Showing papers in "Inventiones Mathematicae in 1991"

Invariants of 3-manifolds via link polynomials and quantum groups

Abstract: The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory

Limit laws for Random matrices and free products

Abstract: In earlier articles we studied a kind of probability theory in the framework of operator algebras, with the tensor product replaced by the free product. We prove here that free random variables naturally arise as limits of random matrices and that Wigner's semicircle law is a consequence of the central limit theorem for free random variables. In this way we obtain a non-commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra, Wigner's law being given by the trace of the spectral measure of the selfadjoint component of this operator

Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system

Abstract: We prove that, if the initial data has moments inv higher than three, then the solution of Vlasov-Poisson has also moments inv higher than three. We deduce from this different regularity results on the local density, the force field or the solution itself. Also we give a new uniqueness result, and new regularity results for solutions satisfying only the energy andL ∞ bounds. Our proofs are based on a new representation formula and logarithmic estimates for the force field.

A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras

Abstract: Some generalizations of the Lusztig-Lascoux-Schutzenberger operators for affine Hecke algebras are considered. As corollaries we obtain Lusztig's isomorphisms from affine Hecke algebras to their degenerate versions, a “natural” interpretation of the Dunkl operators and a new class of differential-difference operators generalizing Dunkl's ones and the Knizhnik-Zamolodchikov operators from the two dimensional conformal field theory.

Symplectic manifolds with contact type boundaries

Abstract: An example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ω) is symplectomorphic to a neighbourhood ofS2n−1 in standard Euclidean space, and if ω vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB2n.

Nonlinear elliptic equations on compact riemannian manifolds and asymptotics of Emden equations

Abstract: We study the asymptotics and the global solutions of the following Emden equations: −Δu=λe u in a 3-dim domain (λ>0) or −Δu=u q +l|x|−2 u (q>1) in anN-dim domain. Precise behaviour is obtained by the use of Simon's results on analytic geometric functionals. In the case of the first equation, or the second equation with l=0 andq=(N+1)/(N−3) (N>3), we point out how the asymptotics are described via the Moebius group onS N−1. For a conformally invariant equation −Δu=ɛ|u|4/(N−2) u+l|x|−2 u(ɛ=±1) we prove the existence of a new type of solution of the formu(x)=|x|(2−N)/2ω(Γ(Ln|x|)(x/|x|)) where ω is defined onS N−1 and Γ∈C ∞ (ℝ;O(N)). Finnally, we extend and simplify the results of Gidas and Spruck on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Licherowicz-Weitzenbock formula.

Periods of modular forms and Jacobi theta functions

Abstract: co where L ( f s) denotes the L-series o f f ( = analytic continuation of ~ = 1 a:( l ) l s ) . The Eichler-Shimura-Manin theory tells us that the maple--, r: is an injection from the space Sg of cusp forms of weight k on F to the space of polynomials of degree _-< k 2 and that the product of the nth and ruth coefficients of r: is an algebraic multiple of the Petersson scalar product ( f , f ) i f f is a Hecke eigenform and n and m have opposite parity. More precisely, for each integer l > 1 the polynomial in two variables

Bounding the complexity of simplicial group actions on trees

Abstract: We shall state the main result of this paper in terms of group actions on simplicial trees. Suppose that a group G acts simplicially on a tree T without inversions. For brevity we say that Tis a G-tree. Then the orbit space T/G is a graph whose vertices and edges correspond to G-equivalence classes of vertices and edges in T. Each vertex and edge in T/G is labeled by the stabilizer of a representative of the corresponding equivalence class. This label, a subgroup of G, is well-defined only up to conjugation in G (for details, see [7] or [8]). Thus T/G is a graph of groups whose fundamental group is G. We are interested in finding a number ?(G), depending only on G, so that for every G-tree T, the graph T/G has no more than ?(G) vertices and edges. Some remarks are in order.

The heat equation and harmonic maps of complete manifolds

Abstract: 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Mean-value inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Gradient estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Short time solutions of the parabolic harmonic map equation . . . . . . . . . . . . . . . 17 4 Harmonic maps: K = 0 and x = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Harmonic maps: 2(M) > 0 and x = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Harmonic maps: Small total energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Proof of the ionization conjecture in a reduced Hartree-Fock model

Abstract: The ionization conjecture for atomic models states that the ionization energy and maximal excess charge are bounded by constants independent of the nuclear charge. We prove this for the Hartree-Fock model without the exchange term.

