We develop a version of Freĭman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates.

Our work complements three other recent approaches to developing non-abelian versions of Freĭman's theorem by Breuillard and Green, Fisher, Katz and Peng, and Tao.

We consider finite dimensional representations of the dihedral group D2p over an algebraically closed field of characteristic two where p is an odd prime and study the degrees of generating and separating polynomials in the corresponding ring of invariants. We give an upper bound for the degrees of the polynomials in a minimal generating set that does not depend on p when the dimension of the representation is sufficiently large. We also show that p + 1 is the minimal number such that the invariants up to that degree always form a separating set. We also give an explicit description of a separating set.

The commutant of a loop is the set of elements which commute with all of the elements in the loop. The commutant of a Moufang loop is a subloop, but it has been an open problem to classify the Moufang loops for which the commutant is normal. It was S. Doro [3] who conjectured that a Moufang loop, under certain conditions, has a normal commutant. We settle this conjecture here by proving that the commutant of any Moufang loop is always a normal subloop.

We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have non-vanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.

We consider two problems concerning signs of Fourier coefficients of classical modular forms, or equivalently Hecke eigenvalues: first, we give an upper bound for the size of the first sign-change of Hecke eigenvalues in terms of conductor and weight; second, we investigate to what extent the signs of Fourier coefficients determine an unique modular form. In both cases we improve recent results of Kowalski, Lau, Soundararajan and Wu. A part of the paper is also devoted to generalized rearrangement inequalities which are utilized in an alternative treatment of the second question.

For functions F satisfying a certain submultiplicativity condition and polynomials Q1, . . ., Qk in [X], Nair and Tenenbaum obtained an upper bound on the short sum with an implicit dependency on the discriminant of Q1 . . . Qk. We obtain a similar upper bound uniform in the discriminant.

Let k be an algebraically closed field of any characteristic except 2, and let G = GLn(k) be the general linear group, regarded as an algebraic group over k. Using an algebro-geometric argument and Dynkin–Kostant theory for G we begin by obtaining a canonical form for nilpotent Ad(G)-orbits in (k) which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi,j) ↦ (xn+1−j,n+1−i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups GUn(q) for all prime powers q.

We consider a tamely ramified abelian extension of local fields of degree n, without assuming the presence of the nth roots of unity in the base field. We give an explicit formula which computes the local reciprocity map in this situation.

We give sufficient conditions for the existence of detecting elements for the Lusternik–Schnirelmann category in proper homotopy. As an application we determine the proper LS category of some semistable one-ended open 3-manifolds.

Let k be an algebraically closed field of ch(k) = 0 and G be a simple simply connected algebraic group G over k. By using results about cohomological invariants, we compute the coniveau spectral sequence for classifying spaces BG.

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. For any homotopy given by a composite homotopy data triple we define a complete invariant of nanophrases. This invariant is used to show that equivalence of nanophrases under such a homotopy can be calculated just by using the homotopies given by its prime factors.

Cromwell and Marar present an analysis of semi-regular (generic) surfaces with a single triple point and connected self-intersection set. Six of their surfaces are the projective plane, including Boy's surface and Steiner's surface. We build on their work by incorporating twists similar to that of Apery's immersion of the projective plane and show that with a few additional surfaces, all such generic maps of the projective plane are now identified.

Let G denote a topological group. In this paper the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that provide a filtration of the classifying space BG. In particular this setting gives rise to a single space constructed out of all the spaces of ordered commuting n–tuples of elements in G. Basic properties of these constructions are discussed, including the homotopy type and cohomology when the group G is either a finite group or a compact connected Lie group. For a finite group the construction gives rise to a covering space with monodromy related to a delicate result in group theory equivalent to the odd-order theorem of Feit–Thompson. The techniques here also yield a counting formula for the cardinality of Hom(π, G) where π is any descending central series quotient of a finitely generated free group. Another application is the determination of the structure of the spaces B(2, G) obtained from commuting n-tuples in G for finite groups such that the centralizer of every non–central element is abelian (known as transitively commutative groups), which played a key role in work by Suzuki on the structure of finite simple groups.

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.

The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

We establish some general results on universal models in Topos Theory and show that the investigation of such models can shed light on problems of definability in Logic as well as on De Morgan's law and the law of excluded middle for Grothendieck toposes.

We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard–Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an L-space, as defined by Ozsváth and Szabó.

We are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just defined a field of tangent convex cones (Cx)x ∈ M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function.

Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of Cx. We find connections with topics pertaining to Hamilton–Jacobi equations, and make use of tools and results issued from weak KAM theory.

We will construct differential forms on the embedding spaces Emb(j, n) for n − j ≥ 2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(j, n) by Emb(j, n), the homotopy fiber of the inclusion Emb(j, n) ↪ Imm(j, n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(j, n) and Emb(j, n). In particular we obtain nontrivial cohomology classes (for example, in H3(Emb(2, 5))) of higher degrees than those of the first nonvanishing homotopy groups.

The purpose of this paper is to compare some classical positions of convex bodies. We provide exact quantitative results which show that the minimal surface area position and the minimal mean width position are not necessarily M-positions. We also construct examples of unconditional convex bodies of minimal surface area that exhibit the worst possible behavior with respect to their mean width or their minimal hyperplane projection.

Fix g a Hecke–Maass form for SL3(). In the family of holomorphic newforms f of fixed weight and large prime level q, we find the average value of the product . From this we derive a result on the simultaneous non-vanishing of these L-functions at the central point.