Showing papers in "Acta Mathematica in 2010"

Monge-Ampère equations in big cohomology classes

Abstract: We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kahler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L1+e-density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kahler–Einstein volume forms with minimal singularities on varieties of general type.

Quantum cohomology of G/P and homology of affine Grassmannian

Abstract: 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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH *(G/P) of a flag variety is, up to localization, a quotient of the homology H *(Gr G ) of the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. For the case G = B, we use Mihalcea’s equivariant quantum Chevalley formula for QH *(G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology. Our main results extend to the torus-equivariant setting.

Geodesics in large planar maps and in the Brownian map

Abstract: We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.

The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture

Abstract: We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery and Tao. The proof uses the polynomial method introduced by Dvir.

Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis

Abstract: 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. Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the L p -boundedness of the associated maximal operator $ \mathcal{M} $ , and a conjecture by Iosevich and Sawyer which relates the L p -boundedness of $ \mathcal{M} $ to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3. In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an $ {L^p}\left( {{\mathbb{R}^3}} \right) - {L^2}(S) $ Fourier restriction theorem for S.

The Fourier spectrum of critical percolation

Abstract: 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.

On the regularization of conservative maps

Abstract: We show that smooth maps are C 1-dense among C 1 volume-preserving maps.

Primes in tuples II

Abstract: We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, lim inf n!1 pnC1 pn logpn D0:

Discrete series characters for affine Hecke algebras and their formal degrees

Abstract: 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).

The mean field traveling salesman and related problems

Abstract: 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.

Dimension of quasicircles

Abstract: 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.

Prime and almost prime integral points on principal homogeneous spaces

Abstract: We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.

Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane

Abstract: 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 $$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}} $$ This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K Astala in [2] and answers in the positive a conjecture of K Astala in op cit

Erratum to: “Distinguished varieties”

Abstract: 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.