The Quarterly Journal of Mathematics - Advance Access

HOMOLOGICAL DIMENSIONS OF KOTHE ALGEBRAS

Pirkovskii, A. Yu. — Sun, 15 Nov 2009 21:46:10 PST

Given a metrizable Köthe algebra (P), we compute the global dimension, the weak global dimension, the bidimension and the weak bidimension of (P) in terms of the Köthe set P.

ASYMPTOTICALLY CONICAL ASSOCIATIVE 3-FOLDS

Lotay, J. D. — Thu, 12 Nov 2009 22:30:40 PST

Given an associative 3-fold N in R⁷ which is asymptotically conical with generic rate < 1, we show that the moduli space of deformations of N is locally homeomorphic to the kernel of a smooth map between smooth manifolds. Moreover, the virtual dimension of the moduli space is computed and shown to be non-negative for > –1, whereas N is expected to be isolated for ≤ –1.

AFFINE HALL-LITTLEWOOD FUNCTIONS FOR AFormula AND SOME CONSTANT TERM IDENTITIES OF CHEREDNIK-MACDONALD-MEHTA TYPE

Viswanath, S. — Mon, 02 Nov 2009 00:01:33 PST

We study t-analogs of string functions for integrable highest weight representations of the affine Kac–Moody algebra A⁽¹⁾₁. We obtain closed form formulas for certain t-string functions of levels 2 and 4. As corollaries, we obtain explicit identities for the corresponding affine Hall–Littlewood functions, as well as higher level generalizations of Cherednik's Macdonald and Macdonald–Mehta constant term identities.

APPROXIMATE AMENABILITY OF SCHATTEN CLASSES, LIPSCHITZ ALGEBRAS AND SECOND DUALS OF FOURIER ALGEBRAS

Choi, Y., Ghahramani, F. — Thu, 29 Oct 2009 09:03:52 PDT

Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for approximate amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we prove that bounded approximate amenability of the second dual of a Fourier algebra implies that it is finite-dimensional. Some other results for related algebras are obtained.

FIXED POINTS OF HOLOMORPHIC TRANSFORMATIONS OF OPERATOR BALLS

Ostrovskii, M. I., Shulman, S., Turowska, L. — Wed, 21 Oct 2009 01:35:26 PDT

A new technique for proving fixed-point theorems for families of holomorphic transformations of operator balls is developed. One of these theorems is used to show that a bounded group representation in a real or complex Hilbert space is orthogonalizable or unitarizable (that is similar to an orthogonal or unitary representation), respectively, provided the representation has an invariant indefinite quadratic form with finitely many negative squares.

BANACH-LIE ALGEBRAS WITH EXTREMAL ELEMENTS

Lopez, A. F. — Wed, 14 Oct 2009 08:10:25 PDT

In this paper strongly prime Banach-Lie algebras with extremal elements are described. They turn out to be natural extensions of the classical Banach-Lie algebras of compact operators on Hilbert spaces.

SPARSE VARIANCE FOR PRIMES IN ARITHMETIC PROGRESSION

Brudern, J., Wooley, T. D. — Fri, 18 Sep 2009 01:17:04 PDT

An analogue of the Montgomery–Hooley asymptotic formula is established for the variance of the number of primes in arithmetic progressions, in which the moduli are restricted to the values of a polynomial.

UNIVERSAL INEQUALITIES FOR EIGENVALUES OF THE VIBRATION PROBLEM FOR A CLAMPED PLATE ON RIEMANNIAN MANIFOLDS

Xia, C. — Fri, 04 Sep 2009 01:26:14 PDT

Eigenvalues of the vibration problem for a clamped plate on compact Riemannian manifolds with boundary (possibly empty) are studied. Universal bounds on eigenvalues of the vibration problem for a clamped plate on compact domains in a complex projective space, a minimal submanifold of a Euclidean space or of a unit sphere are obtained and in particular, an explicit upper bound for the (k + 1)th eigenvalue of the vibration problem for a clamped plate on such objects in terms of its first k eigenvalues will be given.

