Homotopy theory is in the midst of a renaissance as its usefulness in other areas of mathematics is becoming increasingly recognised. From polyhedral products in geometric group theory, to combinatorial methods in group theory, to topological properties of manifolds and spaces of embeddings, to Morava $K$-theory in symplectic geometry, to simplicial methods in topological data analysis, to a host of applications in mathematical physics - and these name only a few areas of interaction - homotopy theory over the past ten to fifteen years has significantly extended its reach. This program will assemble a wide range of leading experts to discuss the state of the art in current research, both in terms of core fundamental problems and crossover to other areas, and explore exciting new directions.

 

The program will also be a memorial for Fred Cohen who passed away recently in January 2022. Fred was a homotopy theorist of striking versatility and profound depth, engaging tirelessly with anyone who wanted to talk mathematics. The range of problems and areas on which he worked is scarcely believable. Highlights include his work with Lada and May on the homology of iterated loop spaces that has become foundational; with Moore and Neisendorfer he proved a magnificent result determining the odd primary exponent of the homotopy groups of spheres; with Bahri, Bendersky and Gitler, and building on work of Grbić and Theriault, he founded the systematic study of polyhedral products, which unify multiple constructions from disparate areas, and proved several fundamental properties. Fred also did important work on configuration spaces, braid groups, spaces of commuting elements in Lie groups, function spaces, mapping class groups, and combinatorial group theory. He was very much ahead of his time in seeing connections between homotopy theory and other areas of mathematics, and the program.

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Clay Mathematics Institute