Showing papers in "Annals of Mathematics in 2010"

Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation

Abstract: Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L 2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L 2 data and critical diffusion (-Δ) 1/2 are locally smooth for any space dimension.

Linear equations in primes

Abstract: Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N →∞, for the number of integer points n ∈ ℤ d ∩ K for which the integers ψ 1 (n), ... , ψ t (n) are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ 1 , ... , ψ t are affinely related; this excludes the important "binary" cases such as the twin prime or Goldbach conjectures, but does allow one to count "nondegenerate" configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowers-norm conjecture (GI(s)) and the Mobius and nilsequences conjecture (MN(s)), where s ∈ {1,2,...} is the complexity of the system and measures the extent to which the forms ψ i depend on each other. The case s = 0 is somewhat degenerate, and follows from the prime number theorem in APs. Roughly speaking, the inverse Gowers-norm conjecture GI(s) asserts the Gowers U s+1 -norm of a function f: [N] → [-1, 1] is large if and only if f correlates with an s-step nilsequence, while the Mobius and nilsequences conjecture MN(s) asserts that the Mobius function μ is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity. These conjectures have long been known to be true for s = 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s = 2 in two papers of the authors. Thus our results in the case of complexity s ≤ 2 are unconditional. In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 ≤ N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.

Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model

Abstract: We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.

Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Abstract: Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N(xi − xj)), where x = (x1, . . ., xN) denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrodinger equation. Suppose that the initial data ψN,0 satisfies the energy condition h ψN,0, H k N ψN,0i ≤ C k N k for k = 1,2, . . .. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψN,t are also asymptotically factorized and the one particle orbital wave function solves the Gross- Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψN,0 is assumed in a stronger sense. AMS Subject Classification Number: 81V70, 81T18, 35Q55

On the classification of finite-dimensional pointed Hopf algebras

Abstract: We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elementsG.A/ is abelian such that all prime divisors of the order of G.A/ are> 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.

The global stability of Minkowski space-time in harmonic gauge

Abstract: We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time.

On a class of II1 factors with at most one Cartan subalgebra

Abstract: We prove that the normalizer of any diffuse amenable subalgebra of a free group factor L(F-r) generates an amenable von Neumann subalgebra. Moreover, any II1 factor of the form Q (circle times) over barL(F-r), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure-preserving action of a free group F-r, 2 <= r <= infinity, on a probability space (X, mu) is profinite then the group measure space factor L-infinity (X) F-r has unique Cartan subalgebra, up to unitary conjugacy.

KAM for the nonlinear Schrödinger equation

Abstract: We consider the $d$-dimensional nonlinear Schrodinger equation under periodic boundary conditions: $-i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is an analytic function with $\hat V$ real, and $F$ is a real analytic function in $\Re u$, $\Im u$ and $x$. (This equation is a popular model for the 'real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\ep=0$ the equation is linear and has time--quasi-periodic solutions $u$, $$ u(t,x)=\sum_{a\in A}\hat u(a)e^{i(|a|^2+\hat V(a))t}e^{i\sc{a,x}} \quad (|\hat u(a)|>0), $$ where $A$ is any finite subset of $\Z^d$. We shall treat $\omega_a=|a|^2+\hat V(a)$, $a\inA$, as free parameters in some domain $U\subset\R^{A}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If $|\ep|$ is sufficiently small, then there is a large subset $U'$ of $U$ such that for all $\omega\in U'$ the solution $u$ persists as a time--quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients.

Noise stability of functions with low influences: Invariance and optimality

Abstract: In this paper, we study functions with low influences on product probability spaces. The analysis of Boolean functions f {-1, 1}/sup n/ /spl rarr/ {-1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly "smoothed"; this extension is essential for our applications to "noise stability "-type problems. In particular; as applications of the invariance principle we prove two conjectures: the "Majority Is Stablest" conjecture [29] from theoretical computer science, which was the original motivation for this work, and the "It Ain't Over Till It's Over" conjecture [27] from social choice theory. The "Majority Is Stablest" conjecture and its generalizations proven here, in conjunction with the "Unique Games Conjecture" and its variants, imply a number of (optimal) inapproximability results for graph problems.

