MathSciDoc: An Archive for Mathematician ∫

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 We prove an unpublished result of D. Peterson which states that the quantum cohomology$QH$^{*}($G$/$P$) of a flag variety is, up to localization, a quotient of the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 $$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $$

In the article “Distinguished varieties”, Lemma 4.6 was incorrect. We prove a different lemma that can then be used to prove the original Theorem 4.1.

We show that smooth maps are$C$^{1}-dense among$C$^{1}volume-preserving maps.

Consider the indicator function$f$of a 2-dimensional percolation crossing event. In this paper, the Fourier transform of$f$is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension ${\frac{31}{36}}$ almost surely, and the corresponding dimension in the half-plane is ${\frac{5}{9}}$ . It is also proved that critical bond percolation on the square grid has exceptional times almost surely. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.

We introduce the$generic central character$of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types ${E_{n}^{(1)}}$ , n=6, 7, 8) with$arbitrary positive parameters$and we prove an explicit product formula for their formal degrees (in all cases).

We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala’s conjecture that the Hausdorff dimension of a$k$-quasicircle is at most 1+$k$^{2}.