Showing papers in "Annals of Mathematics in 1987"

Hecke algebra representations of braid groups and link polynomials

Abstract: By studying representations of the braid group satisfying a certain quadratic relation we obtain a polynomial invariant in two variables for oriented links. It is expressed using a trace, discovered by Ocneanu, on the Hecke algebras of type A. A certain specialization of the polynomial, whose discovery predated and inspired the two-variable one, is seen to come in two inequivalent ways, from a Hecke algebra quotient and a linear functional on it which has already been used in statistical mechanics. The two-variable polynomial was first discovered by Freyd-Yetter, Lickorish-Millet, Ocneanu, Hoste, and Przytycki-Traczyk.

A global uniqueness theorem for an inverse boundary value problem

Abstract: In this paper, we show that the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n ? 3. From a physical point of view, we show that an isotropic conductivity can be determined by steady state measurements at the boundary.

Factoring integers with elliptic curves

Abstract: This paper is devoted to the description and analysis of a new algorithm to factor positive integers. It depends on the use of elliptic curves. The new method is obtained from Pollard's (p - 1)-method (Proc. Cambridge Philos. Soc. 76 (1974), 521-528) by replacing the multiplicative group by the group of points on a random elliptic curve. It is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2, where p is the least prime dividing n and K is a function for which log K(x) = /(2 + o(1))log x log log x for x -x o. In the worst case, when n is the product of two primes of the same order of magnitude, this is

Dehn surgery on knots

Abstract: In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that

Metrics with exceptional holonomy

Abstract: It is proved that there exist metrics with holonomy G2 and Spin(7), thus settling the remaining cases in Berger's list of possible holonomy groups. We first reformulate the "holonomy H" condition as a set of differential equations for an associated H-structure on a given manifold. We collect the needed algebraic facts about G2 and Spin(7). We then apply the machinery of over-determined partial differential equations (in the form of the Cartan-Kahler theorem) to prove the existence of solutions whose holonomy is G2 or Spin(7). We also provide explicit examples and some information about the "generality" of the space of

Hardy spaces and the Neumann problem in L^p for laplace's equation in Lipschitz domains

Abstract: On donne des resultats optimaux pour la resolubilite du probleme de Neumann dans des domaines de Lipschitz a donnees dans L p . On obtient des resultats du point extreme correspondant pour des espaces de Hardy

Negatively curved manifolds, elliptic operators, and the Martin boundary

Abstract: Consider a Riemannian manifold M of dimension n > 2 and assume that M is complete, simply connected and that its sectional curvatures are bounded between two negative constants. In a recent paper [7], M. Anderson and R. Schoen have proved that the Martin boundary of M, with respect to the Laplace-Beltrami operator A, is naturally homeomorphic to Soo(M) the sphere at infinity of M. Here, we shall first establish a similar result for a wide class of elliptic operators on M, giving at the same time a new proof of the theorem of Anderson and Schoen; our proof relies heavily on several concepts of potential theory, but on the other hand avoids the quite difficult calculations of [7], and perhaps makes the result more understandable. Roughly speaking, we show that if Y? is a second order elliptic operator defined over M and has a "uniformly" nice behaviour on each ball of radius 1 in M, then dQ(M), the Y-Martin boundary of M, is S,,(M) provided that the following condition holds: There is a positive Y?+ cI-superharmonic function on M for some positive ?. If instead, we only assume the existence of the Green function for Y' (a minimal requirement when studying the Y-Martin boundary) then d,(M) may be very different from So,(M), even for a rotation invariant operator Y on the unit ball of Rn equipped

The N-particle scattering problem: asymptotic completeness for short-range systems

Abstract: On demontre la completude asymptotique pour des systemes de mecanique quantique a courte portee constitues d'un nombre arbitraire de particules

Families of Rational Maps and Iterative Root-Finding Algorithms

Abstract: In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms. In the context of conformal dynamics, our main result is the following: a stable algebraic family of rational maps is either trivial (all its members are conjugate by Mobius transformations), or affine (its members are obtained as quotients of iterated addition on a family of complex tori). Our classification of generally convergent algorithms follows easily from this result. As another consequence of rigidity, we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices. This implies that bounded analytic functions nearly separate points on the moduli space of a rational map.

