Interpretable Groups, Stably Embedded Sets, and Vaughtian Pairs

The paper concerns sufficiently saturated structures M overa countable language with a unary predicate P. It is shown thatif P(M)is stably embedded and there are no Vaughtian pairs withrespect to P, then an infinite group is interpretable over M(in an infinitary sense of ‘interpretable’). Also,it is shown that if M is -categorical, f:DP is a 0-definablemap with finite fibres, and P(M) is stably embedded but D isnot, then some infinite group is first-order interpretable overM.     

Continuity of Derivations, Intertwining Maps, and Cocycles from Banach Algebras

Let A be a Banach algebra, and let E be a Banach A-bimodule.A linear map S:AE is intertwining if the bilinear map is continuous, and a linear map D:AE is a derivation if ¹D=0,so that a derivation is an intertwining map. Derivations fromA to E are not necessarily continuous. The purpose of the present paper is to prove that the continuityof all intertwining maps from a Banach algebra A into each BanachA-bimodule follows from the fact that all derivations from Ainto each such bimodule are continuous; this resolves a questionleft open in [1, p. 36]. Indeed, we prove a somewhat strongerresult involving left- (or right-) intertwining maps.     

Weyl's Theorem, a-Weyl's Theorem, and Local Spectral Theory

Necessary and sufficient conditions are given for a Banach spaceoperator with the single-valued extension property to satisfyWeyl's theorem and a-Weyl's theorem. It is shown that if T orT^* has the single-valued extension property and T is transaloid,then Weyl's theorem holds for f(T)for every fH((T)). When T^*has the single-valued extension property, T is transaloid andT is a-isoloid, then a-Weyl's theorem holds for f(T) for everyfH((T)). It is also proved that if T or T^* has the single-valuedextension property, then the spectral mapping theorem holdsfor the Weyl spectrum and for the essential approximate pointspectrum.     

Symmetric Groups, Wreath Products, Morita Equivalences, and Broue's Abelian Defect Group Conjecture

It is shown that for any prime p, and any non-negative integerw less than p, there exist p-blocks of symmetric groups of defectw, which are Morita equivalent to the principal p-block of thegroup S_p S_w. Combined with work of J. Rickard, this provesthat Broué's abelian defect group conjecture holds forp-blocks of symmetric groups of defect at most 5.     

Characterization of the Mod 3 Cohomology of the Compact, Connected, Simple, Exceptional Lie Groups of Rank 6

It is shown that the mod 3 cohomology of a 1-connected, homotopyassociative mod 3 H-space that is rationally equivalent to theLie group E₆ is isomorphic to that of E₆ as an algebra. Moreover,it is shown that the mod 3 cohomology of a nilpotent, homotopy-associativemod 3 H-space that is rationally equivalent to E₆, and whosefundamental group localized at 3 is non-trivial, is isomorphicto that of the Lie group Ad E₆ as a Hopf algebra over the mod3 Steenrod algebra. It is also shown that the mod 3 cohomologyof the universal cover of such an H-space is isomorphic to thatof E₆ as a Hopf algebra over the mod 3 Steenrod algebra. 2000Mathematics Subject Classification 57T05, 57T10, 57T25.     

Linking, Legendrian Linking and Causality

The set N of all null geodesics of a globally hyperbolic (d+ 1)-dimensional spacetime (M, g) is naturally a smooth (2d– 1)-dimensional contact manifold. The sky of an eventx in M is the subset X of N consisting of all null geodesicsthrough x, and is an embedded Legendrian submanifold of N diffeomorphicto S^{(d – 1)}. It was conjectured by Low that for d = 2two events x and y are causally related if and only if X andY are linked (in an appropriate sense). We use the contact structureand knot polynomial calculations to prove this conjecture incertain particular cases, and suggest that for d = 3 smoothlinking should be replaced with Legendrian linking.     

Schur Superalgebras in Characteristic p, II

It is proved that if a Schur superalgebra is not semisimple,then it is neither cellular nor quasi-hereditary (Theorem 2),and it has infinite global dimension (Corollary 18). The algebraS(m|n, r) with m, n 1 is semisimple if and only if p, the characteristicof the ground field, is zero or greater than r, or when m =n = 1 and p does not divide r. 2000 Mathematics Subject Classification17A70 (primary), 20C30 (secondary).     

Relative K0, Annihilators, Fitting Ideals and Stickelberger Phenomena

When G is abelian and l is a prime we show how elements of therelative K-group K₀(Z_l[G], Q_l give rise to annihilator/Fittingideal relations of certain associated Z[G]-modules. Examplesof this phenomenon are ubiquitous. Particularly, we give examplesin which G is the Galois group of an extension of global fieldsand the resulting annihilator/Fitting ideal relation is closelyconnected to Stickelberger's Theorem and to the conjecturesof Coates and Sinnott, and Brumer. Higher Stickelberger idealsare defined in terms of special values of L-functions; whenthese vanish we show how to define fractional ideals, generalisingthe Stickelberger ideals, with similar annihilator properties.The fractional ideal is constructed from the Borel regulatorand the leading term in the Taylor series for the L-function.En route, our methods yield new proofs, in the case of abeliannumber fields, of formulae predicted by Lichtenbaum for theorders of K-groups and étale cohomology groups of ringsof algebraic integers. 2000 Mathematics Subject Classification11G55, 11R34, 11R42, 19F27.     

