Abstract: For over a decade, I have had the privilege to collaborate and learn from internationally acclaimed, Brooklyn-based choreographer Reggie Wilson and his Fist and Heel Performance Group https://www.fistandheelperformancegroup.org/ .  This engagement has given me new practices for teaching and mentoring, widened my sense of where and how mathematical knowledge is held, and surfaced unexplored areas near the heart of mainstream geometry and topology. In this short talk, I'll try to give a quick sense of how this came about, what it's taught me about being a mathematician, and why I think dance and bodies have more to teach us mathematically.

 I am interested in eigenvalues and eigenfunctions of the Laplacian. One area that I have been active recently is the inverse spectral problem for plane domains. We would like to know whether the eigenvalues of the Laplacian of a bounded smooth domain determine the shape of the domain. I will report on recent developments in this area. 

 Nonlinear elliptic PDEs arise in many physical and geometric contexts, for example in models of soap films, crystal surfaces, and cloud motion. I will discuss some of the questions mathematicians aim to answer about such PDEs, using the minimal surface equation and Monge-Ampere equation as guiding examples.

The notion of viscosity solution was introduced in 1980s by Evans and Crandall/Lions, which is one of the most important developments in the  theory of elliptic equations.  It provides a rigorous mathematical framework to describe the correct ``physical" solution of  first or second order PDEs when classical solutions might not exist. Important examples include first order Hamilton-Jacobi equations or second order degenerate elliptic equations (e.g mean curvature type equations) arising from control theory or front propagation problems in real applications.   In this talk, I will go over basic definitions, some important techniques, fundamental results and interesting examples.