Harmonic analysis on product spaces

Abstract: Rn x R' and commute with the dilations p33 '8(x2, x2) = (61x1, 62x2) for x1 E Rn, x2 E Rm, and 61, 82 > 0, then the corresponding operators are related to many problems in the theory of multiple Fourier integrals (see Zygmund [21]). The maximal operator which commutes with the p31382 is the Jessen, Marcinkiewicz, and Zygmund [14] strong maximal operator Ms defined by 1 Ms(f)(x1, x2) = sup RI A|(Y1, Y2)1 dy1 dy2 (xl, X2)GR JR1

Harmonic maps of the two-sphere into a complex Grassmann manifold II*

Abstract: where qT is a complex-valued one-form, defined up to a factor of absolute value 1. This form qp defines a complex structure on M. For x E M the space f(x) has an orthogonal space f(x) ' of dimension n - k. We denote by [ f(x)] and [ f(x) '] their corresponding projective spaces, of dimensions k - 1 and n - k - 1,

Isomorphisms of p-adic group rings

Abstract: A long-standing problem, first posed by Graham Higman [15] and later by Brauer [4] is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG = ZH implies G 2: H? Many authors have worked on this question, but progress has been difficult [30]. Perhaps the best positive result was that of Whitcomb in 1968 [37], who showed that the implication G = H holds for G metabelian. Dade [9] showed there were counterexamples, even in the metabelian case, if Z were replaced by the family of all fields. Some mathematicians came to believe the problem was a kind of grammatical accident, that counterexamples for Z were surely there, if difficult to find. We ourselves began this investigation looking for counterexamples, but found that they were indeed very difficult to find, much more difficult than we had anticipated. Slowly we began to believe that at least some exciting positive results were possible. In this paper we answer the isomorphism problem for finite p-groups over the p-adic integers Z, = Zcp), and in a very strong way: In the normalized units (augmentation 1) of Z pG there is only one con&gay cZu.ss of groups of order ] G 1. This answers the isomorphism problem in the affirmative (for finite p-groups G, over Z or Zp) and in addition computes the entire Picard group [2] of the category of Z,G-modules in terms of automorphisms of G. Similarly we are able to settle the isomorphism problem for finite nilpotent groups and compute the associated semilocal Picard groups. We also treat more general coefficient domains: namely, integral domains S of characteristic 0 in which no (rational) prime divisor of the group order is invertible, for the SG isomorphism problem, and treat similar semilocal Dedekind domains for the Picard group computations. The Zassenhaus conjecture concerning the rational behavior of group ring automorphisms is verified for the nilpotent case in this general setting (cf. Corollary 3 below). Over Z, we announce a positive answer to the isomorphism

On the convergence of Riesz means on compact manifolds

Abstract: The purpose of this paper is to study the LP mapping properties of the Riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds. In order to be more specific, let us first introduce some notation. Let M be a compact connected C' manifold of dimension n ? 2 and P an elliptic differential operator on M with C00 coefficients. We shall assume that P is self-adjoint with respect to some positive C00 density dx, kept fixed throughout. That is, (Pu, v) = (U, Pv), where, as usual,

Hyperbolicity and the creation of homoclinic orbits

Abstract: We consider one-parameter families {sp,; t E R} of diffeomorphisms on surfaces which display a homoclinic tangency for yt = 0 and are hyperbolic for yt < 0 (i.e., (pf has a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for yt positive. For many of these families, we prove that (p is also hyperbolic for most small positive values of yt (which implies much regularity of the dynamical structure). A main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency.

