Let Gamma \subset {mathbb SO}(3,1) be a lattice. The well known bending deformations, introduced by linebreak Thurston and Apanasov, can be used to construct non-trivial curves of representations of Gamma into {mathbb SO}(4,1) when Gamma \backslash H³ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in H¹(Gamma, R⁴₁). Our main result generalizes this construction of cohomology to the context of "branched" totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in S³ which is not infinitesimally rigid in {mathbb SO}(4,1). The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.

On a compact connected group G, consider the infinitesimal generator -L of a central symmetric Gaussian convolution semigroup (mu_t)_{t > 0}. Using appropriate notions of distribution and smooth function spaces, we prove that L is hypoelliptic if and only if (mu_t)_{t > 0} is absolutely continuous with respect to Haar measure and admits a continuous density x \mapsto mu_t(x), t > 0, such that lim_{t rightarrow 0} t log mu_t(e) = 0. In particular, this condition holds if and only if any Borel measure u which is solution of Lu = 0 in an open set Omega can be represented by a continuous function in Omega. Examples are discussed.

We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above "special function approach" can be described by a classical Heine problem for differential equations that admit polynomial solutions.

The decomposability number of a von Neumann algebra cal M (denoted by dec(cal M)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in cal M. In this paper, we explore the close connection between dec(cal M) and the cardinal level of the Mazur property for the predual cal M_* of cal M, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L₁(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)^*, etc. We show that for any of these von Neumann algebras, say cal M₀, the cardinal number dec(cal M) and a certain cardinal level of the Mazur property of (cal M)_* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number _{cal K}(G) of G and the least cardinality _{cal X}(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)^**.

Let K be a number field, overline{K} an algebraic closure of K and E/K an elliptic curve defined over K. In this paper, we prove that if E/K has a K-rational point P such that 2P \neq O and 3P \neq O, then for each sigma \in Gal(overline{K}/K), the Mordell–Weil group E(overline{K}^sigma) of E over the fixed subfield of overline{K} under sigma has infinite rank.

We characterize diametrically maximal and constant width sets in C(K), where K is any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A characterization of diametrically maximal sets in ell₁³ is also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets in c₀(I), for every I, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces.

In a previous article, we studied the distribution of "low-lying" zeros of the family of quadratic Dirichlet L-functions assuming the Generalized Riemann Hypothesis for all Dirichlet L-functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval (-2,2). However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic L-functions. In this note, we make use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.

The Banach convolution algebras l¹(omega) and their continuous counterparts L¹(mathbb R⁺, omega) are much studied, because (when the submultiplicative weight function omega is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of "nice" weights omega, the only closed ideals they have are the obvious, or "standard", ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in l¹(omega). His proof was successfully exported to the continuous case L¹(mathbb R⁺, omega) by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in l¹(omega) and L¹(mathbb R⁺, omega). The new proof is based on the idea of a "nonstandard dual pair" which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in L¹(mathbb R⁺, omega) containing functions whose supports extend all the way down to zero in (mathbb R⁺), thereby solving what has become a notorious problem in the area.

Selick and Wu gave a functorial decomposition of Omega Sigma X for path-connected, p-local CW-complexes X which obtained the smallest nontrivial functorial retract A^min(X) of Omega Sigma X. This paper uses methods developed by the second author in order to extend such functorial decompositions to the loops on coassociative co-H spaces.