UNIFORMLY CONVEX-TRANSITIVE FUNCTION SPACES

Rambla-Barreno, F., Talponen, J. — Thu, 27 Aug 2009 23:28:20 PDT

We introduce a property of Banach spaces, called uniform convex-transitivity, which falls between almost transitivity and convex transitivity. We will provide examples of uniformly convex-transitive spaces. This property behaves nicely in connection with some vector-valued function spaces. As a consequence, we obtain some new examples of convex-transitive Banach spaces.

THE TERNARY GOLDBACH PROBLEM WITH PRIMES FROM ARITHMETIC PROGRESSIONS

Tolev, D. I. — Tue, 18 Aug 2009 00:05:24 PDT

We establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.

NON-KAHLER SYMPLECTIC MANIFOLDS WITH TORIC SYMMETRIES

Lin, Y., Pelayo, A. — Tue, 04 Aug 2009 23:36:28 PDT

Drawing on the classification of symplectic manifolds with coisotropic principal orbits by Duistermaat and Pelayo, in this note we exhibit families of compact symplectic manifolds, such that: (i) no two manifolds in a family are homotopically equivalent; (ii) each manifold in each family possesses Hamiltonian, and non-Hamiltonian, toric symmetries; (iii) each manifold has odd first Betti number and hence it is not a Kähler manifold. This can be viewed as an application of the aforementioned classification.

H-CONTACT UNIT TANGENT SPHERE BUNDLES OF EINSTEIN MANIFOLDS

Chun, S. H., Park, J. H., Sekigawa, K. — Fri, 31 Jul 2009 22:48:45 PDT

We study the geometric properties of a base manifold whose unit tangent sphere bundle equipped with the standard contact metric structure is H-contact. We shall prove that a necessary and sufficient condition for the unit tangent sphere bundle of an Einstein manifold to be H-contact is that the base manifold is 2-stein. As its applications, we give an explicit classification of such base manifolds in the special cases of irreducible symmetric spaces or of Kähler–Einstein manifolds with non-negative sectional curvature. Further, we provide examples illustrating the non-homogeneous situations in dimension four.

EXPONENTIAL SUMS WITH CONSECUTIVE MODULAR ROOTS OF AN INTEGER

Shparlinski, I. E. — Sat, 11 Jul 2009 04:15:23 PDT

J. Bourgain and the author have recently estimated exponential sums with consecutive modular roots ^1/n (mod p), where is of multiplicative order t ≥ p modulo a prime p (for some fixed > 0) and n runs through the integers in the interval [M + 1, M + N] with gcd(n, t) = 1. However, the saving in that bound against the trivial estimate has not been made explicit. It is shown here that for t ≥ p^1/2+ one can obtain a fully explicit bound for such exponential sums.

SINGULAR DRESSING ACTIONS ON HARMONIC MAPS

Correia, N., Pacheco, R. — Sat, 11 Jul 2009 04:15:23 PDT

In this paper we prove that any harmonic map from a two-sphere S² into an arbitrary compact semisimple matrix Lie group G may be reduced to a constant by using the singular dressing actions introduced in (M. J. Bergvelt and M. A. Guest, Action of loop groups on harmonic maps, Trans. Amer. Math. Soc. 326 (1991), 861–886); this reduction induces a factorization of into flag factors S² -> G, and the singular dressing actions are produced from curves of simple factors (rational loops having a minimum number of singularities, whose dressing action can be computed explicitly) for G^C. A version of this result for an arbitrary inner symmetric space G/K is established. We also prove generating theorems for the rational loops of the fundamental representations of Sp (n)^C and SU (n)^C: in both cases the class of generators is slightly larger than the class of simple factors.

SEMIFIELDS AS FREE MODULES

Al-Ali, M. I. — Sat, 27 Jun 2009 00:16:26 PDT

The aim of this paper is to prove that, if S is a finite semifield over a finite field, and E is an elementary abelian 2-group of automorphisms, then E acts freely on S. Moreover, if E acts freely of rank 1 and if S has even order, then |E| ≤ 4.

ANNIHILATORS OF PERMUTATION MODULES

Doty, S., Nyman, K. — Thu, 04 Jun 2009 04:49:08 PDT

Permutation modules are fundamental in the representation theory of symmetric groups S_n and their corresponding Iwahori–Hecke algebras H = H(S_n). We find an explicit combinatorial basis for the annihilator of a permutation module in the ‘integral’ case—showing that it is a cell ideal in Murphy's cell structure of H. The same result holds whenever H is semisimple, but may fail in the non-semisimple case.

