Josh Southerland

About

I am a graduate student in Mathematics at the University of Washington being advised by Jayadev Athreya. I did my undergraduate in Mechanical Engineering at Columbia University in New York City. After graduating, I worked in New York City for seven years at BuroHappold Engineering. While I was working I started taking math classes at my alma mater, and found some phenomenal guidance. Thank you Professor Gallagher, Professor Khovanov, Professor Weinstein, and Professor Diogo. If you love math and want to talk, or are curious about the nexus of engineering, sustainability, and climate change, feel free to reach out.

j s o u t h e r AT u w . e d u

Research Interests

are currently developing! My current interests center around homogeneous dynamics and translation surfaces, where I am beginning to investigate questions motivated by quantum mechanics. Luc Hillairet and others have been instrumental in constructing a Laplacian for a special class of translation surfaces, square-tiled surfaces, and I hope to understand this construction and look for ways to further the connection between the quantum mechanics realm (eigenfunctions of the Laplacian) and the classical mechanics realm (geodesics).

More recently, I have begun to explore a concrete manifestation of this correspondence on hyperbolic surfaces. Anke Pohl, Don Zagier, and others have established a connection between eigenfunctions of the hyperbolic Laplacian and geodesics on the surfaces. Underlying this construction are transfer operators, which gives us a way to connect a discrete dynamical system (along geodesics), which on the modular curve is related to continued fraction expansions, to eigenfunctions of the Laplacian on that hyperbolic surface.

I am also studying representation theory, and in particular, how the representation theory of $SL_2(\mathbb{R})$ gives us a way to understand certain spectral properties of the Laplacian on hyperbolic surfaces. Daniel Bump has written several good surveys and a book on automorphic forms that I have found helpful.