Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.     

Duality theorems for convex problems of Lagrange with time-delay

Convex problems of Lagrange with time-delay in the state areexamined. The state x(t) is constrained in a convex set X(t)for each t. Optimal trajectories satisfy certain subdifferentialconditions which involve ‘multipliers’ that solvesome dual, convex problem.     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.     

The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ^–1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ^–1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a ‘symplecticblowup’ of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X.     

Direct Sums of Operator Spaces

It is proved that if X and Y are operator spaces such that everycompletely bounded operator from X into Y is completely compactand Z is a completely complemented subspace of X Y, then thereexists a completely bounded automorphism : X Y X Y with completelybounded inverse such that Z = X₀ Y₀, where X₀ and Y₀ are completelycomplemented subspaces of X and Y, respectively. If X and Yare homogeneous, the existence is proved of such a under aweaker assumption that any operator from X to Y is strictlysingular. An upper estimate is obtained for ||||_cb||^–1||_cbif X and Y are separable homogeneous Hilbertian operator spaces.Also proved is the uniqueness of a ‘completely unconditional’basis in X Y if X and Y satisfy certain conditions.     

Solvability of symmetric word equations in positive definite letters

Let S(X, B) be a symmetric (‘palindromic’) wordin two letters X and B. A theorem due to Hillar and Johnsonstates that for each pair of positive definite matrices B andP, there is a positive definite solution X to the word equationS(X, B)=P. They also conjectured that these solutions are finiteand unique. In this paper, we resolve a modified version ofthis conjecture by showing that the Brouwer degree of such anequation is equal to 1 (in the case of real matrices). It followsthat, generically, the number of solutions is odd (and thusfinite) in the real case. Our approach allows us to addressthe more subtle question of uniqueness by exhibiting equationswith multiple real solutions, as well as providing a secondproof of the result of Hillar and Johnson in the real case.     

Derangements and Eigenvalue-Free Elements in Finite Classical Groups

Permutations that have no fixed points have been known for avery long time as ‘derangements’. Under that headingRouse-Ball [10, p. 46] puts the matter in the following charmingway: ‘Suppose you have written a letter to each of n differentfriends, and addressed the n corresponding envelopes. In howmany ways can you make the regrettable mistake of putting everyletter into a wrong envelope?’ He traces this problemback to 1713 and since then it has occurred in one form or anotherin many elementary texts on probability theory (see for example[12, p. 57] and references cited by Rouse-Ball). Let p_n be theprobability that all the letters are put into the wrong envelope.It is well known that p_ne^–1 as n, and that, moreover,convergence is very fast. In fact, , so that . A derangement can be thought of as a fixed-point-free elementof the symmetric group Sym(n). In this paper we turn our attentionto eigenvalue-free elements of finite linear groups. Since aneigenvector of a linear transformation X in a vector space Vcorresponds to a fixed point of X in the projective space whosepoints are the 1-dimensional subspaces of V, eigenvalue-freeinvertible matrices correspond to derangements in projectivegroups.     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

Banach Spaces Whose Algebras of Operators are Unitary: A Holomorphic Approach

An element u of a norm-unital Banach algebra A is said to beunitary if u is invertible in A and satisfies ||u|| = ||u^–1||= 1. The norm-unital Banach algebra A is called unitary if theconvex hull of the set of its unitary elements is norm-densein the closed unit ball of A. If X is a complex Hilbert space,then the algebra BL(X) of all bounded linear operators on Xis unitary by the Russo–Dye theorem. The question of whetherthis property characterizes complex Hilbert spaces among complexBanach spaces seems to be open. Some partial affirmative answersto this question are proved here. In particular, a complex Banachspace X is a Hilbert space if (and only if) BL(X) is unitaryand, for Y equal to X, X* or X** there exists a biholomorphicautomorphism of the open unit ball of Y that cannot be extendedto a surjective linear isometry on Y. 2000 Mathematics SubjectClassification 46B04, 46B10, 46B20.     

Approximately Local Derivations

Certain linear operators from a Banach algebra A into a BanachA-bimodule X, which are called approximately local derivations,are studied. It is shown that when A is a C^*-algebra, a Banachalgebra generated by idempotents, a semisimple annihilator Banachalgebra, or the group algebra of a SIN or a totally disconnectedgroup, bounded approximately local derivations from A into Xare derivations. This, in particular, extends a result of B.E. Johnson that ‘local derivations on C^*-algebras arederivations’ and provides an alternative proof of it.     

The Convex Hull of Extremal Vector-Valued Continuous Functions

Let T be a completely regular space and X a strictly convexn-dimensional real space. We prove that every continuous functionfrom T into the closed unit ball of X can be expressed as anaverage of eight continuous functions from T into the sphereof X if and only if dim (T) n–1, where dim(T) denotesthe covering dimension of T. The proof we give can be used toprove the same fact, without hypotheses on T, when X is infinite-dimensional,although in this case it has been proved recently that a betterresult can be obtained.     

