The Quarterly Journal of Mathematics - Advance Access

SEMIFIELDS AS FREE MODULES

Al-Ali, M. I. — 2009-06-27

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. — 2009-06-04

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. — 2009-06-02

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. — 2009-05-29

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. — 2009-05-28

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. — 2009-05-15

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. — 2009-05-03

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. — 2009-04-29

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. — 2009-04-02

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. — 2009-03-28

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. — 2009-03-19

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. — 2009-03-17

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. — 2009-03-17

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. — 2009-03-16

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. — 2009-03-03

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. — 2009-02-26

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. — 2009-02-21

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. — 2009-02-21

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. — 2009-02-19

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. — 2009-02-17

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. — 2009-02-10

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. — 2009-02-01

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. — 2009-01-21

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. — 2009-01-09

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. — 2009-01-07

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. — 2009-01-07

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. — 2009-01-02

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. — 2008-12-19

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. — 2008-12-16

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. — 2008-12-12

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. — 2008-12-05

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. — 2008-11-18

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. — 2008-11-14

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. — 2008-11-04

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. — 2008-11-02

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. — 2008-10-29

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. — 2008-10-29

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. — 2008-08-10

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.

MULTIPLICATIVE CHARACTER SUMS WITH TWICE-DIFFERENTIABLE FUNCTIONS

Banks, W. D., Shparlinski, I. E. — 2008-08-02

For a nontrivial multiplicative character modulo p, we bound character sums taken on the integer parts of a real-valued, twice-differentiable function f whose second derivative decays at an appropriate rate. For the special case that f(x) = x with some positive real number , our bounds extend recent results of several authors.

HYPERBOLIC SECTIONS IN SEIFERT-FIBERED SURFACE BUNDLES

Ichihara, K., Motegi, K. — 2008-08-02

Let M be a small Seifert fiber space which has also a structure of surface bundle Fx[0, 1]/{(x, 0)=(f(x), 1)} over the circle, where f: F->F is a monodromy map with non-empty fixed point set. A typical example of such a manifold appears as the result of 0-surgery on a torus knot. For each section in M, we have a ‘projection’ in F in a natural way. We give a condition assuring that the given section in M is hyperbolic in terms of the ‘projection’ in the fiber surface. By translating the result, we give a condition to obtain pseudo-Anosov automorphisms of once punctured surfaces from a periodic automorphism.

EXAMPLES OF FREE ACTIONS ON PRODUCTS OF SPHERES

Hambleton, I., Unlu, O. — 2008-08-02

We construct a non-abelian extension of S¹ by Z/ 3x Z 3x 3, and prove that acts freely and smoothly on S⁵xS⁵. This gives new actions on S⁵xS⁵ for an infinite family P of finite 3-groups. We also show that any finite odd-order subgroup of the exceptional Lie group G₂ admits a free smooth action on S¹¹xS¹¹. This gives new actions on S¹¹xS¹¹ for an infinite family E of finite groups. We explain the significance of these families P, E for the general existence problem, and correct some mistakes in the literature.

SUMS OF OPERATOR LOGARITHMS

Clark, S. — 2008-07-17

Let A and B be a pair of resolvent commuting invertible sectorial operators. We shall show that, under Kalton–Weis-type conditions, the operator log A + log B is closed and equal to log (AB).

FLOWS OF G2-STRUCTURES, I

Karigiannis, S. — 2008-07-14

This is a foundational paper on flows of G₂-structures. We use local coordinates to describe the four torsion forms of a G₂ and derive the evolution equations for a general flow of a G₂-structure on a 7-manifold M. Specifically, we compute the evolution of the metric g, the dual 4-form and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernández–Gray.

As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G₂-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G₂-geometry.

CONTACT 5-MANIFOLDS WITH SU(2)-STRUCTURE

De Andres, L. C., Fernandez, M., Fino, A., Ugarte, L. — 2008-07-14

We consider 5-manifolds with a contact form arising from a hypo structure, which we call hypo-contact. We provide existence conditions for such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU(3)-structure. For half-flat manifolds with a Killing vector field X preserving the SU(3)-structure we study the geometry of the orbits space. Moreover, we describe the solvable Lie algebras admitting a hypo-contact structure. This allows us to exhibit examples of Sasakian -Einstein manifolds, as well as to prove that such structures give rise to new metrics with holonomy SU(3) and G₂.

