We have divided students into the following four working groups.
Each group will give a presentation during the second week of the
minicourse, as indicated on the schedule.
Bernoulli's solution of the brachistochrone problem was an early triumph in
the calculus of variations, and others (including Euler) subsequently succeeded in solving
isoperimetric problems using similar indirect methods.
This group will study these indirect methods
and their application to the brachistochrone
and isoperimetric problems.
Group: Le, Simon
A dominant technique in early variational approaches was Dirichlet's principle,
which lost favor after some mathematicians (notably Weierstrass) pointed out its
weaknesses. This group will discuss Dirichlet's principle, its flaws, and its
salvation via direct minimization methods (e.g., the role of lower semicontinuity and
the identification of appropriate function spaces).
Group: Greer, Koizumi, Messer
Many interesting problems in ordinary differential equations involve
Sturm-Liouville operators, and the spectra of such operators have an
elegant variational characterization. This characterization extends to higher-dimensional
problems as well, where it often plays a crucial role.
This group will discuss variational techniques for Sturm-Liouville problems.
Group: Cormani, Rudd, Ryham
Dirichlet's principle concerns minimizers of
functionals; it has proven extremely useful to study other critical points of
functionals, such as saddle points. An early success in this direction was
Ljusternik and Schnirelman's work on geodesics, which spawned the study of global
analysis. This group will study geodesics,
starting with the simple problem of geodesics on a sphere, and then discuss
problems related to Ljusternik--Schnirelman theory.
Group: Bisgard, Chu, Sim