Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L ²-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is a unital C*-algebra (usually the full group C*-algebra C*() of the fundamental group =₁(M) of a manifold M).     

Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics for standard pseudo-hermitian spaces H_p,q. Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of H_p,q. We study the topology of the projective quadrics and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2ⁿ⁻¹, 2ⁿ⁻¹). The spinorial group Spin U(p,q) is imbedded into SU(2ⁿ⁻¹, 2ⁿ⁻¹). At last, we study the natural imbeddings of the projective quadrics      

Disconnectedness of sublevel sets of some Riemannian functionals

LetM be a compact manifold of dimension greater than four. Denote byRiem(M) the space of Riemannian structures onM (i.e. of isometry classes of Riemannian metrics onM) endowed with the Gromov-Hausdorff metric. LetRiem (M) Riem(M) be its subset formed by all Riemannian structures such that vol()=1 andinj() , whereinj() denotes the injectivity radius of.We prove that for all sufficiently small positive the spaceRiem (M) is disconnected. Moreover, if is sufficiently small, thenRiem (M) is representable as the union of two non-empty subsetsA andB such that the Gromov-Hausdorff distance between any element ofA and any element ofB is greater than/9. We also prove a more general result with the following informal meaning: There exist two Riemannian structures of volume one and arbitrarily small injectivity radius onM such that any continuous path (and even any sequence of sufficiently small jumps) in the space of Riemannian structures of volume one onM connecting these Riemannian structures must pass through Riemannian structures of injectivity radius uncontrollably smaller than the injectivity radii of these two Riemannian structures.These results can be generalized for at least some four-dimensional manifolds. The technique used in this paper can also be used to prove the disconnectedness of many other subsets of the space of Riemannian structures onM formed by imposing various constraints on curvatures, volume, diameter, etc.This work was partially supported by the New York University Research Challenge Fund grant, by NSF grant DMS 9114456 and by the NSERC operating grant OGP0155879.     

Symplectic manifolds with contact type boundaries

Summary 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, ) is symplectomorphic to a neighbourhood ofS ^2n–1 in standard Euclidean space, and if vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB ²ⁿ.Oblatum 19-III-1990Partially supported by NSF grant no: DMS 8803056     

Deformations of branched Lefschetz Pencils

LetM be a projective manifold of dimension 3 andH ¹ (M, )=0. We will show that a deformation of a codimension one singular foliation arising from the fibers of a generic meromorphic map of the formf ^p /g ^q ,p, q>0 has a meromorphic first integral of the same type.Supported by Conacyt: 3398 E9307.     

Minimax tests for convex cones

Let (P : R ^p ) be a simple shift family of distributions onR ^p , and letK R ^p be a convex cone. Within the class of nonrandomized tests ofK versusR ^p K, whose acceptance regionA satisfiesA=A+K, a test with minimal bias is constructed. This minimax test is compared to a likelihood ratio type test, which is optimal with respect to a different criterion. The minimax test is mimicked in the context of linear regression and one-sided tests for covariance matrices.     

Numerical decomposition of a convex function

Given then×p orthogonal matrixA and the convex functionf:R ⁿR, we find two orthogonal matricesP andQ such thatf is almost constant on the convex hull of ± the columns ofP, f is sufficiently nonconstant on the column space ofQ, and the column spaces ofP andQ provide an orthogonal direct sum decomposition of the column space ofA. This provides a numerically stable algorithm for calculating the cone of directions of constancy, at a pointx, of a convex function. Applications to convex programming are discussed.This work was supported by the National Science and Engineering Research Council of Canada (Grant No. A3388 and Summer Grant).     

Cohomology of compact hyperkähler manifolds and its applications

We announce the structure theorem for theH ²(M)-generated part of cohomology of a compact hyperkähler manifold. This computation uses an action of the Lie algebra so(4,n–2) wheren=dimH ²(M) on the total cohomology space ofM. We also prove that every two points of the connected component of the moduli space of holomorphically symplectic manifolds can be connected with so-called twistor lines — projective lines holomorphically embedded in the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.     

Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix

Let Agl(n, C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess(p, A) of the complete flag manifold consisting of those flags S ₁ S _n–1 in ⁿ which satisfy the condition AS _i S _i+p ,for all i. We show that if A has distinct eigenvalues, then Hess(p, A) is smooth and connected. The odd Betti numbers of Hess(p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers. In the case where the eigenvalues of A have distinct moduli, |₁|<<|₁|, these results are applied to determine the dimension and topology of the submanifold of U(n) consisting of those unitary matrices P for which A ₀=P ^-1 AP is in Hessenberg form and for which the diagonal entries of the QR-iteration initialized at A ₀ converge to a given permutation of ₁,_n.Research partially supported by the National Science Foundation under Grant ECS-8696108.     