TWISTOR SPACES, PLURIHARMONIC MAPS AND HARMONIC MORPHISMS

Simoes, B. A., Svensson, M. — 2008-07-10

The application of twistor methods to construct harmonic morphisms has proved to be a fruitful approach in the 4-dimensional case, where a variety of examples and, in some cases, even a complete classification of harmonic morphisms have been found. In this paper, we generalize this construction to obtain higher-dimensional analogues of these maps. We also prove several results on twistor lifts of pluriharmonic and (1, 1)-geodesic maps.

SOME Z/2-GRADED REPRESENTATION THEORY

Parshall, B. J., Scott, L. L. — 2008-06-18

In representation theory, the existence of a Z⁺-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category O associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p≥n, though this grading would suffice to establish Lusztig's character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Astérique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces Z/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full Z⁺-grading, while being possibly much easier to find. We define the notion of a Z/2-based Kazhdan–Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan–Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan–Lusztig theories, Tôhoku Math. J. 45 (1993), 511–534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Extⁿ-groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

ETA-INVARIANTS FROM MOLIEN SERIES

Degeratu, A. — 2008-06-14

We look at the orbifold Cⁿ/ with a finite subgroup of U(n) from two perspectives: from a differential point of view it is a non-compact orbifold with boundary at infinity S^2n–1/, while from an algebraic point of view it is a scheme with coordinate ring the -invariant polynomials in n variables. The main result is a relation between the -invariant of the boundary (an analytical object) and the Molien series of the singularity (an algebraic object).

A QUADRIC WITH ARITHMETIC PAUCITY

Blomer, V., Brudern, J. — 2008-06-10

The quadric given by the equations x₁²+x₂²+x₃² = y₁²+y₂²+y₃², x₁+x₂+x₃ = y₁+y₂+y₃ has almost all its solutions with prime coordinates on the diagonals. This is shown in quantitative form. A similar statement holds for integral solutions whose coordinates can be written as the sum of two squares.

FREIHEITSSATZE FOR ONE-RELATOR QUOTIENTS OF SURFACE GROUPS AND OF LIMIT GROUPS

Howie, J., Saeed, M. S. — 2008-06-10

Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups. Two of the proofs are new even in that restricted context.

LARGE INDECOMPOSABLE MINIMAL GROUPS

Shelah, S., Strungmann, L. — 2008-06-10

Assuming V=L we prove that there exist indecomposable almost-free minimal groups of size for every regular cardinal below the first weakly compact cardinal. This is to say that there are indecomposable almost-free torsion-free abelian groups of cardinality which are isomorphic to all of their finite index subgroups.

THE EULER OBSTRUCTION AND BRUCE-ROBERTS' MILNOR NUMBER

De Goes Grulha, N. — 2008-06-10

In this work we determine relations between the local Euler obstruction of an analytic function f and the Milnor number of f defined by Bruce and Roberts for functions on singular spaces.

ON MORITA THEORY FOR SELF-DUAL MODULES

Willems, W., Zimmermann, A. — 2008-06-10

Let G be a finite group and k be a field of characteristic p. It is known that a kG-module V carries a non-degenerate G-invariant bilinear form b if and only if V is self-dual. We show that whenever a Morita bimodule M that induces an equivalence between two blocks such as B(kG) and B(kH) of group algebras kG and kH is self-dual, then the correspondence preserves self-duality. Even more, if the bilinear form on M is symmetric, then, for p odd, the correspondence preserves the geometric type of simple modules. In characteristic 2, this holds also true for projective modules.

THE GAP BETWEEN LOCAL MULTIPLIER ALGEBRAS OF C*-ALGEBRAS

Argerami, M., Farenick, D., Massey, P. — 2008-06-05

The local multiplier algebra M_loc(A) of a C*-algebra A has the property that M_loc (A) M_loc(M_loc(A)). In this paper we show that there is a separable liminal C*-algebra A such that the inclusion is proper.