Proceedings of the London Mathematical Society - recent issues

Artin braid groups and homotopy groups

Li, J., Wu, J. — Mon, 19 Oct 2009 08:26:28 PDT

We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as summands of the centres of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. The results give new connections between the braid groups and the general higher homotopy groups of spheres.

The curvature invariant for a class of homogeneous operators

Misra, G., Shyam Roy, S. — Mon, 19 Oct 2009 08:26:28 PDT

For an operator T in the class B_n(), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle E_T determine the unitary equivalence class of the operator T. In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator T B_n(). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in B_n(D). For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc D are determined from the simultaneous unitary equivalence class at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples. One of our main results is that the curvature along with its covariant derivative of order (0, 1) at 0 determines the equivalence class of generic homogeneous Hermitian holomorphic vector bundles over the unit disc.

Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup

Bludov, V. V., Glass, A. M. W. — Mon, 19 Oct 2009 08:26:29 PDT

We use permutation groups to give necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable. We obtain several consequences answering previously posed problems and also prove the right-orderable analogues of the Higman Embedding Theorem and the Boone–Higman Theorem.

Cohomogeneity one disk bundles with normal homogeneous collars

Schwachhofer, L. J., Tapp, K. — Mon, 19 Oct 2009 08:26:29 PDT

We consider cohomogeneity one homogeneous disk bundles and address the question when these admit a nonnegatively curved invariant metric with normal collar, that is, such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in {2, 3, 4, 6, 8}. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.

Solution of the polynomial moment problem

Pakovich, F., Muzychuk, M. — Mon, 19 Oct 2009 08:26:29 PDT

In this paper we give a solution of the following ‘polynomial moment problem’ which arose about 10 years ago in connection with Poincaré's center-focus problem: for a given polynomial P(z) to describe polynomials q(z) orthogonal to all powers of P(z) on a segment [a, b].

On the self-similarity problem for ergodic flows

Fraczek, K., Lemanczyk, M. — Mon, 19 Oct 2009 08:26:30 PDT

Given an ergodic flow (T_t)_{t R} we study the problem of its self-similarities, that is, we want to describe the set of s R for which the original flow is isomorphic to the flow (T_st)_{t R}. The problem is examined in some classes of special flows over irrational rotations and over interval exchange transformations. In particular, translation flows on translation surfaces are considered: we prove that under the weak mixing condition the set of self-similarities has Lebesgue measure zero. For von Neumann special flows over irrational rotations given by Diophantine numbers, this set is shown to be equal to {1}, while for horocycle flows a weak convergence in case of some singular (with respect to the volume measure) measures is shown to give rise to some new equidistribution result. The problem of self-similarity is also studied from the spectral point of view, especially in the class of Gaussian systems.

Geometric criteria for Landweber exactness

Hollander, S. — Mon, 19 Oct 2009 08:26:30 PDT

The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks arising from Landweber exact maps of rings. Using as extra input only Lazard's theorem and Cartier's classification of p-typical formal group laws, this result is then applied to deduce many of the main results concerning Landweber exactness in stable homotopy theory and to compute the Bousfield classes of certain BP-algebra spectra. The second criterion can be regarded as a generalization of the Landweber exact functor theorem, and we use it to give a proof of the original theorem.

Building blocks of etale endomorphisms of complex projective manifolds

Nakayama, N., Zhang, D.-Q. — Mon, 19 Oct 2009 08:26:30 PDT

Étale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly étale rational endomorphisms of weak Calabi–Yau varieties.

A solution to the Douglas-Rudin problem for matrix-valued functions

Barclay, S. — Mon, 19 Oct 2009 08:26:31 PDT

We solve the noncommutative Douglas–Rudin problem, showing that any log-integrable essentially bounded square matrix-valued function f can be written in the form h*g, where h and g lie in H . Extensions to other L ^p spaces, with norm bounds on the factors of f, are also provided.

