Bulletin of the London Mathematical Society - recent issues

The fundamental group of a p-compact group

Dwyer, W. G., Wilkerson, C. W. — 2009-05-21

We compute the fundamental group of a connected p-compact group in terms of the map from the homology of the classifying space of a maximal torus to the homology of the classifying space of its normalizer.

Intermediate convergents and a metric theorem of Khinchin

Haynes, A. K. — 2009-05-21

A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function f defined on the positive integers and a real number x, and form the partial sums s_n of f evaluated at the partial quotients a₁, ..., a_n in the continued fraction expansion for x. Does the sequence {s_n/n} have a limit as n -> ? In 1935 Khinchin proved that the answer is yes for almost every x, provided that the function f does not grow too quickly. In this article we are going to explore a natural reformulation of this problem in which the function f is defined on the rationals and the partial sums in question are over the intermediate convergents to x with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above.

Integral means and boundary limits of Dirichlet series

Saksman, E., Seip, K. — 2009-05-21

This paper deals with the boundary behaviour of functions in the Hardy spaces ^p for ordinary Dirichlet series. The main result, answering a question of Hedenmalm, shows that the classical Carlson theorem on integral means does not extend to the imaginary axis for functions in , that is, for the ordinary Dirichlet series in H of the right half-plane. We discuss an important embedding problem for ^p, the solution of which is only known when p is an even integer. Viewing ^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.

On theta functions of order 4

Kopeliovich, Y., Pauly, C., Serman, O. — 2009-05-21

We prove that the fourth powers of theta functions with even characteristics form a basis of the space H⁰(A, O_A(4))₊ of even theta functions of order 4 on a principally polarized Abelian variety (A, ) without a vanishing theta-null.

Fields with measure and automorphism

Tomasic, I. — 2009-05-21

We consider a difference field (K, ) such that finite-dimensional definable sets over K can be compared in size, or measured. Let k be the fixed field of the automorphism . We show that curves of genus 1 defined over k are approximately the size of the affine line over k, an ‘approximative version’ of the Riemann hypothesis for curves of genus 1.

The regular part of symmetric forms associated with second-order elliptic differential expressions

Vogt, H. — 2009-05-21

Let be a (not necessarily closable) positive symmetric form associated with a second-order elliptic differential expression. We show that the regular part of (in the sense of B. Simon) can be obtained by modifying the coefficients of suitably; in particular, the regular part is again associated with a second-order elliptic differential expression.

Sufficiency of jets with line singularities

Brodersen, H. — 2009-05-21

Let z: (R ^{n + 1}, 0) -> (R, 0) be an r-jet with a singular set containing a 1-dimensional manifold L. Let be the set of homeomorphism germs h : (R ^{n + 1}, 0) -> (R ^{n + 1}, 0) leaving L invariant. Let be the set of C^r germs, f : (R ^{n + 1}, 0) -> (R , 0), with singular sets containing L. We say that z is sufficient in if any two f and g in with are -equivalent. In this paper we give necessary and sufficient conditions in terms of Lojasiewicz inequalities for such a jet z to be sufficient in .

Stability of projective Poincare and Picard bundles

Biswas, I., Brambila-Paz, L., Newstead, P. E. — 2009-05-21

Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers, and let M denote the moduli space of stable vector bundles on X of rank n and determinant , where is a fixed line bundle of degree d. If n and d have a common divisor, then there is no universal vector bundle on X x M. We prove that there is a projective bundle on X x M with the property that its restriction to X x {E} is isomorphic to P(E) for all E M and that this bundle (called the projective Poincaré bundle) is stable with respect to any polarization; moreover its restriction to {x} x M is also stable for any x X. We also prove stability results for bundles induced from the projective Poincaré bundle by homomorphisms PGL(n) -> H for any reductive H. We further show that there is a projective Picard bundle on a certain open subset M' of M for any d > n(g–1) and that this bundle is also stable. Also, we obtain new results on the stability of the Picard bundle even when n and d are coprime.

