Let G be a reductive group, and let U, $U^{-}$ be the unipotent radicals of a pair of opposite parabolic subgroups P, $P^{-}$. We prove that the DG categories of $U((t))$-equivariant and $U^{-}((t))$-equivariant D-modules on the affine Grassmannian ${\mathrm{Gr}}_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian defined in a recent work by Finkelberg, Krylov, and Mirković. We study various properties of the mentioned nearby cycles, and in particular compare them with the nearby cycles studied in works by Schieder. We also generalize our results to the Beilinson–Drinfeld Grassmannian ${\mathrm{Gr}}_{G,X^I}$ and to the affine flag variety ${\mathrm{Fl}}_G$.

This version of the paper contains fewer appendices than the version submitted to arXiv.

We improve the upper bound for diagonal Ramsey numbers to

$$R(k+1,k+1)\le exp(-c{(logk)}^2)\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)$$

for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal “effective quasirandomness” results about convergence of graphs. This optimality represents a natural barrier to improvement.

Given any connected compact orientable surface, a pair of mapping classes is said to be procongruently conjugate if it induces a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any pseudo-Anosov mapping class of a connected closed orientable surface, several topological features are confirmed to depend only on the procongurent conjugacy class of the mapping class, including the stretching factor, the topological type of the prong singularities, the transverse orientability of the invariant foliations, and the isomorphism type of the symplectic Floer homology.