Richard Montgomery

The falling cat problem asks: "How does a cat, dropped from upside down, land on her feet,
and moreover, do so optimally?" I will tell the story of how the falling cat
problem led me inexorably to quantum mechanics of highly charged particles near the zero-locus
of a magnetic field. Here is the abbreviated version of the story. The falling cat problem is a problem in optimal control, more precisely in subRiemannian geometry.
Within those geometries exist certain remarkable geodesics with no analogues in Riemannian geometry.
They are called singular geodesics. Their simplest realization arises
in the Kaluza-Klein description of the motion of a planar particle in a magnetic field near the zero locus of the field.
The existence of singular geodesics is a classical phenomenon in the calculus of variations. Do singular geodesics have
a quantum analogue? The simplest incarnation of that question is the Quantum Mechanical of the title. I will fill in some details of this
story. Time permitting, I will remark on other areas within the geometric analysis of PDE
where singular geodesics have been important.