Exotic Spheres from p-adic Cohomology Theories

A smooth, oriented n-manifold is called a homotopy sphere if it is homeomorphic, but not necessarily diffeomorphic, to the standard n-sphere. In dimensions n>4

, one often studies the group Θn of homotopy spheres up to orientation-preserving diffeomorphism, with group operation given by connected sum. I will give a leisurely introduction to the telescope conjecture in stable homotopy theory, and explain how its failure gives new lower bounds on the complexity of Θn. To disprove the telescope conjecture, we construct invariants capable of distinguishing many diffeomorphism classes of exotic spheres: interestingly, key finiteness properties of these invariants are proved in part using intuitions and ideas from prismatic cohomology in p-adic arithmetic geometry. As time permits, I will discuss how homotopy theory suggests generalizations of fundamental theorems about prismatic F-gauges, such as the Lagrangian refinement of Tate duality highlighted in Bhatt's ICM address. I will explain a purely homotopy theoretic construction of Drinfeld's formal group on the Cartier--Witt stack. The talk is based on joint projects with Burklund, Carmeli, Devalapurkar, Levy, Raksit, Schlank, Wilson, Yanovski, and Yuan.