The zeros of derivatives of entire functions and the Polya-Wiman Conjecture

Abstract: An entire function f(x) which assumes only real values on the real axis is said to be a real entire function. Thus, if f is a real entire function, then its Maclaurin coefficients are all real, and consequently the zeros of f are symmetrically located with respect to the real axis. One of the intuitive principles underlying the theory of distribution of zeros of successive derivatives of real entire functions was formulated by Polya [P4] in his celebrated survey article entitled On the zeros of the derivatives of a function and its analytic character. In this work [P4, p. 181] Polya states, "The real axis seems to exert an influence on the complex zeros of f(n)(z); it seems to attract these zeros when the order is less than 2, and it seems to repel them when the order is greater than 2." Indeed, the first confirmation of this principle was made by Alander [A2] (see also P6lya [P2]) who in 1930 showed that the Polya-Wiman conjecture is true if the order of f(x) is less than 2/3. In [Wil] and [Wi2] Wiman established the validity of this conjecture for functions f(x) of order at most 1, while in 1937 Polya [P3] proved it for functions of order less than 4/3. A general survey of these results and related conjectures may be found in [P2] and [P4]. (For more recent surveys of this and related areas of research see Boas [B2] and Prather [Pr].) It seems that no progress has been made on the P6lya-Wiman conjecture since the publication of Polya's 4/3 theorem in 1937. Our proof of the conjecture is self-contained and does not rely on the results cited above. In Section 2 we will (1) introduce some definitions and notations, (2) state some of the properties of functions in the Laguerre-P6lya class, and (3) prove the Polya-Wiman conjecture (Theorem 3). In the proof of Theorem 3 we

Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients of bounded symmetric domains

Abstract: In the theory of locally symmetric spaces of negative Ricci curvature it has been a classical problem to study the extent to which the topology of the manifold determines the geometry, and, in the Hermitian case, the complex structure. The rigidity theorem of Mostow [26] asserts that for a compact locally symmetric Riemannian manifold of negative Ricci curvature, the manifold is determined up to isometry and normalizing constants by its fundamental group among the class of such manifolds, with the obvious exceptions involving compact Riemann surfaces. The same theorem for locally symmetric Riemannian manifolds of finite volume and negative Ricci curvature was proved by Prasad [27] in case of rank 1 and included in the super-rigidity theorem of Margulis [13] in the case of rank ? 2. In the class of compact locally symmetric Hermitian manifolds of negative Ricci curvature, Calabi-Vesentini [5] and Borel [3] proved vanishing theorems of certain cohomology groups which imply in particular that for X locally irreducible of complex dimension> 2, there exists no non-trivial deformation of X as a complex manifold. In this direction Siu ([32], [33]) proved the strong rigidity of the Kdhler manifold X in the sense that any compact Kkhler manifold M homotopic to X is necessarily biholomorphic or conjugate-biholomorphic to X. If M is also locally symmetric by assumption then it follows from the uniqueness of Kihler-Einstein metrics of negative Ricci curvature (Yau [37]) that the (conjugate-) biholomorphism is in fact an isometry up to a normalizing constant. It is thus natural to ask

Non-uniqueness in hyperbolic Cauchy problems

Abstract: where t E R and x E Ri', we say that it possesses the uniqueness property at a point (to, xo) if, for any solution u E C'(V) of (1), V being a neighborhood of (t0o xO) such that u 0 on V nf {t < to }, it results that u 0 on a neighborhood of (to, xo). The first uniqueness results for Equation (1) were supplied by the classical theorems of Holmgren [12], Carleman [6] and Calderon [5]; more refined versions of these theorems are the following ones:

Probleme de la couronne pour des domaines pseudoconvexes a bord lisse

Abstract: soit verifiee dans U? De facqon equivalente, si _' designe le spectre de l'algebre uniforme HOO(Q), Q2 est-il dense dans k muni de la topologie de Gelfand? Le probleme precedent a ete resolu affirmativement pour n = 1 et ?2 le disque unite par L. Carleson [1]. Des domaines plus generaux ont ete etudies par L. Carleson [2], J. Garnett et P. Jones [6]. Nous renvoyons le lecteur 'a la monographie de J. Garnett [5] et a son expose [7] pour une etude de ces problemes ainsi que pour une bibliographie extensive. Signalons cependant qu'il n'est pas connu s'il existe un domaine de C pour lequel le probleme de la couronne a une reponse negative. B. Cole, voir [8], a construit une surface de Riemann W telle que W est non dense dans le spectre A de H?(9?). Pour n ? 2 il existe, voir [11], un domaine de Runge U C C C2 ayant les proprietes suivantes: i) U = U c ~2, U # ~2 OUi ~2 designe le bidisque de C2. ii) Toute fonction de H?(U) se prolonge en une fonction de H(?(A2).

Some counterexamples in nonselfadjoint algebras

Abstract: In this note a construction is given of a special operator T on Hilbert space. A free variable in this construction is a weakly closed subspace tY of operators. By appropriate choice of J, we give counterexamples to several well-known