Quadratisation de Certaines Structures de Poisson

On sait associer à certaines structures de Poisson surRⁿ, de 1-jet nul en 0, des actions de R² sur Rⁿ, donnéespar le ‘rotationnel’ de leur partie quadratiqueet un autre champ de vecteurs. Lorsque ces actions sont ‘nonrésonantes’ et ‘hyperboliques’, onmontre que ces structures sont ‘quadratisables’,en ce sens qu'il existe des coordonnées dans lesquelles,elles sont quadratiques. Dans le cas de la dimension 3, nosrésultats mènent à la ‘non-dégénérescence’générique des structures de Poisson quadratiquesà rotationnels inversibles. We can associate with some Poisson structures defined on Rⁿwith a zero 1-jet at zero, actions from R² on Rⁿ, given by the‘curl’ of their quadratic part and another vectorfield. Assuming that those actions are ‘hyperbolics’and without ‘resonances’, we give a normal formfor those structures. On R³, we prove that every quadratic Poissonstructure with invertible curl, is generically ‘non degenerate’.     

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

Our first result is a ‘sum-product’ theorem forsubsets A of the finite field F_p, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums     

On Principal Blocks of p-Constrained Groups

A theorem of K. W. Roggenkamp and L. L. Scott shows that fora finite group G with a normal p-subgroup containing its owncentralizer, any two group bases of the integral group ringZG are conjugate in the units of Z_pG. Though the theorem presentsitself in the work of others and appears to be needed, thereis no published account. There seems to be a flaw in the proof,because a ‘theorem’ appearing in the survey [K.W. Roggenkamp, ‘The isomorphism problem for integral grouprings of finite groups’, Progress in Mathematics 95 (Birkhäuser,Basel, 1991), pp. 193--220], where the main ingredients of aproof are given, is false. In this paper, it is shown how toclose this gap, at least if one is only interested in the conclusionmentioned above. Therefore, some consequences of the resultsof A. Weiss on permutation modules are stated. The basic stepsof which any proof should consist are discussed in some detail.In doing so, a complete, yet short, proof of the theorem isgiven in the case that G has a normal Sylow p-subgroup. 2000Mathematical Subject Classification: primary 16U60; secondary20C05.     

Strongly Cohen-Macaulay Ideals of Small Second Analytic Deviation

In this paper, the strongly Cohen-Macaulay ideals of secondanalytic deviation one are characterized in terms of the depthproperties of the powers of the ideal in the ‘standardrange’. This provides an explanation of the behaviourof certain ideals that have appeared in the literature. 2000Mathematics Subject Classification 13H10 (primary), 13C40, 13D02,13D25 (secondary).     

Extensions of Isometric Dual Representations of Semigroups

Let T be a dual representation of a suitable subsemigroup Sof a locally compact abelian group G by isometries on a dualBanach space X=(X_*)^*. It is shown that (X, T) can be extendedto a dual representation of G on a dual Banach space Y containingX, and that this extension can be done in a canonical way. Inthe case of a representation by *-monomorphisms of a von Neumannalgebra, the extension is a representation of G by *-automorphismsof a von Neumann algebra.     

Homology and derived p-series of groups

We give homological conditions that ensure that a group homomorphisminduces an isomorphism modulo any term of the derived p –series, in analogy to Stallings's 1963 result for the p-lowercentral series. In fact, we prove a stronger theorem that isanalogous to Dwyer's extensions of Stallings’ results.It follows that spaces that are _p-homology equivalent have isomorphicfundamental groups modulo any term of their p-derived series.Various authors have related the ranks of the successive quotientsof the p-lower central series and of the derived p-series ofthe fundamental group of a 3-manifold M to the volume of M,to whether certain subgroups of ₁(M) are free, to whether finiteindex subgroups of ₁(M) map onto non-abelian free groups, andto whether finite covers of M are ‘large’ in variousother senses.     

The Kissing Numbers of Convex Bodies -- A Brief Survey

Nearly four hundred years ago, the cubic close-packing of equalspheres in R³ was discovered by Kepler [21], in which each spheretouches 12 others. In 1694, Gregory and Newton discussed thefollowing thirteen spheres problem. Can a rigid material spherebe brought into contact with 13 other such spheres of the samesize? Gregory believed ‘yes’, while Newton thought‘no’. 1991 Mathematics Subject Classication 11H31,52C17.     

On the independence of expansions of algebraic numbers in an integer base

Let b 2 be an integer. According to a conjecture of ÉmileBorel, the b-adic expansion of any irrational algebraic numberbehaves in some respects ‘like a random sequence’.We give a contribution to the following related problem: let and ' be irrational algebraic numbers, then prove that theirb-adic expansions either have the same tail, or behave in somerespects ‘like independent random sequences’.     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).