ICM 2026 Invited Section Lecture - Nicolas Perkowski
Keywords
Summary
151 words
Critical Evaluation
Value of the Information & Strength of the Argument
The talk provides a novel framework for singular SPDEs by leveraging the structure of divergence-free nonlinearities and a coercivity condition, allowing treatment of supercritical equations where classical techniques fail. The argumentation is rigorous and builds on well-established theories (Di Perna-Lions, Ambrosio, Figalli), with clear explanations of the necessity of each condition. The speaker carefully distinguishes where classical expansion methods break down and how energy Lagrangian flows offer a different route. The value lies in potentially unifying several particle system scaling limits and providing a basis for quantitative convergence results.
Scientific Rigor, Source Quality, Title Accuracy
The scientific rigor is high; the presentation is formal and references multiple key works in the field (e.g., Di Perna-Lions, Ambrosio, Figalli, Gubinelli). However, no URLs are provided in the description, and the talk does not cite specific articles with full references, making source verification difficult. The title adequately captures the content, and the lecture stays focused on the announced topic. The talk appears to present original research and synthesizes previous results without deviating.
178 words
Title / Content Match
The title precisely reflects the lecture's content, which focuses on energy Lagrangian flows for singular SPDEs and their applications.
Quality & Reliability
8/10
The lecture presents rigorous mathematical constructions with references to established theories, but claim verification is not possible from the transcript alone. The argumentation is logically sound and based on known results, though highly technical.
Key Moments
Markers derived by PSI from the transcript: the creator did not define chapters.
- Introduction to particle systems and motivation for fluctuation models
- Formulation of singular SPDEs and examples (KPZ, Dean-Kawasaki)
- Scaling subcriticality and existing methods (regularity structures, paracontrolled distributions)
- Classical regular Lagrangian flows and their applicability to supercritical equations
- Extension to infinite dimensions and challenges with distribution-valued nonlinearities
- Definition of energy Lagrangian flows and the energy estimate
- Existence result via approximation and compactness
- Uniqueness via duality and commutator estimates
- Applications to coupled Burgers and surface quasi-geostrophic equations
- Invariant measures and quantitative approximation results
Contribution & Novelties
The talk introduces a new notion of energy Lagrangian flows that extends classical Lagrangian flow theory to singular SPDEs. This allows handling scaling critical or supercritical equations under probabilistic subcriticality conditions, avoiding the need for expansions used in regularity structures. The main novelty is the addition of an energy estimate to the definition of stochastic Lagrangian flows, which enables uniqueness proofs. The approach has applications to hydrodynamics and numerical analysis, including derivation of invariant measures for open and multi-layer KPZ equations.
Pour aller plus loin :
- Kardar–Parisi–Zhang equation — Central example of a singular SPDE; the talk references its derivative, Burgers equation.
- Regularity structure — Related theory for singular SPDEs, contrasted with energy Lagrangian flows.
- Burgers’ equation — The stochastic Burgers equation is the main example discussed in the talk.
130 words
Radar Profile
The radar profile shows very high technical level and information quantity, with slightly lower scores for quality and reliability due to lack of independent verification. This reflects a dense, high-level mathematical talk intended for specialists, with strong content but limited accessibility.