simple mathematical model that captures an essential feature of the system is
-u''=au^+-bu^-, u(0)=u(\pi)=0,
where $u^+,u^-$ are the positive and negative components of the displacement function $u$. This is an asymmetric version of Hooke's Law and can be studied effectively (up to a point) using techniques from an undergraduate ODEs course. In the late 1970s Svatopluk Fucik and Norm Dancer independently realized the importance of this model, and its generalizations, and initiated a rigorous analysis of the eigenvalue pairs $(a,b)$, and the solvability of the associated problem
\[ -u''=au^+-bu^-+g(x,u), u(0)=u(\pi)=0. \]
This problem is called a {\em resonance} problem if $(a,b)$ is an eigenvalue pair, and a {\em nonresonance} problem if not.

In this talk I will give an introduction to the problems described above and will then present some recent advances. In particular I will discuss how the eigenvalue pairs $(a,b)$ can be characterized variationally, i.e., as minima or minimax values, and how that characterization aids in understanding resonance and nonresonance problems.