Proceedings of the London Mathematical Society - Advance Access

Zeros of p-adic forms

Heath-Brown, D. R. — Thu, 12 Nov 2009 05:13:30 PST

A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.

Manifolds of semi-negative curvature

Conde, C., Larotonda, G. — Tue, 10 Nov 2009 04:53:12 PST

This paper studies the metric structure of manifolds of semi-negative curvature. Explicit estimates on the geodesic distance and sectional curvature are obtained in the setting of homogeneous spaces G/K of Banach–Lie groups, and a characterization of convex homogeneous submanifolds is given in terms of the Banach–Lie algebras. A splitting theorem via convex expansive submanifolds is proved, inducing the corresponding splitting of the Banach–Lie group G. The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces, and manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class of nonpositively curved spaces. Several well-known results, such as the existence and uniqueness of best approximations from convex closed sets, or the Bruhat–Tits fixed-point theorem, are shown to hold in this setting, without dimension restrictions. Finally, these notions are used to study the structure of the classical Banach–Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such a setting.

Homotopy, homology, and GL2

Miemietz, V., Turner, W. — Tue, 27 Oct 2009 06:29:52 PDT

We define weak 2-categories of finite-dimensional algebras with bimodules, along with collections of operators O_{(c, x)} on these 2-categories. We prove that special examples O_p of these operators control all homological aspects of the rational representation theory of the algebraic group GL₂, over a field of positive characteristic. We prove that when x is a Rickard tilting complex, the operators O_{(c, x)} honour derived equivalences in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight Z₊-gradings on Schur algebras S(2, r), and the existence of braid group actions on the derived categories of blocks of these Schur algebras.

Formal completions of Neron models for algebraic tori

Demchenko, O., Gurevich, A., Xarles, X. — Tue, 27 Oct 2009 06:29:49 PDT

We calculate the formal group law that represents the completion of the Néron model for an algebraic torus over Q split in a tamely ramified abelian extension. To that end, we introduce an analogue of the fixed part of a formal group law with respect to a group action and give a method to compute its Honda p-types.

Affine interval exchange maps with a wandering interval

Marmi, S., Moussa, P., Yoccoz, J.-C. — Thu, 08 Oct 2009 08:14:31 PDT

For almost all interval exchange maps (i.e.m.) T₀, with combinatorics of genus g ≥ 2, we construct affine i.e.m. T that are semi-conjugate to T₀ and have a wandering interval.

Closed ideal structure and cohomological properties of certain radical Banach algebras

Ghahramani, F., Read, C. J., Willis, G. A. — Sat, 03 Oct 2009 03:13:07 PDT

We further the study of a class of singly generated radical Banach algebras (sometimes called LRRW (Loy, Read, Runde and Willis) algebras after the four authors involved in the original paper) that have compact multiplication and are weakly amenable. First, we characterize the closed ideal structure of these algebras. The closed ideals of an LRRW algebra are identified, and the lattice of closed ideals is shown to be isomorphic to the unit interval. Then we show that LRRW algebras are not approximately amenable and have global homological dimension greater than 1. Furthermore, epimorphisms onto these algebras and derivations from them are continuous.

Wilson's map operations on regular dessins and cyclotomic fields of definition

Jones, G. A., Streit, M., Wolfart, J. — Tue, 25 Aug 2009 07:21:40 PDT

Dessins d’enfants can be seen as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, even an algebraic structure as a projective algebraic curve defined over a number field. Combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve, such as the field of moduli or the field of definition. However, apart from a few series of examples, very few general results concerning such correspondences are known. As a step in this direction, we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields, based on Wilson's operations on maps: these leave invariant the graph but change the cyclic ordering of edges around the vertices, therefore they change the embeddings, and hence the dessins, and hence the conformal and algebraic structure of the underlying curves. Under suitable assumptions, satisfied by many series of regular dessins, these changes coincide with the effect of Galois conjugation. This coincidence allows one to draw conclusions about Galois orbits and fields of definition of dessins. The possibilities afforded by these techniques, and their limitations, are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs.

Manifolds of Hilbert space projections

Levene, R. H., Power, S. C. — Mon, 24 Aug 2009 03:37:32 PDT

The Hardy space H² (R) for the upper half-plane together with a multiplicative group of unimodular functions u() = exp(i(₁₁ + ... +_n_n)), with Rⁿ, gives rise to a manifold M of orthogonal projections for the subspaces u() H² (R) of L² (R). For classes of admissible functions _i the strong operator topology closures of M and M M are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals (E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43 (Princeton University Press, Princeton, NJ, 1993)) and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary equivalence. In particular, the Fourier–Plancherel 2-sphere and the hyperbolic 3-sphere of Katavolos and Power (A. Katavolos and S. C. Power, Translation and dilation invariant subspaces of L²(R), J. reine angew. Math. 552 (2002) 101–129) appear as distinguished special cases admitting non-trivial unitary automorphism groups, which are explicitly described.

