We will introduce some basic notions of the Calderon problems and in particular their formulations on Riemann surfaces. Amongst all geometric settings for such problems, results on Riemann surfaces have been most satisfying. We will describe state of the art techniques in solving such problems, which involves existence theory for holomorphic Morse functions with real boundary conditions. Using this result and by proving a class of Carleman estimates, we will construct complex geometric optic solutions which can be used to solve the Calderon problems in two dimensions. If time permits, we will also look at inverse problems in higher dimensions, and briefly discuss why the same technique have limitations on general Riemannian manifolds.

T cells are specialised immune cells responsible for killing pathogens. How these cells are able to self-regulate, however, is unknown. Inspired by analogies with electrical circuts, we develop a mathematical model to infer how immune dynamics are driven by “input signals”. We find that T cell expansion may be a change-detection system, both in response to checkpoint and antigen expression, which has implications for immune therapies against cancer.