Homotopy theory and arithmetic: a dialogue

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Questions in number theory abound, some going back to the ancients. While deceptively simple to state, they can be exceedingly difficult to solve. In contrast, in topology (a subject that explores the part of geometry that is insensitive to perturbation) it can be difficult to state problems, but there are numerous established techniques for solving them. It is the purpose of this proposal to use techniques in topology to solve long-standing open problems in number theory, and to use results in number theory to establish new and interesting projects in topology. The central themes are that topology provides an incredibly flexible tool for doing computations of objects of importance in number theory, and that number theory provides very interesting examples of objects useful to topologists.Specifically, this proposal aims to establish number-theoretic conjectures of Malle and others in the setting of function fields using computations of the homology of Hurwitz spaces of branched covers. We will further develop invariants of Lefschetz fibrations using this machinery. In the opposite direction, we will use the arithmetic of etale cohomology to re-imagine the theory of p-compact groups as a form of arithmetic geometry. These techniques should, for instance, lead to a further development of the representation theory of these objects. Lastly, we will pursue computations in stable homotopy theory using arithmetic and geometric techniques via the chromatic homotopy theory program.