Vacant set of random interlacements and percolation

Abstract: We introduce a model of random interlacements made of a countable collection of doubly infinite paths on ℤ d ,d > 3. A nonnegative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder (ℤ/Nℤ) d-1 x ℤ by simple random walk, or the set of points visited by simple random walk on the discrete torus (ℤ/Nℤ) d at times of order u N d . In particular we study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite connected translation invariant random subset of ℤ d . We introduce a critical value u * such that the vacant set percolates for u u * . Our main results show that u * is finite when d > 3 and strictly positive when d > 7.

A classification of SL(n) invariant valuations

Abstract: A classification of upper semicontinuous and SL(n) invariant valuations on the space of n-dimensional convex bodies is established As a consequence, complete characterizations of cen t ro-affine and L p affine surface areas are obtained The proofs make use of a new SL(n) shaping process for convex bodies

Transformations of elliptic hypergeometric integrals

Abstract: We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, we also consider their “Type II” integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, and satisfying the analogues of the Macdonald conjectures. Finally, we discuss some transformations of Type II-style integrals. In particular, we find that adding two parameters to the Type II integral gives an integral invariant under an appropriate action of the Weyl group E_7.

A family of Calabi-Yau varieties and potential automorphy

Abstract: We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j-invariant defined over a totally real field.

Uniqueness for the signature of a path of bounded variation and the reduced path group

Abstract: We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is a unique path that is tree reduced The set of these paths is the Reduced Path Group It is a continuous analogue of the group of reduced words The signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature In this way, we extend Chen’s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parameterisation in the general setting It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients As a second theme we give quantitative versions of Chen’s theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra

Multiplicity one theorems

Abstract: In the local, characteristic 0, non-Archimedean case, we consider distributions on GL(n + 1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies multiplicity at most one for restrictions from GL(n + 1) to GL(n). Similar theorems are obtained for orthogonal or unitary groups.

Dyson’s ranks and Maass forms

Abstract: which were defined by Ramanujan and Watson decades ago. In his last letter to Hardy dated January 1920 (see pages 127-131 of [27]), Ramanujan lists 17 such functions, and he gives 2 more in his “Lost Notebook” [27]. In his paper “The final problem: An account of the mock theta functions” [32], Watson defines 3 further functions. Surprisingly, much remains unknown about these enigmatic series. Ramanujan’s claims about their analytic properties remain open, and there is even debate concerning the rigorous definition of such a function. Despite these seemingly problematic issues, Ramanujan’s mock theta functions indeed possess many striking properties, and they have been the subject of an astonishing number of important works (for example, see [5, 6, 7, 8, 12, 13, 14, 18, 19, 20, 23, 27, 28, 32, 33, 35, 36] to name a few). Watson predicted this high level of activity in his 1936 Presidential Address to the London Mathematical Society with his prophetic words (see page 80 of [32]):

On the ergodicity of partially hyperbolic systems

Abstract: Pugh and Shub [PS3] have conjectured that essential accessibility implies ergodicity, for a C 2 , partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satsified by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for C 1+γ partially hyperbolic systems.

Surface group representations with maximal Toledo invariant

Abstract: We develop the theory of maximal representations of the fundamental group π 1 (Σ) of a compact connected oriented surface Σ (possibly with boundary) into Lie groups G of Hermitian type. For any homomorphism p: π 1 (Σ) → G, we define the Toledo invariant T(Σ, p), a numerical invariant which has both topological and analytical interpretations. We establish important properties of T(Σ, ρ), among which continuity, uniform boundedness on the representation variety, additivity under connected sum of surfaces and congruence relations mod ℤ. We thus obtain information about the representation variety as well as striking geometric properties of maximal representations, that is representations whose Toledo invariant achieves the maximum value. Moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle. We define a rotation number function for general locally compact groups and study it in detail for groups of Hermitian type. Properties of the rotation number, together with the existence of boundary maps, lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic.

Mass equidistribution for Hecke eigenforms

Abstract: We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors.

Sparse equidistribution problems, period bounds and subconvexity

Abstract: We introduce a "geometric" method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg L-functions; and to bounding Fourier coefficients of automorphic forms.

On the formation of singularities in the critical $O(3)$ $\sigma$-model

Abstract: We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from ℝ 2+1 Minkowski space into the sphere S 2 . We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis are done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k ≥ 4. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) in that it persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.