The effect of convolving families of L-functions on the underlying group symmetries

Duenez, E., Miller, S. J. — Mon, 19 Oct 2009 08:26:31 PDT

Let {F_N} and {G_M} be families of primitive automorphic L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N, M -> , the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N and G_M agrees with that of the eigenvalues near 1 of matrices in G₁ and G₂, respectively, as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level density. (We also study convolved families of the form f x G_M for a fixed f.) Under natural assumptions on the families (which hold in many cases), we can associate to each family L of L-functions a symmetry constant cL equal to 0, 1, or–1 if the corresponding low-lying zero statistics agree with those of the unitary symplectic, or orthogonal group, respectively. Our main result is that cFxG=cF·cG: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f x G_M. We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with positive rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N, M -> , as lower-order terms).

Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials

Kozlovski, O., van Strien, S. — Wed, 12 Aug 2009 15:08:04 PDT

We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by-product we prove that the Julia set of a non-renormalizable polynomial with only hyperbolic periodic points is locally connected, and the Branner–Hubbard conjecture. The main tools are the enhanced nest construction (developed in a previous joint paper with [Rigidity for real polynomials, Ann. of Math. (2) 165 (2007) 749–841.]) and a lemma of Kahn and Lyubich (for which we give an elementary proof in the real case).

Hodge theory for G2-manifolds: intermediate Jacobians and Abel-Jacobi maps

Karigiannis, S., Leung, N. C. — Wed, 12 Aug 2009 15:08:04 PDT

We study the moduli space M of torsion-free G₂-structures on a fixed compact manifold M⁷, and define its associated universal intermediate Jacobian J. We define the Yukawa coupling and relate it to a natural pseudo-Kähler structure on J.

We consider natural Chern-Simons-type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang-Mills connections), and also deformed Donaldson-Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G₂-analogues of Abel-Jacobi maps.

Improving L2 estimates to Harnack inequalities

Filippas, S., Moschini, L., Tertikas, A. — Wed, 12 Aug 2009 15:08:04 PDT

We consider operators of the form L = – L – V, where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain with Dirichlet boundary conditions. We allow the boundary of to be made of various pieces of different codimension. We assume that L has a generalized first eigenfunction of which we know two-sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator L, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates.

Fundamental groups of symmetric sextics II

Degtyarev, A. — Wed, 12 Aug 2009 15:08:04 PDT

We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with the set of inner singularities 2A₈ or A₁₇. We also compute the fundamental groups of a number of other sextics, both of and not of torus type. The groups found are simplest possible, that is, Z₂*Z₃ and Z₆, respectively.

Deformation theory of asymptotically conical coassociative 4-folds

Lotay, J. D. — Wed, 12 Aug 2009 15:08:04 PDT

Suppose that a coassociative 4-fold N in R⁷ is asymptotically conical (AC) to a cone C with rate < 1. If [–2, 1) is generic, then we show that the moduli space of coassociative deformations of N that are also AC to C with rate is a smooth manifold, and we calculate its dimension. If < – 2 and generic, then we show that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds, and we give a lower bound for its expected dimension. We also derive a test for when N will be planar if < – 2 and we discuss examples of AC coassociative 4-folds.

Geometry of nilmanifolds with left-invariant complex structure and deformations in the large

Rollenske, S. — Wed, 12 Aug 2009 15:08:04 PDT

The relation between nilmanifolds with left-invariant complex structure and iterated principal holomorphic torus bundles is clarified and we give criteria under which deformations in the large are again of such type. As an application we obtain a fairly complete picture in complex dimension 3.

Inversion formulas for elliptic functions

Cooper, S. — Wed, 12 Aug 2009 15:08:04 PDT

The aim of this work is to give a unified treatment of the fundamental formulas in Ramanujan's theories of elliptic functions to alternative bases. Our approach relies on well-known results from the theory of theta functions, such as the sum of four squares and sum of eight squares theorems, and their cubic analogues. We prove four inversion theorems, one being classical and the other three belonging to Ramanujan's theories to alternative bases. The connections with iterative means and the corresponding transformation formulas for hypergeometric functions are also established.