Hochschild homology and global dimension

Bergh, P. A., Madsen, D. — 2009-05-21

We prove that, for certain classes of graded algebras (Koszul, local and cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided that the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.

The generating condition for coalgebras

Iovanov, M. C. — 2009-05-21

For a ring R, the properties of being (left) self-injective or being a cogenerator for the left R-modules do not imply one another, and the two combined give rise to the important notion of pseudo-Frobenius-rings. For a coalgebra C, (left) self-projectivity implies that C is a generator for right comodules and the coalgebras with this property are called right quasi-co-Frobenius; however, whether the converse implication is true is an open question. We provide an extensive study of this problem. We show that this implication does not hold, by giving a large class of examples of coalgebras having the ‘generating property’. In fact, we show that any coalgebra C can be embedded in a coalgebra C that generates its right comodules, and, if C is local over an algebraically closed field, then C can be chosen local as well. We also give some general conditions under which the implication ‘C-projective (left) C generator for right comodules’ does work, and such conditions are when C is right semiperfect or when C has finite coradical filtration.

'High spots' theorems for sloshing problems

Kulczycki, T., Kuznetsov, N. — 2009-05-21

We investigate several 2D and 3D cases of the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. In particular, for a domain W R² (canal's cross-section), where W = F B and F (cross-section of the free surface of fluid) is an interval of the x-axis, whereas B (bottom's cross-section) is the graph of a negative function, the following result is proved. The fundamental eigenfunction u₁ of the sloshing problem (the corresponding eigenvalue is simple) has monotonic traces on F and B; moreover, u₁ attains its maximum and minimum values at the end-points of F. It is established that for the 2D (3D) ice-fishing problem with a single (circular) hole, the function u₁ (both fundamental eigenfunctions) attains its maximum value at an inner point of F. A relationship between the high spots and hot spots theorems is considered.

Schwarz lemma for the tetrablock

Edigarian, A., Zwonek, W. — 2009-05-21

We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results from Abouhajar et al. [‘A Schwarz lemma for a domain related to mu-synthesis’, J. Geom. Anal. 17 (2007) 717–750]. The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper.

The equivariant Euler characteristic of real Coxeter toric varieties

Henderson, A., Lehrer, G. — 2009-05-21

Let W be a Weyl group, and let T_W be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of W, and its weight lattice. The real locus T_W(R) is a smooth, connected, compact manifold with a W-action. We give a formula for the equivariant Euler characteristic of T_W(R) as a generalised character of W. In type A_n–1 for n odd, one obtains a generalised character of Sym_n whose degree is (up to sign) the nth Euler number.

On the definition of pseudospectra

Shargorodsky, E. — 2009-05-21

It is well known that the -pseudospectrum of a bounded linear operator defined with the help of a strict inequality is equal to the union of the spectra of all perturbed operators with perturbations that have norms strictly less than . The aim of the paper is to show that the same is not true in the case of non-strict inequalities.

Twisted Alexander polynomials and representation shifts

Silver, D. S., Williams, S. G. — 2009-05-21

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.

Capitulation for locally free class groups of orders of group algebras over number fields

Greither, C., Johnston, H. — 2009-05-21

We prove a capitulation result for locally free class groups of orders of group algebras over number fields. This result allows some control over ramification, and so, as a corollary, we obtain an ‘arithmetically disjoint capitulation result’ for the Galois module structure of rings of integers.

Local well-posedness of nonlinear dispersive equations on modulation spaces

Benyi, A., Okoudjou, K. A. — 2009-05-21

By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the nonlinear Schrödinger, nonlinear wave and nonlinear Klein–Gordon equations with Cauchy data in modulation spaces M_0,s^p,1.

