Research Interests
Mathematical Physics
As a mathematician, I seek rigorous explanation and proof for the fascinating phenomena of quantum physics. Research in the area of mathematical physics allows me to explore a wide variety of mathematical topics. My primary interests lie in areas connected to quantum mechanics and quantum chaos.
Quantum Graphs
My dissertation explores the spectral properties of quantum graphs with symmetry. Quantum graphs fall in the intersection of self-adjoint spectral theory and boundary value problems on graphs. Because they offer a simple model of a system with complex topology, quantum graphs can provide an excellent model for studying quantum phenomena which are challenging to analyze in higher dimensions. Some of their applications include:
Other mesoscopic and nanoscopic systems and phenomena
I am interested in studying quantum graphs which exhibit some sort of symmetry. The first class of graphs that I studied are known as circulant graphs, which consist of an interwoven collection of cycles. Circulant graphs derive their name from their adjacency matrices, which are circulant matrices.
More recently, I've been studying Cayley graphs, which encode the abstract structure of a group. I've been focusing on Cayley graphs of finite groups, specifically ones with two generators of order at least 3.
