Axel Ivan Saenz Rodriguez
Probability, Integrability, and Universality: I work on probability models with many particles that have weak interactions. This means that a single particle only interacts directly with its immediate neighbors and the accumulation of these local interactions govern the dynamics of the whole system in a non-trivial manner. In particular, when the number of particles increases, there are more local interactions that influence the global system and it makes the dynamics of the system highly complex. In general, one may not hope to "solve exactly" such complicated systems, but the exception are systems that are "integrable" (i.e. one may write explicit formulas describing the dynamics of the model). More importantly and somewhat surprising, when there are infinite number particles, it has been observed in the laboratory and in the calculations that the behaviour of integrable models is "universal" (under certain conditions) and this means that other models, which may not be solved exactly, behave very similar to an integrable model. This allows to solve for a wide class models by just solving one special model.
Specific Topics: Asymmetric Simple Exclusion Process (ASEP) on the ring, Schur processes, coordinate Bethe ansatz, Eynard-Orantin topological recursion. Painleve equations.
Check out some simulations on my YouTube Channel.