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2024 Clay Research Award

Newton and Thorne

James Newton and Jack Thorne

A Clay Research Award is made to James Newton (Oxford) and Jack Thorne (Cambridge) in recognition of their remarkable proof of the existence of the symmetric power functorial lift for Hilbert modular forms.

The conjecture that the symmetric powers of automorphic representations associated to classical and Hilbert modular forms should themselves be automorphic is one of the fundamental conjectures of the programme introduced by Langlands in the late 1960’s, indeed it was cited by Langlands as a prototype test case for his conjectures. Building on earlier work of Clozel and Thorne, Newton and Thorne have written a series of ingenious papers proving this result using a very detailed application of modularity lifting results to the associated Galois representations. The proof marks a milestone in work on the Langlands programme.

Paul Nelson

Paul Nelson

A Clay Research Award is made to Paul Nelson (Aarhus) in recognition of his groundbreaking contributions to the analytic theory of automorphic forms.  His work has resulted in the first convexity-breaking bounds for a large class of L-functions on the critical line (including all the standard ones of GL(n)). This marks a signficant advance in a field initiated one hundred years ago by Hermann Weyl in the context of the Riemann Zeta function.  Nelson analyzes L-values via certain associated automorphic periods. His powerful approach involves many ingredients, including a refinement of the orbit method that he developed in earlier work with Venkatesh, and an analysis of the geometric side of an appropriate Relative Trace Formula, which facilitates the use of Amplification.

Maths + Cancer Podcast

PROMYS Europe Executive Director, Vicky Neale, hosts the podcast series Maths + Cancer, exploring the extraordinary and unexpected ways mathematics and statistics is being applied to cancer research across Oxford and beyond.  Throughout the series, Vicky talks to a range of experts to find out more about the role of mathematics and statistics in cancer research, as well as to discover more about the people behind the work.

2022 Clay Research Award

Søren Galatius and Oscar Randal-Williams

A Clay Research Award is made to Søren Galatius (University of Copenhagen) and Oscar Randal-Williams (University of Cambridge) for their profound contributions to the understanding of high dimensional manifolds and their diffeomorphism groups; they have transformed and reinvigorated the subject.

In a celebrated trilogy of papers on moduli spaces of high-dimensional manifolds, they established homology stability for diffeomorphism groups of manifolds in even dimensions 2n>4, where the stabilization is performed by taking connected sums with an increasing number of copies of the product of two n-dimensional spheres. They explained how to compute the limiting homology in terms of a specific spectrum, and beyond their explicit results they developed an array of ideas that have stimulated waves of subsequent advances in the field.

In a later series of papers with Alexander Kupers, Galatius and Randal-Williams pioneered an entirely new approach to homological stability results, developing the sophisticated and powerful theory of cellular Ek algebras. They proved the worth of their theory by settling longstanding conjectures in K-theory, by improving Quillen’s stability results for the homology of general linear groups over finite fields, and by establishing the first general results for the homology of mapping class groups of surfaces outside the stable range.

John Pardon

A Clay Research Award is made to John Pardon of Princeton University in recognition of his wide-ranging and transformative work in geometry and topology, particularly his ground-breaking achievements in symplectic topology.

Pardon’s work displays a remarkable clarity of vision, sustained through extensive projects in which he develops conceptually sophisticated theories that cast long-standing problems into settings where they become tractable. His novel treatment of the theory of moduli spaces of pseudo-holomorphic curves, leading to an elegant construction of virtual fundamental cycles and chains, provides a compelling example.

In a series of papers, Ganatra, Pardon and Shende developed a powerful and subtle descent theory for Fukaya categories, crafted to facilitate diverse applications. This theory has already led to significant advances in representation theory and mirror symmetry, as well as symplectic topology.

Photo by  Denise Applewhite

John Pardon

A 2022 Clay Research Award was made to John Pardon of Princeton University in recognition of his wide-ranging and transformative work in geometry and topology, particularly his ground-breaking achievements in symplectic topology.   

Pardon’s work displays a remarkable clarity of vision, sustained through extensive projects in which he develops conceptually sophisticated theories that cast long-standing problems into settings where they become tractable. His novel treatment of the theory of moduli spaces of pseudo-holomorphic curves, leading to an elegant construction of virtual fundamental cycles and chains, provides a compelling example.

In a series of papers, Ganatra, Pardon and Shende developed a powerful and subtle descent theory for Fukaya categories, crafted to facilitate diverse applications. This theory has already led to significant advances in representation theory and mirror symmetry, as well as symplectic topology.

Photo by  Denise Applewhite

Oscar Randal-Williams

A 2022 Clay Research Award was made to Søren Galatius (University of Copenhagen) and Oscar Randal-Williams (University of Cambridge) for their profound contributions to the understanding of high dimensional manifolds and their diffeomorphism groups; they have transformed and reinvigorated the subject.

In a celebrated trilogy of papers on moduli spaces of high-dimensional manifolds, they established homology stability for diffeomorphism groups of manifolds in even dimensions 2n>4, where the stabilization is performed by taking connected sums with an increasing number of copies of the product of two n-dimensional spheres. They explained how to compute the limiting homology in terms of a specific spectrum, and beyond their explicit results they developed an array of ideas that have stimulated waves of subsequent advances in the field.

In a later series of papers with Alexander Kupers, Galatius and Randal-Williams pioneered an entirely new approach to homological stability results, developing the sophisticated and powerful theory of cellular Ek algebras. They proved the worth of their theory by settling longstanding conjectures in K-theory, by improving Quillen’s stability results for the homology of general linear groups over finite fields, and by establishing the first general results for the homology of mapping class groups of surfaces outside the stable range.

Søren Galatius

A 2022 Clay Research Award was made to Søren Galatius (University of Copenhagen) and Oscar Randal-Williams (University of Cambridge) for their profound contributions to the understanding of high dimensional manifolds and their diffeomorphism groups; they have transformed and reinvigorated the subject.

In a celebrated trilogy of papers on moduli spaces of high-dimensional manifolds, they established homology stability for diffeomorphism groups of manifolds in even dimensions 2n>4, where the stabilization is performed by taking connected sums with an increasing number of copies of the product of two n-dimensional spheres. They explained how to compute the limiting homology in terms of a specific spectrum, and beyond their explicit results they developed an array of ideas that have stimulated waves of subsequent advances in the field.

In a later series of papers with Alexander Kupers, Galatius and Randal-Williams pioneered an entirely new approach to homological stability results, developing the sophisticated and powerful theory of cellular Ek algebras. They proved the worth of their theory by settling longstanding conjectures in K-theory, by improving Quillen’s stability results for the homology of general linear groups over finite fields, and by establishing the first general results for the homology of mapping class groups of surfaces outside the stable range.

Photo:  Ditte Valente/Elite Forsk

Scattering amplitudes and vertex algebras

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.

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Speaker: Kevin Costello (Perimeter Institute)

(some) Things you always wanted to know about Mathematical General Relativity

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Speaker: Igor Rodnianski (Princeton University)

The André–Grothendieck period conjecture over function fields

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Speaker: Jacob Tsimerman (University of Toronto)

Vanishing theorems in algebraic geometry

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.

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

The work of Bhargav Bhatt

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Speaker: Peter Scholze (University of Bonn)

The work of Søren Galatius and Oscar Randal-Williams

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Speaker: Ulrike Tillmann (University of Oxford and INI)