Turbulence, representations, and trace-preserving actions

Kerr, D., Li, H., Pichot, M. — Thu, 20 Aug 2009 01:44:32 PDT

We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X, µ) and on the hyperfinite II₁ factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X, µ) is generically turbulent.

Harmonic analysis on a finite homogeneous space

Scarabotti, F., Tolli, F. — Mon, 17 Aug 2009 05:57:42 PDT

In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, we introduce three types of spherical functions. Then we consider the composition of two permutation representations, giving a noncommutative generalization of the Gelfand pair associated to the ultrametric space; actually, we study the more general notion of crested product. Finally, we consider the exponentiation action, generalizing the decomposition of the Gelfand pair of the Hamming scheme; actually, we study a more general construction that we call wreath product of permutation representations, suggested by the study of finite lamplighter random walks. We give several examples of concrete decompositions of permutation representations and several explicit ‘rules’ of decomposition.

Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres

Hu, Z., Neufang, M., Ruan, Z.-J. — Mon, 17 Aug 2009 05:57:42 PDT

We study multiplier algebras for a large class of Banach algebras which contains the group algebra L₁(G), the Beurling algebras L₁(G, ), and the Fourier algebra A(G) of a locally compact group G. This study yields numerous new results and unifies some existing theorems on L₁(G) and A(G) through an abstract Banach algebraic approach. Applications are obtained on representations of multipliers over locally compact quantum groups and on topological centre problems. In particular, five open problems in abstract harmonic analysis are solved.

Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

Hino, M. — Fri, 14 Aug 2009 00:56:39 PDT

We introduce the concept of index for regular Dirichlet forms by means of energy measures, and discuss its properties. In particular, it is proved that the index of strong local regular Dirichlet forms is identical with the martingale dimension of the associated diffusion processes. As an application, a class of self-similar fractals is taken up as an underlying space. We prove that first-order derivatives can be defined for functions in the domain of the Dirichlet forms and their total energies are represented as the square integrals of the derivatives.

Combinatorial complexity in o-minimal geometry

Basu, S. — Sun, 09 Aug 2009 21:58:07 PDT

In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of n definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result vastly increases the applicability of results on combinatorial and topological complexity of arrangements studied in discrete and computational geometry. As a sample application, we extend a Ramsey-type theorem due to Alon et al. [Crossing patterns of semi-algebraic sets, J. Combin. Theory Ser. A 111 (2005), 310–326. MR 2156215 (2006k:14108)], originally proved for semi-algebraic sets of fixed description complexity to this more general setting.

Cohomology of finite-dimensional pointed Hopf algebras

Mastnak, M., Pevtsova, J., Schauenburg, P., Witherspoon, S. — Sun, 09 Aug 2009 21:58:07 PDT

We prove finite generation of the cohomology ring of any finite-dimensional pointed Hopf algebra, having abelian group of group-like elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig's small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, as well as many new pointed Hopf algebras. We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. As a consequence we obtain some further information about the structure of the cohomology ring of a finite-dimensional pointed Hopf algebra and its related Nichols algebra.

Primitive permutation groups of bounded orbital diameter

Liebeck, M. W., Macpherson, D., Tent, K. — Wed, 05 Aug 2009 23:12:45 PDT

We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specific cases, there is such a diameter bound. This is proved using recent results on the model theory of pseudofinite fields and difference fields.

Rigidity of measures invariant under semisimple groups in positive characteristic

Einsiedler, M., Ghosh, A. — Wed, 29 Jul 2009 07:10:19 PDT

M. Ratner has conjectured a positive characteristic version of her seminal results classifying orbit closures and invariant measures of unipotent flows on homogeneous spaces. In this paper, we provide a partial answer by establishing a positive characteristic version of her classification result for measures invariant under semisimple groups.

On multiple Bernoulli polynomials and multiple L-functions of root systems

Komori, Y., Matsumoto, K., Tsumura, H. — Mon, 20 Jul 2009 23:50:52 PDT

We define generalized Bernoulli numbers, Bernoulli polynomials and multi-variable L-functions associated with root systems. We prove that the values of those L-functions at positive integers can be expressed in terms of those Bernoulli polynomials, and give an explicit formula for the latter. This result is a character analogue of Witten's volume formula for Witten's zeta-functions of semisimple Lie algebras. Furthermore, we show that the L-functions can be continued meromorphically to the whole space, and satisfy certain functional relations.

