Showing papers in "Annals of Mathematics in 2009"

Ricci curvature for metric-measure spaces via optimal transport

Abstract: We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having \curvature bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form

The Euler equations as a differential inclusion

Abstract: We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in R n with n 2. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

The Ten Martini Problem

Abstract: We prove the conjecture (known as the "Ten Martini Problem" after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies.

Donaldson-Thomas type invariants via microlocal geometry

Abstract: We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.

Moduli of finite flat group schemes, and modularity

Abstract: We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of ℚ 3 . The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations. Introduction.

Regularity of flat level sets in phase transitions

Abstract: We consider local minimizers of the Ginzburg-Landau energy functional ∫1/2|∇u| 2 +1/4(1-u 2 ) 2 dx and prove that, if the 0 level set is included in a flat cylinder then, in the interior, it is included in a flatter cylinder. As a consequence we prove a conjecture of De Giorgi which states that level sets of global solutions of Δu = u 3 - u such that |u| ≤ 1, ∂ n u > 0, lim u(x', x n ) ±1 x n →±∞ are hyperplanes in dimension n ≤ 8.

Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

Abstract: Consider a random sum r)\V\ + • • • + r]nvn, where 771, . . . , rin are independently and identically distributed (i.i.d.) random signs and vi, . . . , vn are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as P(r?i^iH In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v\ , . . . , vn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the

Primes in tuples I

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→∞ Pn+1-Pn/log Pn/log = 0. We will quantify this result further in a later paper.

Lie theory for nilpotent L∞-algebras

Abstract: The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor γ from L ∞ -algebras concentrated in degree > -n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L ∞ algebras (i.e., chain complexes), the functor γ is the Dold-Kan simplicial set.

A combinatorial description of knot Floer homology

Abstract: Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.

Moments of the Riemann zeta function

Abstract: Assuming the Riemann hypothesis, we obtain an upper bound for the moments of the Riemann zeta function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in other families of L-functions.

Curvature of vector bundles associated to holomorphic fibrations

Abstract: Let L be a (semi)-positive line bundle over a Kahler manifold, X, fibered over a complex manifold Y. Assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle E over Y whose fibers over points y are the spaces of global sections over Xy to L ⊗ Kx/y, endowed with the L2-metric, is (semi)-positive in the sense of Nakano. We also discuss various applications, among them a partial result on a conjecture of Griffiths on the positivity of ample bundles.

The geometry of fronts

Abstract: We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails

A new upper bound for diagonal Ramsey numbers

Abstract: We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant C such that r(k + 1,k+1) ≤ k -Clog k/log log k ( 2k k ).

Depth-zero supercuspidal L-packets and their stability

Abstract: In this paper we verify the local Langlands correspondence for pure inner forms of unramied p-adic groups and tame Langlands parameters in \general position". For each such parameter, we explicitly construct, in a natural way, a nite set (\ L-packet") of depth-zero supercuspidal representations of the appropriate p-adic group, and we verify some expected properties of this L-packet. In particular, we prove, with some conditions on the base eld, that the appropriate sum of characters of the representations in our L-packet is stable; no proper subset of our L-packets can form a stable combination. Our L-packets are also consistent with the conjectures of B. Gross and D. Prasad on restriction from SO2n+1 to SO2n [24]. These L-packets are, in general, quite large. For example, Sp2n has an L-packet containing 2 n representations, of which exactly two are generic. In fact, on a quasi-split form, eachL-packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced BruhatTits building. When the group has connected center, every depth-zero generic supercuspidal representation appears in one of these L-packets. We emphasize that there is nothing new about the representations we construct. They are induced from Deligne-Lusztig representations on subgroups of nite

A New Approach to Universality Limits Involving Orthogonal Polynomials

Abstract: We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure /x on [? 1,1] Assume that ji is a regular measure, and is absolutely continuous in an open interval containing some point x Assume moreover, that /z' is positive and continuous at x Then universality for /x holds at x If the hypothesis holds for x in a compact subset of (?1,1), universality holds uniformly for such x Indeed, this follows from universality for the classical Legendre weight We also establish universality in an Lp sense under weaker assumptions on fi

Word maps, conjugacy classes, and a noncommutative Waring-type theorem

Abstract: Let w = w(x\,..., Xd) 1 be a nontrivial group word. We show that if G is a sufficiently large finite simple group, then every element g e G can be expressed as a product of three values of w in G. This improves many known results for powers, commutators, as well as a theorem on general words obtained in [19]. The proof relies on probabilistic ideas, algebraic geometry, and character theory. Our methods, which apply the 'zeta function' $g(s) = X^eirrG XO)~s> giye rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.

The concept of duality in convex analysis, and the characterization of the Legendre transform

Abstract: In the main theorem of this paper we show that any involution on the class of lower semi-continuous convex functions which is order-reversing, must be, up to linear terms, the well known Legendre transform.

Optimality and uniqueness of the Leech lattice among lattices

Abstract: We prove that the Leech lattice is the unique densest lattice in R24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R24 can exceed the Leech lattice's density by a factor of more than 1 + 1.65 10-30, and we give a new proof that Es is the unique densest lattice in R8.

Polya-Schur master theorems for circular domains and their boundaries

Abstract: We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Omega subset of C for arbitrary ...