Parity of the Partition Function in Arithmetic Progressions, II

Let p(n) denote the ordinary partition function. Subbarao conjecturedthat in every arithmetic progression r (mod t) there are infinitelymany integers N = r (mod t) for which p(N) is even, and infinitelymany integers M = r (mod t) for which p(M) is odd. We provethe conjecture for every arithmetic progression whose modulusis a power of 2. 2000 Mathematics Subject Classification 11P83.     

Infinite Simple (2, 3, n)-Groups and Congruence Hulls in the Modular Group

We prove that if n > 66 and (n, 30) = 1, then there existuncountably many infinite simple (2, 3, n)- groups, that is,groups generated by a pair of elements x, y, say, where theorders of x, y and xy are 2, 3 and n, respectively. This extendsprevious results of Schupp and the authors. These results are used to prove the existence of subgroups ofthe modular group with special arithmetic properties. 1991 MathematicsSubject Classification 20F06.     

Operator amenability of Fourier-Stieltjes algebras, II

We give an example of a non-compact, locally compact group Gsuch that its Fourier–Stieltjes algebra B (G) is operatoramenable. Furthermore, we characterize those G for which A *(G),the spine of B (G) as introduced by M. Ilie and N. Spronk, isoperator amenable and show that A *(G) is operator weakly amenablefor each G.     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

Animals, Trees and Renewal Sequences: Corrigendum

IN SECTION 3 of the above we omitted to mention aperiodicity.The period p of the pseudo renewal sequence {a_n: n > 0} isgiven by p = g.c.d. {n > 1: a_n > 0}. We are only concernedwith aperiodic renewal sequences (i.e. where p = 1). As it standsTheorem 3.1 is incorrect and should be restated as: THEOREM 3.1 If a = (a_n: n = 0,1,...) is an aperiodic pseudo-renewalsequence its limit a satisfies g_na^–n > 1 where a^–1 is to be interpreted as; if a = 0.     

Multiquadratic Extensions, Rigid Fields and Pythagorean Fields

Let F be a field of characteristic other than 2. Let F⁽²⁾ denotethe compositum over F of all quadratic extensions of F, letF⁽³⁾ denote the compositum over F⁽²⁾ of all quadratic extensionsof F⁽²⁾ that are Galois over F, and let F^{3} denote the compositumover F⁽²⁾ of all quadratic extensions of F⁽²⁾. This paper showsthat F⁽³⁾ = F^{3} if and only if F is a rigid field, and thatF⁽³⁾ = K⁽³⁾ for some extension K of F if and only if F is Pythagoreanand . The proofs depend mainly on the behavior of quadratic forms over quadratic extensions,and the corresponding norm maps.     

Complete CCC Boolean Algebras, the Order Sequential Topology, and a Problem of Von Neumann

Let B be a complete ccc Boolean algebra and let _s be the topologyon B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonemptyopen set in (B, _s) is topologically dense.
2. It is consistentthat every weakly distributive complete cccBoolean algebracarries a strictly positive Maharam submeasure.
3. The topologicalspace(B, _s) is sequentially compact if and onlyif the genericextension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but notindependent reals. 2000 Mathematics Subject Classification 28A60,06E10 (primary); 03E55, 54A20, 54A25 (secondary).     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

Gromov-Witten Invariants of Flag Manifolds, VIA D-Modules

We present a method for computing the 3-point genus zero Gromov–Witteninvariants of the complex flag manifold G/B from the relationsof the small quantum cohomology algebra QH^*G/B (G is a complexsemisimple Lie group and B is a Borel subgroup). In [3] and[9], at least in the case G = GL_nC, two algebraic/combinatoricmethods have been proposed, based on suitably designed axioms.Our method is quite different, being differential geometricin nature; it is based on the approach to quantum cohomologydescribed in [7], which is in turn based on the integrable systemspoint of view of Dubrovin and Givental.     

Ovoids of PG(3,q), q Even, with a Conic Section

It is shown that if a plane of PG(3,q), q even, meets an ovoidin a conic, then the ovoid must be an elliptic quadric. Thisis proved by using the generalized quadrangles T₂(C) (C a conic),W(q) and the isomorphism between them to show that every secantplane section of the ovoid must be a conic. The result thenfollows from a well-known theorem of Barlotti.     

Property A, partial translation structures, and uniform embeddings in groups

We define the concept of a partial translation structure ona metric space X and show that there is a natural C*-algebraC*() associated with it, which is a subalgebra of the uniformRoe algebra C*_u(X). We introduce a coarse invariant of the metricwhich provides an obstruction to embedding the space in a group.When the space is sufficiently group-like, as determined byour invariant, properties of the Roe algebra can be deducedfrom those of C*(). We also give a proof of the fact that theuniform Roe algebra of a metric space is a coarse invariantup to Morita equivalence.