Research: Partial Differential Equations of Many Body Quantum Mechanics
Undergraduate Institution: Syracuse University
Hometown: Scranton, PA
Why did you choose to come to AMSC?
I chose AMSC because it is uniquely interdisciplinary with close ties to the pure math program. I was undecided about entering a physics, applied or pure math PhD program, and AMSC provided a perfect median.
What are your research interests and why?
My research currently lies at the intersection of analysis of PDEs, kinetic theory, and condensed matter physics. In particular, I’m interested in approximating the Schrödinger equation for many particles -- a high-dimensional PDE that depends on all particle coordinates -- by an “effective” low-dimensional equation in certain limiting regimes. It turns out that a specific kind of perturbation theory (mean-field theory) can be used to model Bose-Einstein condensation, a bizarre phase of matter unique to low-temperatures and integer-spin systems.
I’m interested in this approximation from a mathematical perspective. It drastically reduces the complexity of the Schrödinger equation, but giving a proof of its validity is much more complicated. In general, I think kinetic theory is a very cool, very strange part of pure mathematics. A lot of thought has been generated at the intersection of physics and math.
What do you like to do in your free time?
I enjoy fooling around with graphic design software and 3D modeling software.
What are your favorite parts of the greater Washington DC area?
My favorite parts of the area are the Normal Foster atrium of the American Portrait Gallery, any Smithsonian museum, Patuxent Research refuge, and downtown Bethesda.
What advice would you give to incoming students?
Take your time with classes! It is better to take fewer classes per semester and to work methodically in them. If you feel like your understanding is slipping in a course or on a subject (it will, it always does for everyone), don’t be intimidated. There are many ways to learn, and it's more-or-less random what you will attach meaning to first.
Personally, I didn’t understand PDEs at all until after I took functional analysis in my second year. The surest way for me to pick-up on a subject is to approach it from multiple angles (classes, different books, seminars, chatting with peers) and not take myself too seriously while doing so (taking some proofs for granted, playing with symbols in a heuristic way). Eventually, things will click.