NON-COMMUTATIVE LOCALLY CONVEX MEASURES

Bonet, J., Wright, J. D. M. — Tue, 02 Jun 2009 00:41:20 PDT

We study weakly compact operators from a C*-algebra with values in a complete locally convex space. They constitute a natural non-commutative generalization of finitely additive vector measures with values in a locally convex space. Several results of Brooks, Saîto and Wright are extended to this more general setting. Building on an approach due to Saîto and Wright, we obtain our theorems on non-commutative finitely additive measures with values in a locally convex space, from more general results on weakly compact operators defined on Banach spaces X whose strong dual X' is weakly sequentially complete. Weakly compact operators are also characterized by a continuity property for a certain ‘Right topology’ as in joint work by Peralta, Villanueva, Wright and Ylinen.

ON THE NUMBER OF SQUARES REPRESENTED BY A PRODUCT OF TWO TERNARY QUADRATIC FORMS

Munshi, R. — Fri, 29 May 2009 05:04:19 PDT

In the context of Manin's conjecture it is an important problem to estimate the number of times a ternary quartic form represents a square. In this paper we give good estimates for this counting problem when the quartic form is a product of two ternary quadratic forms.

GROUP ACTION ON GENUS 7 CURVES AND THEIR WEIERSTRASS POINTS

Zomorrodian, R. — Thu, 28 May 2009 02:30:59 PDT

In this work, we generalize the theory of elliptic modular functions, to the case of genus 7. We investigate the equations of all algebraic curves of genus 7, their automorphism groups and their link to modern algebraic geometry and the theory of hyperelliptic curves. We discuss the cyclic covers of any curve of genus 7, the local structure of the moduli space at the corresponding Weierstrass points for each curve. We show that the largest finite group acting as the full automorphism group of a hyperelliptic curve of genus 7 has order 64 and we find its equation. We then obtain all the 3g – 3 = 18 hyperelliptic curves of genus 7 and their full automorphism groups. We discover that there are merely three other finite groups of the order >64 acting on some non-hyperelliptic curves of genus 7. We also obtain the equations of the non-hyperelliptic curves.

DIFFERENTIAL OPERATORS ON AN AFFINE CURVE: IDEAL CLASSES AND PICARD GROUPS

Berest, Y., Wilson, G. — Fri, 15 May 2009 04:52:00 PDT

Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration : R -> Pic X. We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant n which indexes the decomposition of the fibres of into Calogero–Moser spaces. We also study the action of the group Pic D on our fibration; and we explain how to define in the framework of the Grassmannian description of R due to Cannings and Holland.

A GENERIC MULTIPLICATION IN QUANTIZED SCHUR ALGEBRAS

Su, X. — Sun, 03 May 2009 22:54:17 PDT

We define a generic multiplication in quantized Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantized Schur algebras, defined in (A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GL_n, Duke Math. J. 61 (1990), 655–677), a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in (M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147–163). We also prove that the subalgebra of the new algebra gives a geometric realization of a positive part of 0-Schur algebras, defined in (S. Donkin, The q-Schur Algebra, London Mathematical Society Lecture Note Series 253. Cambridge University Press, Cambridge, 1998, x + 179. ISBN: 0-521-64558-1.). Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.

UNIQUENESS OF THE EXTENSION OF 2-HOMOGENEOUS POLYNOMIALS

Galindo, P., Lourenco, M. L. — Wed, 29 Apr 2009 06:01:16 PDT

Homogeneous polynomials of degree 2 on the complex Banach space are shown to have unique norm-preserving extension to the bidual space. This is done by using M-projections and extends the analogous result for c₀ proved by P.-K. Lin.

COMPLEX SUBMANIFOLDS OF ALMOST COMPLEX EUCLIDEAN SPACES

Di Scala, A. J., Vezzoni, L. — Thu, 02 Apr 2009 06:40:23 PDT

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of (R⁴, J), for some almost complex structure J if and only if it is an elliptic curve. Furthermore, we show that any (almost) complex 2n-torus can be holomorphically embedded in (R⁴ⁿ, J) for a suitable almost complex structure J. This allows us to embed any compact Riemann surface in some almost complex Euclidean space and to show many explicit examples of almost complex structures in R²ⁿ, which cannot be tamed by any symplectic form.

