Short- and large-scale limits of stochastic PDEs and interacting particle systems through rigorous renormalization group techniques. KPZ equation. Non-linear sigma model.

Fourier normal ordering algorithm. Complex techniques for fractional Brownian motion. Renormalized Lévy area. Ultraviolet "evanescent" interaction and constructive field theoretic methods.

Log-gas dynamics/generalized Dyson's Brownian motion. Dynamical loop identities. Connection to 1d quantum systems.

Schrödinger-Virasoro algebra and dynamical Schrödinger symmetries. Algebraic and geometric study. Action by reparametrization on equations of mathematical physics. Applications to log-gas dynamics. An application to hard-core bosons. 

1d quantum systems. Superconductivity and Bardeen-Cooper-Schrieffer (BCS) theory.