Variational Questions in Mathematics and Physics

In an overwhelming number of questions in mathematics, physics, engineering and economics and other disciplines one seeks solutions that satisfy certain demands or constraints and importantly, one looks for an optimal one. The mathematical theory that addresses these questions is the calculus of variations. Variational questions are deeply rooted in nature in the sense that systems tend to settle in a state that has energy as low as possible.

An example is an excited atom that falls to its lowest energy state while emitting light. Likewise, a hot liquid tends to an equilibrium with the environment by cooling down. The aim of research in this area is always twofold: describe the state to which a system tends and explain how the system reaches that state.

In the case of the atom this means finding the lowest energy state and describing how it decays to that state; in the case for the liquid this means describing how heat flows from hot to cold. The research will pursue such questions in relation to new and relevant examples. There has been considerable progress in understanding how systems such as gases of colliding particles tend to their equilibrium.

In research the investigator will extend these ideas to quantum mechanical systems and to analyze how they tend to their equilibrium. In quantum mechanics these questions are to a large extent open and their answers will shed some light on various issues in quantum information theory. Another question to be pursued concerns magnetic fields that keep electrons confined to a region.

The challenge will be to find fields that are optimal in some precise sense and to describe how the electrons are distributed. It is expected that the fields’ lines have interesting patterns that resemble the ones created in fusion research where one uses magnetic fields to confine hot plasma. Graduate students will be exposed to and contribute to several questions in research level mathematics.

The principal investigator is expected to significantly impact the community, as evidenced by their role as secretary of the International Association for Mathematical Physics.

The project addresses various questions in the calculus of variations. One question is to study zero modes of the three-dimensional Dirac equation. Zero modes are important in quantum field theory and in the question of proving the stability of matter.

It is well known that if the ‘3/2 norm’ of the magnetic field is small then the magnetic field cannot support a zero mode. A similar result holds for the ‘3 norm’ of the vector potential. The aim is to find sharp necessary conditions on these quantities zero modes exists.

It is conjectured that the optimal fields have field lines that look like the Hopf fibration. In this connection investigations will be pursued concerning possible blow up of solutions of the coupled Maxwell-Pauli equations where the magnetic moment is larger than 2. A different research direction will be to understand the approach to equilibrium in certain quantum mechanical systems.

Such systems can be described by Lindblad equations. In general, not much can be said about the rate of approach to equilibrium but there are interesting Quantum Markov Operators that are analogs of the classical Kac master equation. In these cases, quantitative determination is expected of the rate of approach to equilibrium.

One possibility is to do this by computing the gap of the generator. Another attempt is to prove approach to equilibrium in entropy. This will be achieved by establishing analogs of classical entropy inequalities.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.