THE GRADED CENTER OF THE STABLE CATEGORY OF A BRAUER TREE ALGEBRA

Kessar, R., Linckelmann, M. — Sat, 28 Mar 2009 06:10:00 PDT

We calculate the graded center of the stable category of a Brauer tree algebra. The canonical map from the Tate analogue of Hochschild cohomology to the graded center of the stable category is shown to induce an isomorphism module taking quotients by suitable nilpotent ideals. More precisely, we show that this map is surjective with nilpotent kernel in even degrees, while this map need not be surjective in odd degrees in general.

STRATIFICATION OF UNFOLDINGS OF CORANK 1 SINGULARITIES

Houston, K. — Thu, 19 Mar 2009 07:19:22 PDT

In the study of equisingularity of families of mappings Gaffney introduced the crucial notion of excellent unfoldings. This definition essentially says that the family can be stratified so that there are no strata of dimension 1 other than the parameter axis for the family. Consider a family of corank 1 multi-germs with source dimension less than target. In this paper it is shown how image Milnor numbers can ensure some of the conditions involved in being excellent. The methods used can also be successfully applied to cases where the double point set is a curve. In order to prove the results the rational cohomology description of the disentanglement of a corank 1 multi-germ is given for the first time. Then, using a simple generalization of the Marar–Mond Theorem on the multiple point space of such maps, this description is applied to give conditions which imply the upper semi-continuity of the image Milnor number. From this the main results follow.

ON O-MINIMAL HOMOTOPY GROUPS

Baro, E., Otero, M. — Tue, 17 Mar 2009 06:37:22 PDT

We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with known results on semialgebraic homotopy allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.

AN INTEGRAL REPRESENTATION OF MULTIPLE HURWITZ-LERCH ZETA FUNCTIONS AND GENERALIZED MULTIPLE BERNOULLI NUMBERS

Komori, Y. — Tue, 17 Mar 2009 01:40:50 PDT

A surface integral representation of a multiple generalization of the Hurwitz–Lerch zeta function is given, which is a direct analogue of the well-known contour integral representation of the Riemann zeta function of Hankel's type. From this integral representation, we derive a detailed description of the set of its possible singularities. In addition, we present two formulae for special values of the zeta function at non-positive integers in terms of generalizations of Bernoulli numbers. These results are refinements of previously known ones.

PERIODIC MODULES OF DIMENSION p

Towers, M. — Mon, 16 Mar 2009 07:04:57 PDT

We determine all finite p-groups G such that kG has periodic modules of dimension p, where k is an algebraically closed field of characteristic p, and obtain information about the period of these modules.

STRETCHING CONVEX DOMAINS AND THE HYPERBOLIC METRIC

Banuelos, R., Carroll, T. — Tue, 03 Mar 2009 06:36:03 PST

It is shown that the log-convexity of the density of the hyperbolic metric in a convex planar domain leads to a pointwise comparison between the density of the hyperbolic metric in a convex domain D and that in a domain obtained by stretching D. Applications of this result are given, including estimates for the density of the hyperbolic metric in the domain interior to an ellipse and a lower bound for the density of the hyperbolic metric in a convex domain in terms of the density in a comparison strip. Connections are made with the convexity of related functions on convex regions in space.

A SHORT PROOF OF A THEOREM OF PFITZNER

Fernandez-Polo, F. J., Peralta, A. M. — Thu, 26 Feb 2009 04:52:59 PST

We present a new and shorter proof of the characterization of weak compactness in the dual of a C*-algebra obtained by Pfitzner [H. Pfitzner, Weak compactness in the dual of a C*-algebra is determined commutatively, Math. Ann. 298(2) (1994), 349–371].

THE REPRESENTATION OF THE MAPPING CLASS GROUP OF A SURFACE ON ITS FUNDAMENTAL GROUP IN STABLE HOMOLOGY

Tillmann, U. — Sat, 21 Feb 2009 00:29:41 PST

The natural action of the mapping class group of an orientable or non-orientable surface on its fundamental group induces a group homomorphism into the automorphism group of a free group. In the light of a recent theorem, we determine here the map on stable homology.

