Showing papers in "Annals of Mathematics in 2005"

On fusion categories

Abstract: Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

On the hardness of approximating minimum vertex cover

Abstract: We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of 1.3606, extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and to borrow and extend ideas from several fields.

Basic properties of SLE

Abstract: SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.

Nonconventional ergodic averages and nilmanifolds

Abstract: We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg?fs proof of Szemer?Ledi?fs theorem. The second average is taken along cubes whose sizes tend to +??. For each average, we show that it is sufficient to prove the convergence for special systems, the characteristic factors. We build these factors in a general way, independent of the type of the average. To each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold. From the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density.

Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems

Abstract: We consider the Cauchy problem for a strictly hyperbolic, n × n system in one-space dimension: ut + A(u)ux = 0, assuming that the initial data have small total variation. We show that the solutions of the viscous approximations ut + A(u)ux = euxx are defined globally in time and satisfy uniform BV estimates, independent of e. Moreover, they depend continuously on the initial data in the L 1 distance, with a Lipschitz constant independent of t, e. Letting e → 0, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where A = Df is the Jacobian

The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation

Abstract: We consider the critical nonlinear Schr?odinger equation iut = -.u-|u| 4 N u with initial condition u(0, x) = u0 in dimension N = 1. For u0 . H1, local existence in the time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is, u0 such that limt.T 1.

Well-posedness for the motion of an incompressible liquid with free surface boundary

Abstract: We study the motion of an incompressible perfect liquid body in vacuum This can be thought of as a model for the motion of the ocean or a star The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface This leads to a free boundary problem for Euler?s equations, where the regularity of the boundary enters to highest order We prove local existence in Sobolev spaces assuming a ?physical condition?, related to the fact that the pressure of a fluid has to be positive

Gröbner geometry of Schubert polynomials

Abstract: Given a permutation w ?? Sn, we consider a determinantal ideal Iw whose generators are certain minors in the generic n ?~ n matrix (filled with independent variables). Using ?emultidegrees?f as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal Iw.

A local regularity theorem for mean curvature flow

Abstract: This paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the Gaussian densities are close to 1.

Two dimensional compact simple Riemannian manifolds are boundary distance rigid

Abstract: We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

Higher symmetries of the laplacian

Abstract: We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.

The density of discriminants of quartic rings and fields

Abstract: We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic rings and fields.

The Lyapunov exponents of generic volume-preserving and symplectic maps

Abstract: We show that the integrated Lyapunov exponents of C1 volume-preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents are equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume-preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all C1 symplectic diffeomorphisms on any compact manifold, either the diffeomorphism is Anosov or almost every point has zero as a Lyapunov exponent, with multiplicity at least 2. Finally, given any set S ?? GL(d) satisfying an accessibility condition, for a residual subset of all continuous S-valued cocycles over any measure-preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial. The condition on S is satisfied for most common matrix groups and also for matrices that arise from discrete Schr?Nodinger operators.

A new application of random matrices: Ext$(C^*_{\mathrm{red}}(F_2))$ is not a group

Abstract: In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system Since then, random matrices have played a key role in von Neumann algebra theory (cf [V8], [V9]) The main result of this paper is the following extension of Voiculescui¯s random matrix result: Let (X(n) 1 , , X(n) r ) be a system of r stochastically independent n i? n Gaussian self-adjoint random matrices as in Voiculescui¯s random matrix paper [V4], and let (x1, , xr) be a semi-circular system in a C-probability space Then for every polynomial p in r noncommuting variables lim niaeiA

pX(n) 1 (¥o), , X(n) r (¥o)

= p(x1, , xr), for almost all ¥o in the underlying probability space We use the result to show that the Ext-invariant for the reduced C-algebra of the free group on 2 generators is not a group but only a semi-group This problem has been open since Anderson in 1978 found the first example of a C-algebra A for which Ext(A) is not a group

The uniqueness of the helicoid

Abstract: In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R3. M of finite topology means M is homeomorphic to a compact surface M̂ (of genus k and empty boundary) minus a finite number of points p1, ..., pj ∈ M̂ , called the punctures. A closed neighborhood E of a puncture in M is called an end of M . We will choose the ends sufficiently small so they are topologically S1 × [0, 1) and hence, annular. We remark that M̂ is orientable since M is properly embedded in R3. The simplest examples (discovered by Meusnier in 1776) are the helicoid and catenoid (and a plane of course). It was only in 1982 that another example was discovered. In his thesis at Impa, Celso Costa wrote down the Weierstrass representation of a complete minimal surface modelled on a 3-punctured torus. He observed the three ends of this surface were embedded: one top catenoidtype end1, one bottom catenoid-type end, and a middle planar-type end2 [8]. Subsequently, Hoffman and Meeks [15] proved this example is embedded and they constructed for every finite positive genus k embedded examples of genus k and three ends. In 1993, Hoffman, Karcher and Wei [14] discovered the Weierstrass data of a complete minimal surface of genus one and one annular end. Computer generated pictures suggested this surface is embedded and the end is asymptotic to an end of a helicoid. Hoffman, Weber and Wolf [17] have now given a proof that there is such an embedded surface. Moreover, computer evidence suggests that one can add an arbitrary finite number k of handles to a helicoid to obtain a properly embedded genus k minimal surface asymptotic to a helicoid. For many years, the search went on for simply connected examples other than the plane and helicoid. We shall prove that there are no such examples.

