We examine the number of cycles of length $k$ in a permutation as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows us to study the formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time $t$, and estimates for the cases of shorter cycles.

We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply our previously developed theory to flag varieties, and we obtain new insights into fundamental categories in representation theory. First, we show that one can recover finite Hecke categories (realized by $\mathcal{D}$-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via $S^1$-equivariant localization. Similarly, one can recover $\mathcal{D}$-modules on the nilpotent cone from coherent sheaves on the commuting variety. We also show that the categorical Langlands parameters for real groups studied by Adams, Barbasch, and Vogan and by Soergel arise naturally from the study of loop spaces of flag varieties and their Jordan decomposition (or in an alternative formulation, from the study of local systems on a Möbius strip). This provides a unifying framework that overcomes a discomforting aspect of the traditional approach to the Langlands parameters, namely their evidently strange behavior with respect to changes in infinitesimal character.

We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (resp., solvable, satisfies a nontrivial identity, elementary amenable, of finite decomposition complexity) whenever $H$ also shares those conditions. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.

We prove that for forms of $U\left(3\right)$ which are compact at infinity and split at places dividing a prime $p$, in generic situations the Serre weights of a mod $p$ modular Galois representation which is irreducible when restricted to each decomposition group above $p$ are exactly those previously predicted by Herzig. We do this by combining explicit computations in $p$-adic Hodge theory (based on a formalism of strongly divisible modules and Breuil modules with descent data which we develop here) with a technique that we call weight cycling.