Centralizers in domains of finite Gelfand-Kirillov dimension

Bell, J. P. — 2009-05-21

We study centralizers of elements in domains. We extend a result of the author and Small ‘Centralizers in domains of Gelfand–Kirillov dimension 2’, Bull. London Math. Soc. 36 (2004) 779–785, showing that if A is a finitely generated domain of finite Gelfand–Kirillov (GK) dimension and a A is not algebraic over the extended center of A, then the centralizer of a has GK dimension at most one less than the GK dimension of A. In the case that A is a finitely generated noetherian domain of GK dimension 3 over the complex numbers, we show that the centralizer of an element a A that is not algebraic over the extended center of A satisfies a polynomial identity.

The group of automorphisms of a real rational surface is n-transitive

Huisman, J., Mangolte, F. — 2009-05-21

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces.

Classifying spaces of sporadic groups * (Mathematical Surveys and Monographs 147) * By David J. Benson and Stephen D. Smith: 289 pp., US$85.00, * ISBN 978-0821-84474-8 * (American Mathematical Society, Providence, RI, 2008)

Lazarev, A. — 2009-05-21

Quantum groups: A path to current algebra * (Australian Mathematical Society Lecture Series 19) * By Ross Street: 141 pp., {pound}28.99 (US$57.00), ISBN 0-5216-9524-4 * (Cambridge University Press, Cambridge, 2007)

Launois, S. — 2009-05-21

Honorary Members 2008

2009-05-21

A general formula for the algebraic degree in semidefinite programming

Graf von Bothmer, H.-C., Ranestad, K. — 2009-03-27

In this article, we use a natural desingularization of the conormal variety of (n x n)-symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming.

Harmonic morphisms on heaven spaces

Baird, P., Pantilie, R. — 2009-03-27

We prove that any (real or complex) analytic horizontally conformal submersion from a three-dimensional conformal manifold (M³, c_M) to a two-dimensional conformal manifold (N², c_N) can be, locally, ‘extended’ to a unique harmonic morphism from the H(eaven)-space (H⁴, g) of (M³, c_N) to (N², c_N). Moreover, any positive harmonic morphism with two-dimensional fibres from (H⁴, g) is obtained in this way.

Whitehead moves for G-trees

Clay, M., Forester, M. — 2009-03-27

We generalize the familiar notion of a Whitehead move from Culler and Vogtmann's Outer space to the setting of deformation spaces of G-trees. Specifically, we show that there are two moves, each of which transforms a reduced G-tree into another reduced G-tree, that suffice to relate any two reduced trees in the same deformation space. These two moves further factor into three moves between reduced trees that have simple descriptions in terms of graphs of groups. This result has several applications.

The sum of the Mobius function

Maier, H., Montgomery, H. L. — 2009-03-27

We derive from the Riemann Hypothesis an estimate for M(x) = _n≤xµ(n). This is the first improvement of the bound that Titchmarsh established in 1927.

Regular lattice polytopes and root systems

Montagard, P.-L., Ressayre, N. — 2009-03-27

Consider , a lattice in a real finite-dimensional vector space. Here we are interested in lattice polytopes, that is, convex polytopes with vertices in . Consider the group G of the affine real transformations that map the lattice onto itself. Replacing the group of Euclidean motions by the group G one can define the notion of regular lattice polytopes. More precisely, a lattice polytope is said to be regular if the subgroup of G which preserves the polytope acts transitively on the set of its complete flags. In this paper, we associate to each regular lattice polytope a root system. This association allows us to give a new proof of the classification of regular lattice polytopes recently obtained by Karpenkov.

Mean time exit and isoperimetric inequalities for minimal submanifolds of NxR

Bessa, G. P., Montenegro, J. F. — 2009-03-27

Based on Markvorsen and Palmer's work on mean time exit and isoperimetric inequalities, we establish a slightly better isoperimetric inequality and mean time exit estimates for minimal submanifolds of N x R when N is non-positively curved. We prove isoperimetric inequalities for submanifolds with tamed second fundamental form in Hadamard spaces with bounded sectional curvature. We use mean time exit functions to show that spherically symmetric manifolds with geodesic spheres with exponential volume growth have a positive first eigenvalue.

Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields

Bierme, H., Lacaux, C., Xiao, Y. — 2009-03-27

Let X = {X(t), t R^N} be a Gaussian random field with values in R^d defined by X(t) = (X₁(t), ..., X_d(t)), where X₁, ..., X_d are independent copies of a centered Gaussian random field X₀. Under certain general conditions on X₀, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X^–1(F), where F R^d is a non-random Borel set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and the operator-scaling Gaussian random fields with stationary increments constructed in [H. Biermé, M. M. Meerschaert and H.-P. Scheffler, ‘Operator scaling stable random fields’, Stochastic Process. Appl. 117 (2007) 312–332.].

On a question of Sarkozy and Sos for bilinear forms

Cilleruelo, J., Rue, J. — 2009-03-27

We prove that, if 2 ≤ k₁ ≤ k₂, then there is no infinite sequence A of positive integers such that the representation function r(n) = #{(a, a'): n = k₁a + k₂a', a, a' A} is constant for n large enough. This result completes the previous work of Dirac and Moser for the special case k₁ = 1 and answers a question posed by Sárkozy and Sós.

The Bott inverted infinite projective space is homotopy algebraic K-theory

Spitzweck, M., Ostvaer, P. A. — 2009-03-27

We show a motivic stable weak equivalence between the Bott inverted infinite projective space and homotopy algebraic K-theory.

Higher-order Kato class potentials for Schrodinger operators

Zheng, Q., Yao, X. — 2009-03-27

This paper is concerned with characterizations and approximation properties of higher-order Kato class K(Rⁿ) introduced by Davies and Hinz, as well as the applications to higher-order Schrödinger operators with such potentials.

Minimal polynomial dynamics on the set of 3-adic integers

Durand, F., Paccaut, F. — 2009-03-27

In this paper, the polynomials that have all of their orbits dense in the set of 3-adic integers Z₃ are characterized in terms of their coefficients.

On singular Calogero-Moser spaces

Bellamy, G. — 2009-03-27

Using combinatorial properties of complex reflection groups we show that, if the group W is different from the wreath product and the binary tetrahedral group (labelled G(m, 1, n) and G₄, respectively, in the Shephard–Todd classification), then the generalised Calogero–Moser space X_c associated to the centre of the rational Cherednik algebra H_{0, c}(W) is singular for all values of the parameter c. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety x */W when W is a complex reflection group different from and the binary tetrahedral group (where is the reflection representation associated to W). Conversely, it has been shown by Etingof and Ginzburg that X_c is smooth for generic values of c when W . We show that this is also the case when W is the binary tetrahedral group. A theorem of Namikawa then implies the existence of a symplectic resolution in this case, completing the classification. Finally, we note that the above results, together with the work of Chlouveraki, are consistent with a conjecture of Gordon and Martino on block partitions in the restricted rational Cherednik algebra.

Super-conformal surfaces associated with null complex holomorphic curves

Moriya, K. — 2009-03-27

A correspondence from a null complex holomorphic curve in four-dimensional complex Euclidean space to a super-conformal surface in four-dimensional Euclidean space is defined by the quaternionic theory of surfaces. As an application, a transformation of super-conformal surfaces is defined.

On s-numbers and Weyl inequalities of operators in Banach spaces

Carl, B., Hinrichs, A. — 2009-03-27

Let s = (s_n) be an injective s-number sequence in the sense of Pietsch. We show the following Weyl inequality between geometric means of eigenvalues and s-numbers for a Riesz-operator T: X -> X acting on a (complex) Banach space of weak type 2: for any 0 < ≤ 1 and all n N, we have , where wC₂(X) is the weak cotype 2 constant of X, n colone [n/(1+)] and c() ≤ c₀ (1+1/ ln (1/)) with an absolute constant c₀ ≥ 1.