THETA CHARACTERISTICS AND STABLE HOMOTOPY TYPES OF CURVES

Rondigs, O. — Sat, 21 Feb 2009 00:09:59 PST

Let k be a field and X be a smooth projective curve over k with a rational point. Then X admits a theta characteristic if and only if the motivic stable homotopy type of X splits off the top cell. The constructed splitting lifts the splitting of the motive of X.

ON CUSP FORM COEFFICIENTS IN NONLINEAR EXPONENTIAL SUMS

Sun, Q. — Thu, 19 Feb 2009 19:43:12 PST

Let f be either a holomorphic Hecke eigenform of weight for SL₂(Z) with or a Maass Hecke eigenform for SL₂(Z) with Laplace eigenvalue 1/4 + ². In the latter case, Here K_i is the modified Bessel function of the third kind and e(z) = e^2iz. This paper studied the cancelation of the coefficients (n) or (n) in nonlinear exponential sums with amplitude n, 0 < ≤ 1/2.

DIAMETER BOUNDS AND HITCHIN-THORPE INEQUALITIES FOR COMPACT RICCI SOLITONS

Fernandez-Lopez, M., Garcia-Rio, E. — Tue, 17 Feb 2009 02:14:36 PST

We give lower bounds for the diameter of a compact Ricci soliton depending on the scalar and Ricci curvatures as well as on the range of the potential function, which do not depend on the dimension of the manifold. As an application, sufficient conditions are provided for a four-dimensional compact Ricci soliton to satisfy the Hitchin-Thorpe inequality.

SEMI-GLOBAL INVARIANTS OF PIECEWISE SMOOTH LAGRANGIAN FIBRATIONS

Castano-Bernard, R., Matessi, D. — Tue, 10 Feb 2009 00:40:17 PST

We study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle co-ordinates to these fibrations by defining certain invariants which give a semi-global classification of germs of stitched fibrations. We then describe stitched fibrations with monodromy in terms of these invariants.

TATE-SHAFAREVICH GROUPS AND FROBENIUS FIELDS OF REDUCTIONS OF ELLIPTIC CURVES

Shparlinski, I. E. — Sun, 01 Feb 2009 21:29:36 PST

Let E/Q be a fixed elliptic curve over Q which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, Cojocaru and Duke have obtained an asymptotic formula for the number of primes p≤x such that the reduction of E modulo p has a trivial Tate–Shafarevich group. Recent results of Cojocaru and David lead to a better error term. We introduce a new argument in the scheme of the proof, which gives a further improvement.

MODULI SPACES OF FLAT SU(2)-BUNDLES OVER NON-ORIENTABLE SURFACES

Baird, T. J. — Wed, 21 Jan 2009 23:37:38 PST

We study the topology of the moduli space of flat SU (2)-bundles over a non-orientable surface . This moduli space may be identified with the space of homomorphisms Hom (₁(), SU (2)) modulo conjugation by SU (2). In particular, we compute the (rational) equivariant cohomology ring of Hom (₁(), SU (2)) and use this to compute the ordinary cohomology groups of the quotient Hom (₁(), SU (2))/SU (2). A key property is that the conjugation action is equivariantly formal.

LOCALLY INNER AUTOMORPHISMS OF OPERATOR ALGEBRAS

Sherman, D. — Fri, 09 Jan 2009 04:53:31 PST

In this paper, an automorphism of a unital C*-algebra is said to be locally inner if on any element it agrees with some inner automorphism. We make a fairly complete study of local innerness in von Neumann algebras, incorporating comparison with the pointwise innerness of Haagerup–Størmer. On some von Neumann algebras, including all with separable predual, a locally inner automorphism must be inner. But a transfinitely recursive construction demonstrates that this is not true in general. As an application, we show that the diagonal sum descends to a well-defined map on the automorphism orbits of a unital C*-algebra if and only if all its automorphisms are locally inner.

