Le travail ci-dessous developpe sur quelques points les tex:tes fondamentaux de C.L. Siegel [13[ et de K. Ramachandra [2). Remerclements C'est au Max Planck Institut de Bonn que la plus grande part des resultats (th. 2 et 3, ex:ception faite du point 3 d et th. 4 et 5) ont ete soit rectiges soit con~s. La rectaction definitive de ce travail a eu lieu ä l'Institut Fourier de Grenoble durant l'hiver 1990. Le th. 1 tel qu'il apparait ici, et le corollaire du th. 6 cf. identite (13), sont nouveaux. On trouvera une rectaction detailleedes th. 2 et 3 dans [51 et, parmi d'autres resultats, des th. 4, 5 et 6 dans [7). Que tous mes collegues et les deux equipes de secretartat recoivent ici mes remerciements les plus chaleureux. 2 1) On pose e( x) = e 1rix, x E C. Pour L un reseau complex:e, on note une base positivement olientee de L = lw + lw c'est-ä-dire teile que 1 2 On definit alors une forme modulaire .,.p> de poids 1 par 1](2)(w) ~fn (21l"i)ql/12 IJ ( - qn)2 1 { w2 n>l 1 12 q = e(W) , q 1 = e(W/12) , W = wt!w2 .

This is the 13th annual volume of papers based on lectures given at the Seminaire des Nombres de Paris. The results presented here by an international group of mathematicians reflect recent work in many areas of number theory and should form a basis for further discussion on these topics.

The second of two volumes presenting papers from an international conference on analytic number theory. The two volumes contain 50 papers, with an emphasis on topics such as sieves, related combinatorial aspects, multiplicative number theory, additive number theory, and Riemann zeta-function.

A four-day conference, "Functional Analysis on the Eve of the Twenty First Century," was held at Rutgers University, New Brunswick, New Jersey, from October 24 to 27, 1993, in honor of the eightieth birthday of Professor Israel Moiseyevich Gelfand. He was born in Krasnye Okna, near Odessa, on September 2, 1913. Israel Gelfand has played a crucial role in the development of functional analysis during the last half-century. His work and his philosophy have in fact helped to shape our understanding of the term "functional analysis" itself, as has the celebrated journal Functional Analysis and Its Applications, which he edited for many years. Functional analysis appeared at the beginning of the century in the classic papers of Hilbert on integral operators. Its crucial aspect was the geometric interpretation of families of functions as infinite-dimensional spaces, and of op erators (particularly differential and integral operators) as infinite-dimensional analogues of matrices, directly leading to the geometrization of spectral theory. This view of functional analysis as infinite-dimensional geometry organically included many facets of nineteenth-century classical analysis, such as power series, Fourier series and integrals, and other integral transforms.

In topological measure theory, Radon measures are the most important objects. In the context of locally compact spaces, there are two equivalent canonical definitions. As a set function, a Radon measure is an inner compact regular Borel measure, finite on compact sets. As a functional, it is simply a positive linear form, defined on the vector lattice of continuous real-valued functions with compact support. During the last few decades, in particular because of the developments of modem probability theory and mathematical physics, attention has been focussed on measures on general topological spaces which are no longer locally compact, e.g. spaces of continuous functions or Schwartz distributions. For a Radon measure on an arbitrary Hausdorff space, essentially three equivalent definitions have been proposed: As a set function, it was defined by L. Schwartz as an inner compact regular Borel measure which is locally bounded. G. Choquet considered it as a strongly additive right continuous content on the lattice of compact subsets. Following P.A. Meyer, N. Bourbaki defined a Radon measure as a locally uniformly bounded family of compatible positive linear forms, each defined on the vector lattice of continuous functions on some compact subset.

This monograph explores the geometry of the local Langlands conjecture. The conjecture predicts a parametrizations of the irreducible representations of a reductive algebraic group over a local field in terms of the complex dual group and the Weil-Deligne group. For p-adic fields, this conjecture has not been proved; but it has been refined to a detailed collection of (conjectural) relationships between p-adic representation theory and geometry on the space of p-adic representation theory and geometry on the space of p-adic Langlands parameters. This book provides and introduction to some modern geometric methods in representation theory. It is addressed to graduate students and research workers in representation theory and in automorphic forms.

On Convex Combinations of Unitary Operators in C*-Algebras.- Approximately Inner Derivations, Decompositions and Vector Fields of Simple C*-Algebras.- Derivations in Commutative C*-Algebras.- Representation of Quantum Groups.- Automorphism Groups and Covariant Irreducible Representations.- Proper Actions of Groups on C*-Algebras.- On the Baum-Connes Conjecture.- On Primitive Ideal Spaces of C*-Algebras over Certain Locally Compact Groupoids.- On Sequences of Jones' Projections.- The Powers' Binary Shifts on the Hyperfinite Factor of Type II1.- Index Theory for Type III Factors.- Relative Entropy of a Fixed Point Algebra.- Jones Index Theory for C*-Algebras.- Three Tensor Norms for Operator Spaces.- Extension Problems for Maps on Operator Systems.- Multivariable Toeplitz Operators and Index Theory.- On Maximality of Analytic Subalgebras Associated with Flow in von Neumann Algebras.- Reflections Relating a von Neumann Algebra and Its Commutant.- Normal AW*-Algebras.

Pseudodifferential methods are central to the study of partial differential equations, because they permit an "algebraization." The main purpose of this book is to set up an operational calculus for operators defined from differential and pseudodifferential boundary values problems via a resolvent construction. A secondary purposed is to give a complete treatment of the properties of the calculus of pseudodifferential boundary problems with transmission, both the first version by Boutet de Monvel (brought completely up to date in this edition) and in version containing a parameter running in an unbounded set. And finally, the book presents some applications to evolution problems, index theory, fractional powers, spectral theory and singular perturbation theory. Thus the book’s improved proofs and modern points of view will be useful to research mathematicians and to graduate students studying partial differential equations and pseudodifferential operators.