Showing papers in "Bulletin of the American Mathematical Society in 1987"

Recursively enumerable sets and degrees

Abstract: TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11. Basic facts. 12. The finite injury priority method. 13. The infinite injury priority method. 14. The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R.

Symplectic groupoids and Poisson manifolds

Abstract: On definit la notion de groupoide symplectique On donne la variete de Poisson de ce groupoide symplectique On presente une construction

Maximum entropy and the moment problem

Abstract: Introduction. The trigonometric moment problem stands at the source of several major streams in analysis. From it flow developments in function theory, in spectral representation of operators, in probability, in approximation, and in the study of inverse problems. Here we connect it also with a group of questions centering on entropy and prediction. In turn, this will suggest a simple approach, by way of orthogonal decomposition, to the moment problem itself. In statistical estimation, one often wants to guess an unknown probability distribution, given certain observations based on it. There are generally infinitely many distributions consistent with the data, and the question of which of these to select is an important one. The notion of entropy has been proposed here as the basis of a principle of salience which has received considerable attention. We will show that, in the context of spectral analysis, this idea is linked to a certain question of prediction by the trigonometric moment problem, and that all three strongly illuminate one another. The phenomena we describe are known, but our object is to unify them conceptually and to reduce the analytic intricacy of the arguments. To this end, we give a completely elementary discussion, virtually free of calculation, which shows that all the facts, including those concerning the moment problem, can be understood as direct consequences of orthogonal decomposition in a finite-dimensional space. We then describe how, in its continuous version, this leads to a view of second-order Sturm-Liouville differential equations, and conclude with some questions concerning the connection between combinatorial ideas and orthogonality in this problem.

Conformal geometry and complete minimal surfaces

Abstract: 1. Introduction. This note applies new ideas from conformai geometry to the study of complete minimal surfaces of finite total curvature in R 3. The two main results illustrate this in complementary ways. Theorem A implies several uniqueness or nonexistence corollaries, while Theorem B constructs new examples, including the first complete immersed nonorientable minimal surfaces with finite total curvature and embedded ends. This construction is based upon the close relationship between these minimal surfaces and certain compact immersed surfaces minimizing the conformally invariant functional W = fH 2 da (Theorems C and D).

The topology and geometry of embedded surfaces of constant mean curvature

Abstract: where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.

The structure of algebraic threefolds: an introduction to Mori's program

Abstract: Introduction. This article intends to present an elementary introduction to the emerging structure theory of higher-dimensional algebraic varieties. Introduction is probably not the right word; it is rather like a travel brochure describing the beauties of a long cruise, but neglecting to mention that the first half of the trip must be spent toiling in the stokehold. Perusal of brochures might give some compensation for lack of royal roads. Having this limited aim in mind, the prerequisities were kept very low. As a general rule, geometry is emphasized over algebra. Thus, for instance, nothing is used from abstract algebra. This had to be compensated by using more results from topology and complex variables than is customary in introductory algebraic geometry texts. Still, aside from some harder results used in occasional examples, only basic notions and theorems are required. Throughout the history of algebraic geometry the emphasis constantly shifted between the algebraic and the geometric sides. The first major step was a detailed study of algebraic curves by Riemann. He approached the subject from geometry and analysis, and gave a quite satisfactory structure theory. Subsequently the German school, headed by Max Noether, introduced algebra to the subject and problems arising from algebraic geometry substantially influenced the development of commutative algebra, especially the works of Emmy Noether and Krull. During the same period the Italian school of Castelnuovo, Enriques, and Severi investigated the geometry of algebraic surfaces and achieved a satisfactory structure theory. Their work, however, lacked the Hilbertian rigor, and

Constant mean curvature surfaces in Euclidean three-space

Abstract: In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in [10] by adopting the more precise and powerful version of the methodology which was developed in [14]. As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in [10] and we produce a very large class of new embedded examples of finite topology.

The Steinberg representation

Abstract: Introduction. Group representations occupy a sort of middle ground between abstract groups and transformation groups, i.e., groups acting in concrete ways as permutations of sets, homeomorphisms of topological spaces, diffeomorphisms of manifolds, etc. The requirement that the elements of a group act as linear operators on a vector space limits somewhat the complexity of the action without sacrificing the depth or applicability of the resulting theory. As in other areas of mathematics, study of linear phenomena may illuminate more general phenomena. The widespread use of group representations in mathematics (as well as in physics, chemistry,... ) does not imply the existence of a single unified subject, however. Nor do practitioners always understand one another's language. Groups come in many flavors: finite, infinite-but-discrete, compact, locally compact, etc. Vector spaces may be finite or infinite dimensional; in the latter case there might be a Hilbert space structure and operators might be required to be unitary. The underlying scalar field may be complex, real, /?-adic, finite,.... One can also make groups act on free modules over rings of arithmetic interest such as Z. Even the study of finite group representations, which probably came first historically, has become somewhat fragmented. Traditionally one considers representations of finite groups by n X n matrices with entries from C. These are the \"ordinary\" representations. But in the late 1930s Richard Brauer began to show the usefulness of \"modular\" representations (with matrix entries lying in a field of prime characteristic) as a tool in the ordinary theory and in the structure theory of finite groups. There is now an active modular industry, with a life of its own, benefiting from recent innovations such as quivers and almost split sequences in the representation theory of finite dimensional algebras (which include group algebras). Study of \"integral\" representations is equally active, motivated by number-theoretic considerations or by questions raised by topologists about integral group rings of fundamental groups.