Diophantine inequalities in function fields

Spencer, C. V. — 2009-03-27

This paper develops the Bentkus–Götze–Freeman variant of the Davenport–Heilbronn method for function fields in order to count F_q[t]-solutions to diagonal Diophantine inequalities in F_q ((1/t)).

Uncountable families of prime z-ideals in C0(R)

Le Pham, H. — 2009-03-27

Denote by c = 2^N₀ the cardinal of continuum. We construct an intriguing family (P: c) of prime z-ideals in C₀ (R) with the following properties:

if f P_i₀ for some i₀ c, then F P_i for all but finitely many i c;

_i!=i₀ P_i P_i₀ for each i₀ c.

As an application, assuming the continuum hypothesis, there exists a homomorphism from C₀ (R) into a Banach algebra whose continuity ideal is not the intersection of any countable family of prime ideals in C₀ (R).

We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type of prime z-ideals in C₀ (R) for any ordinal of cardinality c.

Finitistic dimension through infinite projective dimension

Huard, F., Lanzilotta, M., Mendoza, O. — 2009-03-27

We show that an Artin algebra having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with a vanishing radical cube.

The Steinhaus property and Haar-null sets

Dodos, P. — 2009-03-27

It is shown that, if G is an uncountable Polish group and A G is a universally measurable set such that A^–1A is meager, then the set T_l(A) = {µ P(G): µ(gA) = 0 for all g G} is co-meager. In particular, if A is analytic and not left Haar-null, then 1 Int (A^–1AA^–1A).

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

Jensen, B. T., Su, X., Yu, R. W. T. — 2009-02-20

In this paper, we show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in gl_n(k), thus answering positively in this gl_n(k) case to a question raised independently by Michel Duflo and Dmitri Panyushev. The proof gives an explicit construction, using -filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type A. An example of a seaweed Lie algebra in a simple Lie algebra of type E₈ not admitting an open orbit in its nilpotent radical is given.

The Gabriel-Roiter measure for radical-square zero algebras

Chen, B. — 2009-02-20

Let be a radical-square zero algebra over an algebraically closed field k with radical r, and let be the associated hereditary algebra. There is an explicit functor F: mod -> mod , which induces a stable equivalence. In this paper, it will be proved that the functor F preserves the Gabriel–Roiter (GR) measures and the GR factors. Thus the GR measure for can be studied by the use of F and known facts for hereditary algebras. In particular, the middle terms of the Auslander–Reiten sequences ending at the GR factors and the relationship between the preprojective partition for and the take-off -modules will be investigated.

A reverse Denjoy theorem

Fenton, P. C., Rossi, J. — 2009-02-20

Suppose that C₁ and C₂ are two simple curves joining 0 to , non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, has measure at most 2, where 0 < < . Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = B(|z|, u) for all large z C₁ C₂. Let A_D(r, u) = inf { u(z):z D and | z | = r }. It is shown that if A_D(r, u) = O(1) (or A_D(r, u) = o(B(r, u))), then lim_{r ->} B(r, u)/r^/2 > 0 (or lim_r-> log B(r, u)/log r ≥ /2).

On Bass' question for finitely generated algebras over large fields

Gubeladze, J. — 2009-02-20

Recently Cortiñas–Haesemayer–Walker–Weibel gave affirmative answer to Bass' 1972 question on NK-groups for algebras of essentially finite type over large fields of characteristic 0. Here we give an alternative short proof of this result for algebras of finite type over such fields. Our approach is based on classical techniques in higher K-theory of rings and a direct K_i-analog of an old observation of Murthy–Pedrini, dating back from the same 1972.

Densities for Ornstein-Uhlenbeck processes with jumps

Priola, E., Zabczyk, J. — 2009-02-20

We consider an Ornstein–Uhlenbeck process with values in Rⁿ driven by a Lévy process (Z_t) taking values in R^d with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability rank condition and a mild assumption on the Lévy measure of (Z_t), we prove that the law of the Ornstein–Uhlenbeck process at any time t > 0 has a density on Rⁿ. Moreover, when the Lévy process is of -stable type, (0, 2), we show that such density is a C-function.

