## 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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Abstract: Scattering amplitudes are fundamental quantities in quantum field theory. Vertex algebras are well-developed algebraic objects which encode the structure of two-dimensional conformal field theories. I will describe a surprising relationship between these two topics, which goes by the name “celestial holography”. This relationship gives new techniques for computing scattering amplitudes of Yang-Mills theory. No prior knowledge of quantum field theory will be required for this talk.

**Speaker:** Kevin Costello (Perimeter Institute)

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**Speaker:** Igor Rodnianski (Princeton University)

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**Speaker:** Jacob Tsimerman (University of Toronto)

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Abstract: Vanishing theorems in complex geometry, such as the Kodaira vanishing theorem, play an important role in understanding the structure of complex algebraic varieties. Ultimately, they rely on critical input from Hodge theory. After recalling this story, I will survey some recent and ongoing work giving analogous results in mixed characteristic as well as applications to birational geometry; the key input now comes from p-adic Hodge theory.

**Speaker:** Bhargav Bhatt (IAS/Princeton University and University of Michigan)

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**Speaker:** Peter Scholze (University of Bonn)

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**Speaker:** Ulrike Tillmann (University of Oxford and INI)