GEOMETRY AND ANALYTIC BOUNDARIES OF MARCINKIEWICZ SEQUENCE SPACES

Boyd, C., Lassalle, S. — Wed, 07 Jan 2009 06:12:02 PST

We investigate the geometric structure of the unit ball of the Marcinkiewicz sequence space , giving characterizations of its real and complex extreme points and of the exposed points in terms of the symbol . Using our knowledge of the geometry of we then give necessary and sufficient conditions for a subset of to be a boundary for , the algebra of functions which are uniformly continuous on and holomorphic on the interior of . We show that it is possible for the set of peak points of to be a boundary for yet for not to have a Silov boundary in the sense of Globevnik.

CONSTRUCTION OF CLASS FIELDS OVER IMAGINARY QUADRATIC FIELDS AND APPLICATIONS

Cho, B., Koo, J. K. — Wed, 07 Jan 2009 06:12:01 PST

Let K be an imaginary quadratic field, H_O the ring class field of an order O in K and K_(N) be the ray class field modulo N over K for a positive integer N. In this paper we provide certain general techniques of finding H_O and K_(N) by using the theory of Shimura's canonical models via his reciprocity law, from which we partially extend some results of Schertz (Remark 4.2), Chen-Yui (Remark 4.2, Corollary 4.4), Cox–McKay–Stevenhagen (Corollary 4.5) and Cais–Conrad (Remark 5.3). And, we further reilluminate the classical result of Hasse by means of such a method (Corollary 5.4), and discover how to get one ray class invariant over K from Hasse's two generators (Corollary 5.5) which is different from Ramachandra's invariant [K. Ramachandra, Some applications of Kronecker's limit formulas, Ann. Math. 80 (1964), 104–148].

ASYMPTOTIC UNCONDITIONALITY

Cowell, S. R., Kalton, N. J. — Fri, 02 Jan 2009 08:22:57 PST

We show that a separable real Banach space embeds almost isometrically in a space Y with a shrinking 1-unconditional basis if and only if lim _n->|| x* + x_n*|| = lim _n->||x* – x_n*|| whenever x* X*, is a weak*-null sequence and both limits exist. If X is reflexive then Y can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.

RELATIVE SUPPORT VARIETIES

Bergh, P. A., Solberg, O. — Fri, 19 Dec 2008 07:15:15 PST

We define relative support varieties with respect to some fixed module over a finite-dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite-generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type.

DISTRIBUTION OF ANGLES IN HYPERBOLIC LATTICES

Risager, M. S., Truelsen, J. L. — Tue, 16 Dec 2008 04:55:51 PST

We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result from the study by Boca.

A NOTE ON BELYI'S THEOREM FOR KLEIN SURFACES

Kock, B., Lau, E. — Fri, 12 Dec 2008 05:47:15 PST

Singerman and the first named author have recently developed a real Belyi theory, leaving open a particular case in the proof of Belyi's theorem for Klein surfaces. We answer their question affirmatively by a descent argument which turns out to extend to a much more general context.

KIRWAN SURJECTIVITY IN K-THEORY FOR HAMILTONIAN LOOP GROUP QUOTIENTS

Harada, M., Selick, P. — Fri, 05 Dec 2008 03:30:29 PST

Let G be a compact Lie group and LG be its associated loop group. The main result of this article is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology by R. Bott, S. Tolman and J. Weitsman, Surjectivity for Hamiltonian loop group spaces, Invent. Math. 155 (2004), 225–251, math.DG/0210036. Our proof techniques differ from that of Bott et al. in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy equivalences to obtain the necessary isomorphisms in equivariant K-theory. The main theorem should also be viewed as a first step towards a similar theorem in K-theory for quasi-Hamiltonian G-spaces and their associated quasi-Hamiltonian quotients.

SURFACES WITH CONSTANT MEAN CURVATURE IN RIEMANNIAN PRODUCTS

De Lira, J. H. S., Vitorio, F. A. — Tue, 18 Nov 2008 05:43:01 PST

We prove the existence of holomorphic quadratic differentials for surfaces with parallel mean curvature in some four-dimensional products of space forms. These differentials are then used to characterize spheres with parallel mean curvature immersed into these spaces.