The two possible values of the chromatic number of a random graph

Abstract: Given d ?? (0,??) let kd be the smallest integer k such that d < 2k log k. We prove that the chromatic number of a random graph G(n, d/n) is either kd or kd + 1 almost surely.

Geometrization of 3-dimensional orbifolds

Abstract: This paper is devoted to the proof of the orbifold theorem: If O is a compact connected orientable irreducible and topologically atoroidal 3-orbifold with nonempty ramification locus, then O is geometric (i.e. has a metric of constant curvature or is Seifert fibred). As a corollary, any smooth orientationpreserving nonfree finite group action on S3 is conjugate to an orthogonal action.

Une borne supérieure pour l'entropie topologique d'une application rationnelle

Abstract: Let X be a complex projective manifold and let f be a dominating rational map from X onto X. We show that the topological entropy h(f) of f is bounded from above by the logarithm of its maximal dynamical degree.

Einstein metrics on spheres

Abstract: We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularities to realize spheres (and exotic spheres) as circle orbi-bundles over complex algebraic orbifolds, and lift a Kaehler-Einstein metric from the orbifold to a Sasakian-Einstein metric on the sphere.

On metric Ramsey-type phenomena

Abstract: The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky?s theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any
> 0, every n point metric space contains a subset of size at least n1-
which is embeddable in Hilbert space with O
log(1/
)
 distortion. The bound on the distortion is tight up to the log(1/
) factor. We further include a comprehensive study of various other aspects of this problem.

Roth's theorem in the primes

Abstract: We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes.

Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes

Abstract: The infinite-dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4-parameter family of characters of U(∞). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(∞). The spectral decomposition of a character of U(∞) is described by the spectral measure which lives on an infinite-dimensional space Ω of indecomposable characters. The key idea which allows us to solve the problem is to embed Ω into the space of point configurations on the real line without two points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special ‘integrable’ form and are expressed through the Gauss hypergeometric function. From the analytic point of view, the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by Richard Askey and Peter Lesky. One difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0. We do this by expressing the polynomials in terms of a solution of a discrete Riemann-Hilbert problem and computing the (nonoscillatory) asymptotics of this solution. From the point of view of statistical physics, we study thermodynamic limit of a discrete log-gas system. An interesting feature of this log-gas is that its density function is asymptotically equal to the characteristic function of an interval. Our point processes describe how different the random particle configuration is from the typical ‘densely packed’ configuration. In simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices, similar results were obtained in our papers [BO1], [BO2], [BO4].

Sharp local well-posedness results for the nonlinear wave equation

Abstract: This article is concerned with local well-posedness of the Cauchy problem for second order quasilinear hyperbolic equations with rough initial data. The new results obtained here are sharp in low dimension.

Formation of singularities for a transport equation with nonlocal velocity

Abstract: We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.

A sharp form of Whitney's extension theorem

Abstract: 3. Order relations involving multi-indices 4. Statement of two main lemmas 5. Plan of the proof 6. Starting the main induction 7. Nonmonotonic sets 8. A consequence of the main inductive assumption 9. Setup for the main induction 10. Applying Helly's theorem on convex sets 11. A Calder6n-Zygmund decomposition 12. Controlling auxiliary polynomials I 13. Controlling auxiliary polynomials II 14. Controlling the main polynomials 15. Proof of Lemmas 9.1 and 5.2 16. A rescaling lemma 17. Proof of Lemma 5.3 18. Proofs of the theorems 19. A bound for k# References

A resolution of the K(2)-local sphere at the prime 3

Abstract: We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum LK(2)S 0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E hF where F is a finite subgroup of the Morava stabilizer group and E2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n =2a tp = 3 represents the edge of our current knowledge: n = 1 is classical and at n = 2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic. The problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics. A period of calculation beginning with Serre’s computation of the cohomology of Eilenberg-MacLane spaces and the advent of the Adams spectral sequence culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on periodic phenomena in the homotopy groups of spheres and Ravenel’s nilpotence conjectures. The solutions to most of these conjectures by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of the “chromatic” point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory. Computations passed from center stage, to some extent, although there has been steady work in the wings – most notably by Shimomura and his coworkers, and Ravenel, and more lately by Hopkins and

Ergodic properties of rational mappings with large topological degree

Abstract: Let X be a projective manifold and f : X ?? X a rational mapping with large topological degree, dt > ?Ek.1(f) := the (k . 1)th dynamical degree of f. We give an elementary construction of a probability measure ?Ef such that d.n t (fn).?? ?? ?Ef for every smooth probability measure ?? on X. We show that every quasiplurisubharmonic function is ?Ef -integrable. In particular ?Ef does not charge either points of indeterminacy or pluripolar sets, hence ?Ef is f-invariant with constant jacobian f.?Ef = dt?Ef . We then establish the main ergodic properties of ?Ef : it is mixing with positive Lyapunov exponents, preimages of ?hmost?h points as well as repelling periodic points are equidistributed with respect to ?Ef . Moreover, when dimC X . 3 or when X is complex homogeneous, ?Ef is the unique measure of maximal entropy.

Universal bounds for hyperbolic Dehn surgery

Abstract: This paper gives a quantitative version of Thurston?s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic conemanifold structures, using infinitesimal harmonic deformations and analysis of geometric limits.