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McNair Scholars Program

Ashley Wheeler, Ph.D.


Education: Bachelor of Science in mathematics (May 2008)

Doctor of Philosophy in mathematics from the University of Michigan

McNair Project: Symplectic Topology of Hamiltonian systems with One Degree of Freedom (2007)

Mentor: Ricardo Castaño-Bernard, Ph.D.

Given a symplectic surface, is there a Hamiltonian differential equation, H, with any predetermined equilibrium set? We give a simple example of a non-compact symplectic surface which represents the level sets of solutions to Hamilton's equations for a simple pendulum. We then consider a compact example, the torus. From there, we construct more complicated surfaces; based on the geometric properties of these symplectic surfaces we examine where we can place elliptic and hyperbolic singularities. We then use "gluing" maps to study their Hamiltonian differential equations and identify those equations which are physically meaningful.