Morse theory for fixed points of symplectic diffeomorphisms

Abstract: On demontre le cas particulier suivant de la conjecture de Arnold sur les points fixes d'une deformation exacte φ d'une variete symplectique compacte close P: si π 2 (P)=0 et tous les points fixes de φ sont non degeneres, alors leur nombre est plus grand ou egal a la somme des nombres de Betti de P par rapport aux coefficients Z 2

Generalized exponents via Hall-Littlewood symmetric functions

Abstract: The generalized exponents of finite-dimensional irreducible representations of a compact Lie group are important invariants first constructed and studied by Kostant in the early 1960s. Their actual computation has remained quite enigmatic. What was known ([K] and [Hs, Theorem 1]) suggested to us that their computation lies at the heart of a rich combinatorially flavored theory. This note announces several results all tied together by Theorem 2.3 below, which selects the natural generalizations of the Hall-Littlewood symmetric functions, rather than the irreducible characters, as the best basis of the character ring. Full details will appear elsewhere.

On the decidability of Diophantine problems in combinatorial geometry

Abstract: On montre que la decidabilite algorithmique de la resolubilite des equations diophantiennes dans le corps des nombres rationnels est equivalente a une conjecture de B. Grunbaum sur la coordination rationnelle en geometrie combinatoire

Arguesian lattices which are not linear

Abstract: A linear lattice is one representable by commuting equivalence relations. We construct a sequence of finite lattices An (n > 3) with the properties: (i) An is not linear, (ii) every proper sublattice of An is linear, and (iii) any set of generators for An has at least n elements. In particular, An is then Arguesian for n > 7. This settles a question raised in 1953 by Jónsson.

BMO on the Bergman spaces of the classical domains

Abstract: Let Ü be a bounded symmetric (Cartan) domain with its Harish-Chandra realization in C n [T]. For dv the usual Euclidean volume measure on C n = R n , normalized so that v(Q) = 1, we consider the Hubert space of squareintegrable complex-valued functions L = L(Q,dv) and the Bergman subspace H = H(Q) of holomorphic functions in L. The self-adjoint projection from L onto H is denoted by P. For ƒ, g in L, we consider the multiplication operator Mf on L given by Mjg = f g and the Hankel operator Hf on L given by H f = (I — P)MfP. For ƒ in L, these operators are only densely defined and may be unbounded. The commutator [Mf,P] = MfP — PMf is densely defined on L and may also be unbounded. From the equations

On discrete chamber-transitive automorphism groups of affine buildings

Abstract: 1. Introduction. Let A be the affine building of a simple adjoint algebraic group Q of relative rank > 2 over a locally compact local field K. Let Aut A (resp. E Aut A) denote the group of type-preserving (resp. of all) auto-morphisms of A. Note that E Aut A contains the group $(K) of ÜT-rational points of §. We will be interested in discrete subgroups of Aut A which are chamber-transitive on A. It is extremely rare that such groups exist and, as can therefore be expected, exceptions are interesting phenomena; our purpose is to list them all (see the theorem below). In order to describe them we must first introduce some notation. Let ƒ be a quadratic form in n variables over Q p with coefficients in Z. We let Pfi(/, Z[l/p]) denote the intersection PSO(/, Q p)'nPGL(n, Z[l/p]) within PGL(n,Q p), and similarly PGO(/, Z[l/p]) = PGO(/,Q p) nPGL(n,Z[l/p]). In the following list, T will always be a chamber-transitive subgroup of Aut A. The fundamental quadratic form (over Z) of the root lattice of type A n , B n , E n , normalized so that the long roots have squared length 2, will be denoted by a n ,6 n ,e n , respectively; note that b n is Yl\" x l-(i) Let ƒ = eg, &7,ci6,66*^6* or as, and let A be the affine building of PSO(/,Q 2). Here T can be any group between r min = Pfi(/,Z[l/2]) and r ma x = PGO(/,Z[l/2]) fi Aut A. The quotient r max /r min is elementary abelian of order 1, 1, 1, 4, 2, or 2, respectively, and r max is generated by Train and reflections. (ii) Let ƒ = &5,e6, or b' e = Xa x ? + ^ x h an(l let A be the building of PSO(/,Q 3). The group r max (/) = PGO(/, Z[l/3]) n Aut A has 3, 5, or 9 conjugacy classes of chamber-transitive subgroups T. Passage mod 2 maps r m ax(b5) onto the symmetric group S5, and the preimages in r max (&5) of S5, A5, or a group of order 20 form the 3 desired conjugacy classes of groups T. The forms e& and b' e are rationally equivalent, and hence the buildings they define over Q3 are the \"same\" ; with suitable identifications of buildings and groups, T b = r max (ee) …

Counting Latin rectangles

Abstract: On obtient une formule semblable a celle de J.R. Nechvatal pour l'enumeration des rectangles latins

Quantum field theory in ninety minutes

Abstract: On presente divers aspects de la theorie quantique des champs: la theorie p(φ) 2 a 2 dimensions, la theorie φ 3 4 a 3 dimensions, la theorie de Yang-Mills a 4 dimensions