Trivial centralizers for codimension-one attractors

Fisher, T. — 2009-02-20

We show that if is a codimension-one hyperbolic attractor for a C^r diffeomorphism f, where 2 ≤ r ≤ , and f is not Anosov, then there is a neighborhood U of f in Diff^r(M) and an open and dense set V of U such that any g V has a trivial centralizer on the basin of attraction for .

Commuting holomorphic maps on the spectral unit ball

Costara, C. — 2009-02-20

We prove that if F is a holomorphic map from the open spectral unit ball of a primitive Banach algebra into itself satisfying F(0) = 0, F^' (0) = I and F(x) x = xF(x) for every x, then F is the identity map. Using this, we prove that if A is a semisimple Banach algebra and B is a primitive Banach algebra, then any unital spectral isometry from A onto B which locally preserves commutativity is a Jordan morphism. The same is true when A and B are both assumed to be von Neumann algebras.

Normal families and omitted functions II

Zhang, G., Pang, X., Zalcman, L. — 2009-02-20

Let k ≥ 2 be an integer and let F be a family of functions meromorphic on a domain D in C, all of whose poles are multiple and whose zeros all have multiplicity at least k + 1. Let h be a function meromorphic on D, h 0, . Suppose that for each f F, f^(k)(z) != h(z) for z D. Then F is a normal family on D.

Positive fixed points and fourth-order equations

Cid, J. A., Franco, D., Minhos, F. — 2009-02-20

This work presents sufficient conditions for the existence of at least one positive solution for a nonlinear fourth-order beam equation under Lidstone boundary conditions. The main tool used is a fixed point theorem that essentially combines the monotone iterative technique with fixed point theorems of cone expansion or compression type. The last section contains two examples that complement some related results existent in the literature.

The Fitting ideal problem

Simis, A., Ulrich, B. — 2009-02-20

Let A be a Noetherian local ring and let E be a finitely generated A-module having rank r. In this note one deals with the expected inequality (^rE) ≥ height (Fitt_r(E)), where height (Fitt_r(E)) is the codimension of the rth Fitting ideal of E, and (M) stands for the analytic spread of a module M. One establishes cases where the inequality holds as well as where it fails. A special case where the inequality holds implies the celebrated Zak inequality for the dimension of the image of the Gauss map.

Equivalent birational embeddings

Mella, M., Polastri, E. — 2009-02-20

Let X be a projective variety of dimension r over an algebraically closed field. It is proven that two birational embeddings of X in Pⁿ with n ≥ r + 2 are equivalent up to Cremona transformations of Pⁿ.

Finiteness properties of automorphism groups of right-angled Artin groups

Charney, R., Vogtmann, K. — 2009-02-20

We study the algebraic structure of the outer automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved earlier by the authors and Crisp for 2-dimensional right-angled Artin groups.

On a theorem of K. Schmidt

Feldman, G. M. — 2009-02-20

Let Y be a locally compact group, Aut(Y) be the group of topological automorphisms of Y and P(Y) be the set of continuous positive definite functions on Y which have unit value at the identity. A function P (Y²) is said to be of product type if there are such functions _j P (Y) that (u, v) = ₁(u)₂(v). Define the mapping T: Y² -> Y² by the formula T(u, v) = (A₁ uA₂ v, A₃ u A₄ v), where A_j Aut(Y), and assume that T is a one-to-one transform. K. Schmidt proved: (i) if both (u, v) and (T(u, v)) are of product type, then the functions _j are infinitely divisible; (ii) if Y is Abelian, both (u, v) and (T(u, v)) are of product type, and (u, v) != 0, then the functions _j are Gaussian. We show that statement (i), generally, is not valid, but K. Schmidt's proof holds true if (u, v) != 0. We also give another proof of statement (ii). Our proof uses neither the Levy–Khinchin formula for a continuous infinitely divisible positive definite function nor (i) on which K. Schmidt's proof is based.