Peano's theorem for rough differential equations in infinite-dimensional Banach spaces

Caruana, M. — Thu, 16 Jul 2009 01:55:11 PDT

We present a proof for Peano's theorem for rough differential equations, which is valid in infinite dimensions under an appropriate compactness assumption on the vector fields. Our approach makes full use of Lyons’ Universal Limit Theorem and is based on the construction of a family of rough polynomial approximations, each of which is a concatenation of rough path solutions of different equations.

Theorie ergodique des fractions rationnelles sur un corps ultrametrique

Favre, C., Rivera-Letelier, J. — Sat, 20 Jun 2009 01:13:27 PDT

Résumé

On donne les premiers éléments pour l’étude des propriétés ergodi-ques d’une fraction rationnelle à coefficients dans un corps algébriquement clos et complet pour une norme non archimé-dienne. En particulier, pour une telle fraction rationnelle R on montre l’existence d’une mesure naturelle _R représentant la distribution asymptotique des préimages itérées de chaque point non exceptionnel de R. On montre que cette mesure est (exponentiellement) mélangeante, et qu’elle satisfait au théorème limite central. De plus, on donne une estimation de l’entropie métrique de cette mesure, et de l’entropie topologique de R, qui permettent de caractériser les fractions rationnelles d’entropie topologique nulle.

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure _R which represents the asymptotic distribution of preimages of non-exceptional points. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of _R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.

The true complexity of a system of linear equations

Gowers, W. T., Wolf, J. — Sat, 20 Jun 2009 01:13:27 PDT

In this paper we look for conditions that are sufficient to guarantee that a subset A of a finite Abelian group G contains the ‘expected’ number of linear configurations of a given type. The simplest non-trivial result of this kind is the well-known fact that if G has odd order, A has density and all Fourier coefficients of the characteristic function of A are significantly smaller than (except the one at zero, which equals ), then A contains approximately ³|G|² triples of the form (a, a+d, a+2d). This is ‘expected’ in the sense that a random set A of density has approximately ³|G|² such triples with very high probability. More generally, it was shown by the first author (in the case G = Z_N for N prime, but the proof generalizes) that a set A of density has about ^k|G|² arithmetic progressions of length k if the characteristic function of A is almost as small as it can be, given its density, in a norm that is now called the U^k–1-norm. When investigating linear equations in the primes, Green and Tao found the most general statement that follows from the technique used to prove this result, introducing a notion that they call the complexity of a system of linear forms. They prove that if A has almost minimal U^k+1-norm, then it has the expected number of linear configurations of a given type, provided that the associated complexity is at most k. The main result of this paper is that the converse is not true: in particular there are certain systems of complexity 2 that are controlled by the U²-norm, whereas the result of Green and Tao requires the stronger hypothesis of U³-control. We say that a system of m linear forms L₁, ..., L_m in d variables with integer coeffcients has true complexity k if k is the smallest positive integer such that, for any set A of density and almost minimal U^k+1-norm, the number of d-tuples (x₁, ..., x_d) such that L_i(x₁, ..., x_d) A for every i is approximately ^m|G|^d. We conjecture that the true complexity k is the smallest positive integer s for which the functions L₁^s+1, ... ,L_m^s+1 are linearly independent. Using the ‘quadratic Fourier analysis’ of Green and Tao we prove this conjecture in the case where the complexity of the system (in Green and Tao's sense) is 2, s=1 and G is the group F_pⁿ for some fixed odd prime p. A closely related result in ergodic theory was recently proved independently by Leibman. We end the paper by discussing the connections between his result and ours.

A Maslov cocycle for unitary groups

Kramer, L., Tent, K. — Tue, 16 Jun 2009 07:42:00 PDT

We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew-fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, and so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Chern class.

Spectral properties of matrices associated with some directed graphs

Davies, E. B., Incani, P. A. — Tue, 09 Jun 2009 04:33:40 PDT

We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.

Families of absolutely simple hyperelliptic jacobians

Zarhin, Y. G. — Tue, 09 Jun 2009 04:33:40 PDT

We prove that the jacobian of a hyperelliptic curve y² = (x – t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)

Deformations of finite conformal energy: existence and removability of singularities

Iwaniec, T., Onninen, J. — Thu, 14 May 2009 10:58:54 PDT

This paper features a class of mappings between bounded domains X, Y Rⁿ, having finite n-harmonic energy, such that we have The fundamental question is whether or not the domains X, Y Rⁿ of the same topological type admit a homeomorphism in a given homotopy class having finite energy. The examples of non-existence, somewhat testing our theory, arise when we remove from bounded smooth domains X and Y thin subsets and , referred to as cracks or fractures. We are looking for homeomorphisms of finite energy for which is the cluster set of h over . In general, infinite energy is required in order to increase the dimension of a crack that is, when . Suppose now that a bounded deformation of finite energy is given. Does h extend continuously to X and, if so, is the extension injective on X? We give affirmative answers to these questions.