Free boundaries in optimal transport and Monge-Ampere obstacle problems

Abstract: Given compactly supported 0 f,g 2 L 1 (R n ), the problem of trans- porting a fraction m min{kfkL1,kgkL1} of the mass of f onto g as cheaply as possible is considered, where cost per unit mass transported is given by a cost function c, typically quadratic c(x,y) = |x y| 2 /2. This question is shown to be equivalent to a double obstacle problem for the Monge-Ampere equation, for which sucient conditions are given to guarantee uniqueness of the solution, such as f vanishing on sptg in the quadratic case. The part of f to be transported increases monotonically with m, and if sptf and sptg are separated by a hyperplane H, then this part will separated from the balance of f by a semiconcave Lipschitz graph over the hyperplane. If f = f and g = g are bounded away from zero and infinity on separated strictly convex domains , R n , for the quadratic cost this graph is shown to be a C 1, loc hypersurface in whose normal coincides with the direction transported; the optimal map between f and g is shown to be Holder continuous up to this free bound- ary, and to those parts of the fixed boundary @ which map to locally convex parts of the path-connected target region.

Sur un problème de Gelfond: la somme des chiffres des nombres premiers

Abstract: L'objet de cet article est de repondre a une question posee par Gelfond en 1968 en montrant que la somme des chiffres s q (p) des nombres premiers p ecrits en base q ≥ 2 est equirepartie dans les progressions arithmetiques (excepte pour certains cas degeneres bien connus). Nous montrons egalement que la suite (αs q (p)) ou p decrit l'ensemble des nombres premiers est equirepartie modulo 1 si et seulement si α ∈ ℝ \ ℚ.

The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

Abstract: The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

The classification of Kleinian surface groups, I: models and bounds

Abstract: We give the first part of a proof of Thurston’s Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a “Lipschitz model” for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick part, and to give a priori geometric bounds.

Heegner divisors, l-functions and harmonic weak maass forms

Abstract: Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular L-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of L-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier.

Teichmuller curves, triangle groups, and Lyapunov exponents

Abstract: We construct a Teichmuller curve uniformized by a Fuchsian triangle group commensurable to �(m, n, ∞) for every m, n ≤ ∞. In most cases, for example when m 6 n and m or n is odd, the uniformizing group is equal to the triangle group �(m, n, ∞). Our construction includes the Teichmuller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find billiard tables that generate these Teich- muller curves. We interpret some of the so-called Lyapunov exponents of the Kontsevich-Zorich cocycle as normalized degrees of a natural line bundle on a Teichmuller curve. We determine the Lyapunov exponents for the Teichmuller curves we construct.

An algorithm for computing some Heegaard Floer homologies

Abstract: In this paper, we give an algorithm to compute the hat version of Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtration coming from a null-homologous link in a three-manifold.

Small cancellations over relatively hyperbolic groups and embedding theorems

Abstract: We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable groups. For instance, we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes. In particular, this gives the affirmative answer to the well-known question of the existence of a finitely generated group G other than ℤ/2ℤ such that all nontrivial elements of G are conjugate.

Elliptic functions, Green functions and the mean field equations on tori

Abstract: We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seem to be any direct method to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over the moduli space of flat tori through deformations. The functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori. The general picture is also emerged, though some of the necessary technicality is still to de developed. 1. INTRODUCTION AND STATEMENT OF RESULTS The study of geometric or analytic problems on two dimensional tori is the same as the study of problems on R 2 with doubly periodic data. Such situations occur naturally in sciences and mathematics since early days. The mathematical foundation of elliptic functions was subsequently devel- oped in the 19th century. It turns out that these special functions are rather deep objects by themselves. Tori of different shape may result in very dif- ferent behavior of the elliptic functions and their associated objects. Arith- metic on elliptic curves is perhaps the eldest and the most vivid example. In this paper, we show that this is also the case for certain non-linear par- tial differential equations. Indeed, researches on doubly periodic problems in mathematical physics or differential equations often restrict the study to rectangular tori for simplicity. This leaves the impression that the theory for general tori may resemble much the same way as for the rectangular case. However this turns out to be false. We will show that the solvability of themean field equation dependson the shape of the Green function, which in turn depends on the geometry of the tori in an essential way. Recall that the Green function G(z,w) on a flat torus T = C/Zω1 + Zω2 is the unique function on T × T which satisfies −△zG(z,w) = δw(z) − 1 |T|