Weakly o-minimal expansions of ordered fields of finite transcendence degree

Wencel, R. — 2009-02-20

Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K R is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,≤,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond–Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,≤,+,·) are power bounded.

A Blaschke-type condition and its application to complex Jacobi matrices

Borichev, A., Golinskii, L., Kupin, S. — 2009-02-20

We obtain a Blaschke-type necessary condition on zeros of analytic functions on the unit disc with different types of exponential growth at the boundary. These conditions are used to prove Lieb–Thirring-type inequalities for the eigenvalues of complex Jacobi matrices.

Homotopy counting S1- and S2-valued maps with prescribed dilatation

Liu, L. — 2009-02-20

Let V, W be two compact Riemannian manifolds and # [V, W]_D the number of homotopy classes of maps with dilatation less than or equal to D. It is shown that (c₁D – 1)^b ≤ # [M, S¹]_D ≤ (c₂D + 1)^b, where b = b₁(M) is the first Betti number of M. The second result is that if M is a closed oriented Riemannian 3-manifold, then the number of homotopy classes of algebraically trivial maps M -> S² with dilatation less than D is at most c₃D⁴. This result covers an earlier theorem by Gromov. Finally, we prove that if M is a closed oriented Riemannian 3-manifold with H₁(M, Z) torsion free, then # [M, S²]_D ≤ c₃D⁴ + c₄D^2b+2. The above constants c_i depend on the metrics on the manifolds concerned.

Diophantine approximation on non-degenerate curves with non-monotonic error function

Budarina, N., Dickinson, D. — 2009-02-20

It is shown that a non-degenerate curve in Rⁿ satisfies a convergent Groshev theorem with a non-monotonic error function. In other words it is shown that if a volume sum converges the set of points lying on the curve which satisfy a Diophantine condition has Lebesgue measure zero.

Reduction modulo p of cuspidal representations and weights in Serre's conjecture

Schein, M. M. — 2009-02-20

Let O be the ring of integers of a p-adic field and p its maximal ideal. This paper computes the Jordan–Hölder decomposition of the reduction modulo p of the cuspidal representations of GL₂(O/p^e) for e ≥ 1. An alternative formulation of Serre's conjecture for Hilbert modular forms is then provided.

Global smooth fibrations of R3 by oriented lines

Salvai, M. — 2009-02-20

A smooth fibration of R³ by oriented lines is given by a smooth unit vector field V on R³ all of whose integral curves are straight lines. Such a fibration is said to be nondegenerate if dV vanishes only in the direction of V. Let L be the space of oriented lines of R³ endowed with its canonical pseudo-Riemannian neutral metric. We characterize the nondegenerate smooth fibrations of R³ by oriented lines as the closed (in the relative topology) definite connected surfaces in L. In particular, local conditions on L imply the existence of a global fibration. Besides, for any such fibration the base space is diffeomorphic to the open disc and the directions of the fibers form an open convex set of the two-sphere. We characterize as well, in a similar way, the smooth (possibly degenerate) fibrations.

Douglas Geoffrey Northcott, FRS, 1916-2005

Sharp, R. Y. — 2009-02-20

Handbook of tilting theory * (London Mathematical Society Lecture Note Series 332) * By Lidia Angeleri Hugel, Dieter Happel and Henning Krause: * 472 pp., {pound}46.00/LMS members' price {pound}34.50, ISBN 0-521-68045X * (London Mathematical Society, London, 2007)

Buan, A. B. — 2009-02-20

A geometric approach to free boundary problems * (Graduate Studies in Mathematics 68) * By Luis Caffarelli and Sandro Salsa: 270 pp., US$ 51.00 ISBN 0-8218-3784-2 * (American Mathematical Society, Providence, RI, 2005)

Veretennikov, A. Yu. — 2009-02-20

Prizewinners 2008

2009-02-20