The classification of ϕ-symmetric Sasakian manifolds

A -symmetric spaceM is a complete connected regular Sasakian manifold, that fibers over an Hermitian symmetric spaceN, so that the geodesic involutions ofN lift to define global (involutive) automorphisms of the Sasakian structure onM. In the present paper the complete classification of -symmetric spaces is obtained. The groups of automorphisms of the Sasakian structures and the groups of isometries of the underlying Riemannian metrics are determined. As a corollary, the Sasakian space forms are also determined.     

A geometric programming approach to the Van der Waerden conjecture on doubly stochastic matrices

LetD _p be the set of all doubly stochastic square matrices of orderp i.e. the set of allp × p matrices with non-negative entries with row and column sums equal to unity. The permanent of ap × p matrixA = (a _ij ) is defined byP(A) = _Sp II _i=1 ^p a _i(i) whereS _p is the symmetric group of orderp. Van der Waerden conjectured thatP(A) p !/p ^p for all A AD _p with equality occurring if and only ifA = J _p , whereJ _p is the matrix all of whose entries are equal to 1/p.The validity of this conjecture has been shown for a few values ofp and for generalp under certain assumptions. In this paper the problem of finding the minimum of the permanent of a doubly stochastic matrix has been formulated as a reversed geometric program with a single constraint and an equivalent dual program is given. A related problem of reversed homogeneous posynomial programming problem is also studied.     

Formule d'homotopie entre les complexes de Hochschild et de De Rham

Let k be the field or let M be the space k ⁿ and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hochschild homology H _·(A,A) is isomorphic to the de Rham differential forms over M: this means that the complexes (C _·(A,A),b) and (^·(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy formula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functions over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology.     

An eigenvalue problem for the numerical range of a bounded linear operator

LetX be a complex Lebesgue space with a unique duality mapJ fromX toX ^*, the conjugate space ofX. LetA be a bounded linear operator onX. In this paper we obtain a non-linear eigenvalue problem for (A)=sup{Re: W(A} whereW(A)={J(x)A(x)) : x=1}, under the assumption that (A) and the convex hull ofW(A) for some linear operatorsA onl _p , 2<p<.     

On the Space of Riemannian Metrics on Symplectic and Contact Manifolds

Let A M be the space of all almost Kahlerian smooth metrics on a symplectic manifold M ²ⁿ , such that the fundamental form of each metric coincides with . It is well known that A M is a retractor of the space M of all smooth metrics on M. We show that M is a smooth trivial bundle over A M . A similar fact holds also in the case of a contact manifold.     

Difference sets and inverting the difference operator

For a setA of non-negative numbers, letD(A) (the difference set ofA) be the set of nonnegative differences of elements ofA, and letD ^k be thek-fold iteration ofD. We show that for everyk, almost every set of non-negative integers containing 0 arises asD ^k (A) for someA. We also give sufficient conditions for a setA to be the unique setX such that 0X andD ^k (X)=D ^k (A). We show that for eachm there is a setA such thatD(X)=D(A) has exactly 2 ^m solutionsX with 0X.This work was supported by grants DMS 92-02833 and DMS 91-23478 from the National Science Foundation. The first author acknowledges the support of the Hungarian National Science Foundation under grants, OTKA 4269, and OTKA 016389, and the National Security Agency (grant No. MDA904-95-H-1045).Lee A. Rubel died March 25, 1995. He is very much missed by his coauthors.     

Coupling of the GB set property for ergodic averages

Let (Y,,,T) be an ergodic dynamical system. LetA be an nonempty subset ofL ²() such that , whereA=sup{||sȒt||₂ ,s, tA} andN(A, u) is the smallest number ofL ²()-open balls of radiusu, centered inA, enough to coverA. Let . We prove as a consequence of a more general result, thatC(A) is aGB subset ofL ²().     

Exact solution of linear equations usingP-adic expansions

Summary A method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA for some primep; (ii) using successive refinements to compute an integer vector such that (modp ^m ) for a suitably large integerm; and (iii) deducing the rational solutionx from thep-adic approximation . For matricesA andb with entries of bounded size and dimensionsn×n andn×1, this method can be implemented in timeO(n ³(logn)²) which is better than methods previously used.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (Grant No. A 7171)     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

Galois Theory for the Selmer Group of an Abelian Variety

This paper concerns the Galois theoretic behavior of the p-primary subgroup Sel _A (F) _p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map s _K/F Sel _A (F) _p Sel _A (K) _p ^G(K/F) where F is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B.Mazurand M. Harris.     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.