Subharmonic solutions of Hamiltonian equations on tori

Abstract: Let the torus T2n be equipped with the standard symplectic structure and a periodic Hamiltonian Dt e C3(SX x T2n, R). We look for periodic orbits of the Hamiltonian flow u(t) = J VDt(t, (/)). A subharmonic solution is a periodic orbit with minimal period an integral multiple m of the period of Dt, with m > 1. We prove that if the Hamiltonian flow has only finitely many orbits with the same period as Dt, then there are subharmonic solutions with arbitrarily high minimal period. Thus there are always infinitely many distinct periodic orbits. The results proved here were proved in the nondegenerate case by Conley and Zehnder and in the case n = 1 by Le Calvez.

Holomorphic curves into algebraic varieties

Abstract: This paper establishes a defect relation for algebraically nondegenerate holomorphic mappings into an arbitrary nonsingular complex projective variety V (rather than just the projective space) intersecting possible nonlinear hypersurfaces, extending the result of H. Cartan.

Stable manifolds for an orbitally unstable nonlinear Schrödinger equation

Abstract: (2) −4φ+ αφ = φ. By this we mean that φ > 0 and that φ ∈ C2(R3). It is a classical fact (see Coffman [10]) that such solutions exist and are unique for the cubic nonlinearity. Moreover, they are radial and smooth. Similar facts are known for more general nonlinearities; see e.g., Strauss [45] and Berestycki and Lions [5] for existence and Kwon [30] for uniqueness in greater generality. Clearly, ψ = eitα 2 φ solves (1). We seek an H1-solution ψ of the form ψ = W +R where

Strong cosmic censorship in T-3-Gowdy spacetimes

Abstract: Einstein's vacuum equations can be viewed as an initial value problem, and given initial data there is one part of spacetime, the so-called maximal globally hyperbolic development (MGHD), which is uniquely determined up to isometry. Unfortunately, it is sometimes possible to extend the spacetime beyond the MGHD in inequivalent ways. Consequently, the initial data do not uniquely determine the spacetime, and in this sense the theory is not deterministic. It is then natural to make the strong cosmic censorship conjecture, which states that for generic initial data, the MGHD is inextendible. Since it is unrealistic to hope to prove this conjecture in all generality, it is natural to make the same conjecture within a class of spacetimes satisfying some symmetry condition. Here, we prove strong cosmic censorship in the class of T-3-Gowdy spacetimes. In a previous paper, we introduced a set G(i,c) of smooth initial data and proved that it is open in the C-1 x C-0-topology. The solutions corresponding to initial data in G(i,c) have the following properties. First, the MGHD is C-2-inextendible. Second, following a causal geodesic in a given time direction, it is either complete, or a curvature invariant, the Kretschmann scalar, is unbounded along it (in fact the Kretschmann scalar is unbounded along any causal curve that ends on the singularity). The purpose of the present paper is to prove that G(i,c) is dense in the C-infinity-topology.

On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q)

Abstract: We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations of the absolute Galois group of Q. It is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight.

The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

Abstract: We prove the B. and M. Shapiro conjecture that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This, in particular, implies the following result: If all ramification points of a parametrized rational curve φ: ℂℙ 1 → ℂℙ r lie on a circle in the Riemann sphere ℂℙ 1 , then φ maps this circle into a suitable real subspace ℝℙ r ⊂ ℂℙ r . The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has real spectrum. In Appendix A, we discuss properties of differential operators associated with Bethe vectors in the Gaudin model. In particular, we prove a statement, which may be useful in complex algebraic geometry; it claims that certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of the corresponding Gaudin Hamiltonians is simple. In Appendix B, we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with Lie algebras of types A r , B r and C r .

Blowing up Kähler manifolds with constant scalar curvature, II

Abstract: In this paper we prove the existence of Kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature Kahler metric. In the case where the manifold has nontrivial holomorphic vector fields with zeros, we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature Kahler metrics.

Fitting a Cm-smooth function to data, III

Abstract: Suppose we are given a finite subset E ? Rn and a function f : E ? R. How to extend f to a Cm function F : Rn ? R with Cm norm of the smallest possible order of magnitude? In this paper and in [20] we tackle this question from the perspective of theoretical computer science. We exhibit algorithms for constructing such an extension function F, and for computing the order of magnitude of its Cm norm. The running time of our algorithms is never more than CN log N, where N is the cardinality of E and C is a constant depending only on m and n.

A rigid irregular connection on the projective line

Abstract: In this paper we construct a connection ∇ on the trivial G-bundle on P 1 for any simple complex algebraic group G, which is regular outside of the points 0 and ∞, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point ∞, with slope 1/h, the reciprocal of the Coxeter number of G. The connection ∇, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of l-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These l- adic representations, and their characteristic 0 counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation V of G, and describe the differential Galois group of ∇ as a subgroup of G.

On the dynamics of dominated splitting

Abstract: Let f : M! M be a C 2 dieomorphism of a compact surface. We give a complete description of the dynamics of any compact invariant set having dominated splitting. In particular, we prove a Spectral Decomposition Theorem for the limit set L(f) under the assumption of dominated splitting. Moreover, we describe all the bifurcations that these systems can exhibit and the dierent types of dynamics that could follow for small C r perturbations.