Clay Mathematics Institute

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2022 Clay Research Award

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.

Galatius photo (left): Ditte Valente/Elite Forsk