2025-26

The conference focuses on recent progress in the study of stochastic dynamics, including contributions from a theoretical perspective as well as the mathematical analysis of numerical methods for the efficient simulation of these dynamics, with particular emphasis on their linear response properties.

Please check the program here

https://sites.google.com/view/sineq-finalconference/home/

We would be grateful if you could circulate it among potentially interested students and researchers.

We are looking forward to seeing you during the week.

The renormalization group is one of the most important tools for the description of critical points in theoretical physics. Most mathematically rigorous treatments are based on implementing the renormalization group as a discrete dynamical system, which introduces a number of theoretical complications, and in particular obscures the emergence of full scale invariance, which is one of the most important features of a critical point. In the last few years, however, there have been several new results about solutions of the Polchinski equation (a nonlinear differential equation which implements the renormalization group as a continuous dynamical system) for Fermionic systems. I will present one in particular, the construction of a nontrivial fixed point of a family of continuous renormalization group flows corresponding to certain weakly interacting Fermionic quantum field theories with a parameter in the propagator allowing the scaling dimension to be tuned in a manner analogous to dimensional regularization. 

We investigate a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension $d\ge 3$, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence.

The presentation is based on a joint work with Francesco Grotto and Mario Maurelli.