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.

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.

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.

We explain how non-Archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis, we work over a nonstandard model of the field of complex numbers, which is endowed at the same time with an Archimedean and a non-Archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of Archimedean and non-Archimedean forms which is compatible with integration.

We prove sharp ${\ell }^q{L}^p$ decoupling inequalities for $p,q\in [2,\mathrm{\infty })$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp–Lieb inequalities.

We extend the definition of Khovanov–Lee homology to links in connected sums of $S^1\times S^2$’s and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1\times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following 4-manifolds: $B^2\times S^2$, $S^1\times B^3$, ${\mathbb{CP}}^2$, and various connected sums and boundary sums of these. We deduce that Rasmussen’s invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are nonstandard.

We prove that every finite-dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.