We prove that $$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $$ where$p$_{$n$}denotes the$n$th prime. Since on average$p$_{$n$+1}−$p$_{$n$}is asymptotically log_{$n$}, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences$p$−$p$′ between primes which includes the small gap result above.

Let$G$be a simple and simply-connected complex algebraic group,$P$⊂$G$a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology$QH$^{*}($G$/$P$) of a flag variety is, up to localization, a quotient of the homology$H$_{*}(Gr_{$G$}) of the affine Grassmannian Gr_{$G$}of$G$. As a consequence, all three-point genus-zero Gromov–Witten invariants of$G$/$P$are identified with homology Schubert structure constants of$H$_{*}(Gr_{$G$}), establishing the equivalence of the quantum and homology affine Schubert calculi.

The edges of a complete graph on$n$vertices are assigned i.i.d. random costs from a distribution for which the interval [0,$t$] has probability asymptotic to$t$as$t$→0 through positive values. In this so called pseudo-dimension 1 mean field model, we study several optimization problems, of which the traveling salesman is the best known. We prove that, as$n$→∞, the cost of the minimum traveling salesman tour converges in probability to a certain number, approximately 2.0415, which is characterized analytically.

We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces$S$of finite type in 3-dimensional Euclidean space. For$p$> 2, we prove that if no affine tangent plane to$S$passes through the origin and$S$is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if$p$>$h$($S$), where$h$($S$) denotes the so-called height of the surface$S$(defined in terms of certain Newton diagrams). For non-analytic$S$we obtain the same statement with the exception of the exponent$p$=$h$($S$). Our notion of height$h$($S$) is closely related to A. N. Varchenko’s notion of height$h$($ϕ$) for functions$ϕ$such that$S$can be locally represented as the graph of$ϕ$after a rotation of coordinates.

Let $ E \subset \mathbb{C} $ be a compact set, $ g:\mathbb{C} \to \mathbb{C} $ be a$K$-quasiconformal map, and let 0 <$t$< 2. Let $ {\mathcal{H}^t} $ denote$t$-dimensional Hausdorff measure. Then $$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $$