Massimiliano Gubinelli received a PhD in theoretical physics in 2003 from the University of Pisa. He held positions at the Universities of Pisa, Paris XI, Paris Dauphine and has been Hausdorff Chair at the University of Bonn. He is currently the Wallis professor of Mathematics at the University of Oxford. His interests lie at the crossroads of analysis, probability and mathematical physics. In particular he contributed to develop novel ideas in the analysis of non-linear stochastic ordinary and partial differential equations describing various physical phenomena linked to universality in statistical mechanics and to the Euclidean approach to quantum field theories. He delivered an ICM invited lecture in 2019 and the Lévy lecture at SPA 2019 and at the IMS World Congress in 2020.
This Medallion lecture will be delivered at the 43rd Conference on Stochastic Processes and their Applications (SPA) in Lisbon, Portugal, July 24–28, 2023. Meeting details: https://www.spa2023.org/
Stochastic analysis of rough random fields
In recent years there have been steady advances in our understanding of a broad class of non-Gaussian distributional random fields that can be described via local partial differential equations involving universal source of randomness like a white noise. This line of research extends the fundamentals ideas of Ito beyond the case of diffusion processes and involve a deep interplay between analysis, algebra, probability and eventually geometry. The applications go from the description of the macroscopic universal fluctuations in weakly nonlinear microscopic random systems to the construction of Euclidean quantum field theories below their critical dimension. More generally, these ideas suggests new points of view in the analysis of such random fields via a natural coupling to some Gaussian source of randomness. The partial differential equations involved are usually ill-posed from a classical point of view and in order to tackle them in a fully non-perturbative way one has to develop a fine understanding on how the fluctuations on a wide range of scales are coupled by the non-linearities and how they influence the large scale behavior of solutions. Resolving these challenges also relies on the deep ideas of renormalization theory originating from physics, providing at the same time a framework where to study and implement those ideas in a fully rigorous non-perturbative way.
In this talk I will review some of the progress in these topics and highlight the many open problems and opportunities for new mathematics.