Weights of exponential sums, intersection cohomology, and Newton polyhedra

Abstract: (1.1) Throughout this paper k always denotes a finite field Fq with q elements, and E a prime number not dividing q. The algebraic closure of a field K is denoted by / ( . Let ~b: k--+ C • be a nontrivial additive character, and ~ , the Qt-sheaf on A~ associated to ~ and the Artin-Schreier covering t q t = x. For a morphism f : X --+ A~, with X a scheme of finite type over k, one considers the exponential sum S(f ) = ~Xtk)~b(f(x)). By Grothendieck's trace formula we have

A Frobenius formula for the characters of the Hecke algebras

Abstract: This paper uses the theory of quantum groups and the quantum Yang-Baxter equation as a guide in order to produce a method of computing the irreducible characters of the Hecke algebra. This approach is motivated by an observation of M. Jimbo giving a representation of the Hecke algebra on tensor space which generates the full centralizer of a tensor power of the “standard” representation of the quantum group\(U_q (\mathfrak{s}l(n))\). By rewriting the solutions of the quantum Yang-Baxter equation for\(U_q (\mathfrak{s}l(n))\) in a different form one can avoid the quantum group completely and produce a “Frobenius” formula for the characters of the Hecke algebra by elementary methods. Using this formula we derive a combinatorial rule for computing the irreducible characters of the Hecke algebra. This combinatorial rule is aq-extension of the Murnaghan-Nakayama for computing the irreducible characters of the symmetric group. Along the way one finds connections, apparently unexplored, between the irreducible characters of the Hecke algebra and Hall-Littlewood symmetric functions and Kronecker products of symmetric groups.

Volume et entropie minimale des espaces localement symétriques

Abstract: M. Gromov introduit la notion de volume minimal d'une variete riemannienne: min vol(M)=inf{Vol(M,g)\|courbure sectionnelle de g|≤1}. Les questions naturelles sont alors (i) Quelles varietes M verifient min vol(M)=0,min vol(M):0? (ii) Dans le second cas, existe-t-il des metriques realisant le minimum? Lesquelles? En guise de reponse a la seconde question on peut au mieux esperer que, si M supporte une metrique «naturelle», elle realisera le volume minimal. Ceci conduit a la conjecture suivante posee par M. Gromov: Si M est une variete admettant une metrique hyperbolique le volume fini notee hyp, alors min vol(M)=Vol(M,hyp). Le but de l'article est de donner une reponse partielle a cette conjecture

Propagation of singularities and maximal functions in the plane

Abstract: In this work we mainly generalize Bourgain's circular maximal function to include variable coefficient averages. Our techniques involve a combination of Bourgain's basic ideas plus microlocal analysis. In particular, to see the role of curvature, we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations. We also show that, forp>2, there is local smoothing inL p for solutions to the wave equation.

Rational points on K3 surfaces: A new canonical height

Abstract: A fundamental tenet of Diophantine Geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties. This is certainly true for curves, where the sign of the Euler characteristic of C determines whether the set of rational points on C is finite (X (C) 0). For higher dimensional varieties there are some precise conjectures due to Bombieri, Lang, and Vojta [15] which predict when the rational points on a variety should be finite or degenerate (i.e. not Zariski dense), and some conjectures of Manin et al. [-2, 5] on the distribution of rational points in those cases when they are Zariski dense. But except for abelian varieties, their subvarieties, and some Fano varieties (varieties for which the anticanonical bundle is ample), there are very few general theorems. In this paper we will study the rational points on a certain class of K 3 surfaces defined over a number field K. The moduli space of marked algebraic K3 surfaces is a countable union of 19 dimensional quasi-projective varieties. We are going to look at the 18 dimensional family studied by Wehler 1-17]. Wehler's family consists of K3 surfaces S whose automorphism group Aut(S) contains a subgroup d isomorphic to the free product 7/,2"~ 2 of two cyclic groups of order 2. We will use the geometric information provided by this infinite automorphism group to study the K-rat ional points on S. For any point PeS, we can look at the orbit of P under the action of d,