## The work of Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jérémie Szeftel

## Details

**Speaker:** Isabelle Gallagher (ENS Paris)

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**Speaker:** Isabelle Gallagher (ENS Paris)

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**Speaker:** Ivan Smith (University of Cambridge)

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Abstract: The generalized Ramanujan conjecture predicts that all cuspidal automorphic representations for GL(n) are tempered. A density theorem is a certain quantitative approximation towards the Ramanujan conjecture that in many cases serves as a good substitute. In this talk I will survey results, methods, and applications.

**Speaker:** Valentin Blomer (University of Bonn)

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Abstract: Moduli spaces of solutions to nonlinear elliptic pdes (anti-self-dual connections, monopoles, pseudo-holomorphic curves, etc.) are a fundamental tool in low-dimensional and symplectic topology. I will discuss foundational aspects of moduli spaces of pseudo-holomorphic curves, in particular how to construct their derived structure using moduli functors, as conjectured by Joyce. Key tools include derived manifolds, log smoothness, and stacks.

**Speaker:** John Pardon (Stony Brook University)

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Abstract: Entropy is a key concept in many fields of physics and mathematics (statistical physics, information theory, dynamical systems): although it is always linked to a notion of complexity, it has a variety of definitions. The aim of this presentation is to understand what it can measure, close to equilibrium, in the process of relaxation towards equilibrium and far from equilibrium. A major issue is to know whether it can measure mixing properties.

**Speaker:** Laure Saint Raymond (IHES)

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Abstract: A complex variety with a positive first Chern class is called a Fano variety. The question of whether a Fano variety has a Kähler-Einstein metric has been a major topic in complex geometry since the 1980s. In the last decade, algebraic geometry, or more specifically higher dimensional geometry has played a surprising role in advancing our understanding of this problem. In fact, the algebraic part of this question is one step of a larger project, namely constructing projective moduli spaces that parametrize Fano varieties satisfying the K-stability condition. The latter is exactly the algebraic characterization of the existence of a Kähler-Einstein metric. In the lecture, I will explain the main ideas behind the recent progress of the field.

**Speaker:** Chenyang Xu (Princeton University)

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A Clay Research Award is made to Frank Merle (IHES, Paris), Pierre Raphaël (Cambridge), Igor Rodnianski (Princeton), and Jérémie Szeftel (Sorbonne Université, Paris). The award is made in recognition of their profound contributions to the theory of nonlinear partial differential equations.

Merle, Raphaël, Rodnianski and Szeftel are recognised, in particular, for their groundbreaking advances in the understanding of singular solutions to the fundamental equations of fluid dynamics, including their construction of smooth self-similar solutions for the compressible Euler equation, and families of finite-energy blow-up solutions for both the compressible Euler and Navier-Stokes equations.

They are also recognised for establishing the existence of finite energy blow-up solutions, arising from smooth initial data, for the energy supercritical defocusing nonlinear Schrödinger equation (NLS) in a spectacular work that resolves a longstanding conjecture of Bourgain. Their discovery of a startling connection between the fundamental equations of fluid mechanics and the NLS has opened a new vista in the field of nonlinear partial differential equations.

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A Clay Research Award is made to Frank Merle (IHES, Paris), Pierre Raphaël (Cambridge), Igor Rodnianski (Princeton), and Jérémie Szeftel (Sorbonne Université, Paris). The award is made in recognition of their profound contributions to the theory of nonlinear partial differential equations.

Merle, Raphaël, Rodnianski and Szeftel are recognised, in particular, for their groundbreaking advances in the understanding of singular solutions to the fundamental equations of fluid dynamics, including their construction of smooth self-similar solutions for the compressible Euler equation, and families of finite-energy blow-up solutions for both the compressible Euler and Navier-Stokes equations.

They are also recognised for establishing the existence of finite energy blow-up solutions, arising from smooth initial data, for the energy supercritical defocusing nonlinear Schrödinger equation (NLS) in a spectacular work that resolves a longstanding conjecture of Bourgain. Their discovery of a startling connection between the fundamental equations of fluid mechanics and the NLS has opened a new vista in the field of nonlinear partial differential equations.

Home — 2023

A Clay Research Award is made to Frank Merle (IHES, Paris), Pierre Raphaël (Cambridge), Igor Rodnianski (Princeton), and Jérémie Szeftel (Sorbonne Université, Paris). The award is made in recognition of their profound contributions to the theory of nonlinear partial differential equations.

Merle, Raphaël, Rodnianski and Szeftel are recognised, in particular, for their groundbreaking advances in the understanding of singular solutions to the fundamental equations of fluid dynamics, including their construction of smooth self-similar solutions for the compressible Euler equation, and families of finite-energy blow-up solutions for both the compressible Euler and Navier-Stokes equations.

They are also recognised for establishing the existence of finite energy blow-up solutions, arising from smooth initial data, for the energy supercritical defocusing nonlinear Schrödinger equation (NLS) in a spectacular work that resolves a longstanding conjecture of Bourgain. Their discovery of a startling connection between the fundamental equations of fluid mechanics and the NLS has opened a new vista in the field of nonlinear partial differential equations.

Home — 2023

Home — 2023

**Frank Merle** (IHES, Paris), **Pierre Raphaël **(Cambridge), **Igor Rodnianski **(Princeton), and **Jérémie Szeftel **(Sorbonne Université, Paris). The award is made in recognition of their profound contributions to the theory of nonlinear partial differential equations.

The presentation of the Award will take place at the Clay Research Conference in Oxford in September 2023.

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Pavel Etingof gave the Clay Lecture at the ICMS conference *Various Guides of Reflections Arrangements*, March 16, 2023

**Speaker:** Pavel Etingof (MIT)