ABSOLUTE CONTINUITY ON C*-ALGEBRAS

Saito, K., Wright, J. D. M. — Fri, 14 Nov 2008 06:36:08 PST

In an earlier work the notion of absolute continuity was extended from finitely additive measures to non-commutative C*-algebras. But to obtain a generalisation of the Vitali–Hahn–Saks theorem valid for all C*-algebras it was necessary to introduce ‘weak’ and ‘strong’ absolute continuity. For commutative algebras, these two notions of absolute continuity coincide but, given recent work by Chetcuti and Hamhalter, it is reasonable to ask if there are wider classes of C*-algebras for which weak and strong absolute continuity coincide.We show here that this is not true. If weak and strong absolute continuity coincide for a given algebra then the algebra must be commutative.

THE EULER CLASS OF A SUBSET COMPLEX

Guclukan, A., Yalcin, E. — Tue, 04 Nov 2008 05:11:17 PST

The subset complex (G) of a finite group G is defined as the simplicial complex whose simplices are non-empty subsets of G. The oriented chain complex of (G) gives a ZG-module extension of Z by tilde;, where tilde; is a copy of integers on which G acts via the sign representation of the regular representation. The extension class _G Ext_ZG^|G|–1 (Z, tilde;) of this extension is called the Ext class or the Euler class of the subset complex (G). This class was first introduced by Reiner and Webb [The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), 291–327] who also raised the following question: What are the finite groups for which _G is non-zero?

In this paper, we answer this question completely. We show that _G is non-zero if and only if G is an elementary abelian p-group or G is isomorphic to Z/9, Z/4 x Z/4 or (Z/2)ⁿ x Z/4 for some integer n ≥ 0. We obtain this result by first showing that _G is zero when G is a non-abelian group, then by calculating _G for specific abelian groups. The key ingredient in the proof is an observation by Mandell which says that the Ext class of the subset complex (G) is equal to the (twisted) Euler class of the augmentation module of the regular representation of G.

We also give some applications of our results to group cohomology, to filtrations of modules and to the existence of Borsuk–Ulam type theorems.

ON SOME CONFORMAL MINIMAL 2-SPHERES IN A COMPLEX PROJECTIVE SPACE

Jiao, X., Peng, J. — Sun, 02 Nov 2008 18:45:14 PST

In this paper, the geometry of a linearly full conformal minimal 2-sphere S² immersed in a complex projective space CPⁿ which satisfies various conditions is studied. Let ₁(p) be the first normal space of S² at the point p, and let T_p S² = ₁(p) ₂(p) for p S². We prove that S² is of constant Kähler angle if and only if J₁(p) T_p S² for all p S², where J is the complex structure of CPⁿ. Furthermore, we prove that (i) S² is totally geodesic in CP² if J ₁(p) T_p S² for all p S²; (ii) S² is either a holomorphic curve in CPⁿ or the first element of the Veronese sequence, up to an isometry of CPⁿ, if J₁(p) ₁(p) for all p S²; (iii) S² is totally real if J₁(p) ₂(p) for all p S². It is also proved that S² is either an element of the Veronese sequence in CP² or a totally real curve of constant curvature 1/3 in CP⁴ if its second fundamental form is parallel.

HOCHSCHILD HOMOLOGY AND COHOMOLOGY OF {ell}1(ZFormula)

Choi, Y. — Wed, 29 Oct 2008 21:40:35 PDT

Building on the recent determination of the simplicial cohomology groups of the convolution algebra ¹(Z^k₊) [F. Gourdeau, Z. A. Lykova and M. C. White, A Künneth formula in topological homology and its applications to the simplicial cohomology of ¹(Z^k₊), Studia Math. 166 (2005), 29–54], we investigate what can be said for the cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of the Harrison homology and cohomology in the context of Banach algebras and a development of some of its basic features. As an application of our techniques, we reprove some known results on second-degree cohomology.

A NOTE ON THE SUM OF THE FIRST n PRIMES

Matomaki, K. — Wed, 29 Oct 2008 19:47:41 PDT

We show that the arithmetic mean of the first n primes is an integer for<<N^19/24+ numbers n≤N. This follows from showing that the discrepancy of the sequence consisting of the arithmetic means is<<N^–5/24+.

ON SUMS OF 13 'ALMOST EQUAL' CUBES

Daemen, D. — Sun, 10 Aug 2008 20:26:46 PDT

A simple proof of a special case is presented in Waring's problem on sums of 13 cubes localized close to their average size, which currently seems to be out of reach for the circle method.