An orthogonal test of the L-functions Ratios conjecture

Miller, S. J. — Wed, 12 Aug 2009 15:08:04 PDT

We test the predictions of (a weakened version of) the L-functions Ratios conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N -> through the primes or N = 1 and k -> . We study the main and lower-order terms in the 1-level density. We provide evidence for the Ratios conjecture by computing and confirming its predictions up to a power savings in the family's cardinality, at least for test functions whose Fourier transforms are supported in (–2, 2). We do this both for the weighted and unweighted 1-level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1-level densities differ by a term of size 1/log (k² N). Finally, we show that there is another way of extending the sums arising in the Ratios conjecture, leading to a different answer (although the answer is such a lower-order term that it is hopeless to observe which is correct).

Principal non-commutative torus bundles

Echterhoff, S., Nest, R., Oyono-Oyono, H. — Thu, 11 Jun 2009 09:10:12 PDT

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally RKK-trivial. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to Tⁿ-equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal Tⁿ-bundles with H-flux, as studied by Mathai and Rosenberg which possess ‘classical’ T-duals.

On the role of convexity in functional and isoperimetric inequalities

Milman, E. — Thu, 11 Jun 2009 09:10:12 PDT

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz’ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, q-log-Sobolev inequalities (q [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ) curvature–dimension condition of Bakry–Émery.

Transversality in families of mappings

Wall, C. T. C. — Thu, 11 Jun 2009 09:10:12 PDT

If N and P are smooth manifolds, and Q is a smooth submanifold of P, the set of maps of N to P transverse to Q is residual. However given a family F of smooth mappings, we cannot expect to be able to deform the family to make each member of the family transverse to Q. The development of generic transversality conditions runs in parallel with that of stability theory. We develop a convenient notation, and establish equivalence of different transversality conditions. The first main result expresses equivalence between F inducing a versal unfolding of any multi-germ, multi-transversality of F to contact orbits relative to projection on the parameter space U, and local stability of the projection to U of the pre-image of Q. For global stability we need Q to be closed, the deformation F to be proper, and Q or N compact.

Corresponding results also hold for topological stability in the Thom–Mather sense, and an appropriate notion of topological versality.

To obtain results with Q a stratified subset of P requires an extensive study of theories of contact equivalence relative to a subset of the target. Here we need an analyticity hypothesis on Q; for deeper study of finite determinacy, we need a holonomic condition; even so, the theory is much more complicated. In general, there are properties corresponding to the former ones, but many are weaker. Our second main result, a direct analogue of the first, is only obtained under a hypotheses that F has property (G) at all points and is transverse to Q.

In the topological case, while some results can be obtained, the failure of direct analogues to some basic tools is a bar to further progress.

A class of noncommutative projective surfaces

Rogalski, D., Stafford, J. T. — Thu, 11 Jun 2009 09:10:12 PDT

Let A = _i≥0A_i be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A) = k(X)[t, t^–1; ], where is an automorphism of the integral projective surface X. Then we prove that A can be written as a naïve blowup algebra of a projective surface X birational to X. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in qgr-A will always be in (1-1) correspondence with the closed points of the scheme X.

Representations of Lie superalgebras in prime characteristic I

Wang, W., Zhao, L. — Thu, 11 Jun 2009 09:10:12 PDT

We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p > 2. A superalgebra generalization of the celebrated Kac–Weisfeiler conjecture is formulated, which exhibits a mixture of p-power and 2-power divisibilities of dimensions of modules. We establish the conjecture for basic classical Lie superalgebras.

On pseudo-harmonic maps in conformal geometry

Kokarev, G. — Thu, 11 Jun 2009 09:10:12 PDT

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We discuss existence theorems and obtain an extension of Siu's strong rigidity to Kähler–Weyl geometry. Other applications include topological obstructions to the existence of Kähler–Weyl structures. For example, we show that no co-compact lattice in SO(1, n), n > 2, can be the fundamental group of a compact Kähler–Weyl manifold of certain type.

Constructing smooth manifolds of loop spaces

Stacey, A. — Thu, 11 Jun 2009 09:10:12 PDT

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite-dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists. We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations.

Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

Sela, Z. — Thu, 11 Jun 2009 09:10:12 PDT

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion-free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first-order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion-free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent.