In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree $d>0$ will necessarily have a closed étale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type $G_2$, type $F_4$, or simply connected of type $E_6$. We extend the list of cases where the answer is ‘yes’ to all groups of type $G_2$ and some nonsplit groups of type $F_4$ and $E_6$. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.

Using properties of the Frobenius eigenvalues, we show that, in a precise sense, ‘most’ isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and we show a similar result for ‘most’ isogeny classes. Some global cases are also treated.

Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kähler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field <formula form="inline" disc="math" id="frm14"><formtex notation="AMSTeX"> $X:=\tilde{X}|_V$ on $V$ : \begin{equation} 0\longleftarrow {\cal O}_{V,0} {\buildrel X\over\longleftarrow}\,\Omega_{V,0}^1\,{\buildrel X\over\longleftarrow}\, \cdots \,{\buildrel X\over\longleftarrow}\, \Omega_{V,0}^n\, {\buildrel X\over\longleftarrow}\, \Omega_{V,0}^{n+1}\longleftarrow 0. \end{equation}

The compression theorem is used to prove results for equivariant configuration spaces that are analogous to the well-known non-equivariant results of May, Milgram and Segal.

The uniserial $p$-adic space groups of coclass $r$ play a central role in the classification of the finite $p$-groups of coclass $r$. A practical algorithm to determine up to isomorphism the uniserial $p$-adic space groups of coclass $r$, where $p$ is an odd prime, is described. As an application, these groups are constructed or counted for some values on $p$ and $r$. For example, it is observed that there are 137 299 953 383 uniserial 3-adic space groups of coclass 4.

Let R be a commutative Noetherian ring. Let [Pscr ](R) (respectively, [Iscr ](R)) be the category of all finite R-modules of finite projective (respectively, injective) dimension. Sharp [9] constructed a category equivalence between [Iscr ](R) and [Pscr ](R) for certain Cohen–Macaulay local rings R. Thus many properties about finite modules of finite projective dimension can be connected with those of finite injective dimension through this equivalence.

The integral cohomology rings of the configuration spaces of $n$-tuples of distinct points on arbitrary surfaces (not necessarily orientable, not necessarily compact and possibly with boundary) are studied. It is shown that for punctured surfaces the cohomology rings stabilize as the number of points tends to infinity, similarly to the case of configuration spaces on the plane studied by Arnold, and the Goryunov splitting formula relating the cohomology groups of configuration spaces on the plane and punctured plane to arbitrary punctured surfaces is generalized. Moreover, on the basis of explicit cellular decompositions generalizing the construction of Fuchs and Vainshtein, the first cohomology groups for surfaces of low genus are given.

We study concentration phenomena for the system

\[\varepsilon^2 \Delta v - v - \delta \phi v + \gamma v^{p} =0,\quad \Delta\phi+ \delta v^2=0 \]

in the unit ball $B_1$ of $\mathbb{R}^3$ with Dirichlet boundary conditions. Here $\varepsilon,\ \delta,\ \gamma >0$ and $p>1$. We prove the existence of positive radial solutions $(v_{\varepsilon}, \phi_{\varepsilon})$ such that $v_{\varepsilon}$ concentrates at a distance $({\varepsilon}/{2}) |{\rm log}\, {\varepsilon} |$ away from the boundary $\partial B_1$ as the parameter $\varepsilon$ tends to 0. The approach is based on a combination of Lyapunov–Schmidt reduction procedure together with a variational method.

We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p-block for all primes p in the set. Our main focus is on the simple and quasi-simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

The determination of the modular character tables of the sporadic O'Nan group, its automorphism group and its covering group is completed by the calculation of the 7-modular decomposition numbers. The results are obtained with the assistance of the systems GAP, MOC, and MeatAxe, and by applying new condensation methods.

A noninjective bounded-to-one factor map from an irreducible shift of finite type onto a sofic system can be factored as a composition of other such maps in only finitely many ways (up to isomorphism). This generalizes to factor maps from systems with canonical coordinates to finitely presented dynamical systems.

A simple geometrical argument shows that every pair of projections on a finite-dimensional complex vector space has a common invariant subspace of dimension 1 or 2. The idea extends to certain pairs of projections on an infinite-dimensional Hilbert space H. In particular every projection on H has a reducing subspace, although a finite-dimensional one need not exist. In a final section, the results are extended to the existence of hyperinvariant subspaces for pairs of projections.

JB*-triples occur in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on C*-algebras. These spaces are equipped with a ternary product {·,·,·}, the Jordan triple product, and are essentially geometric objects in that the linear isometries between them are exactly the linear bijections preserving the Jordan triple product (cf. [23]).

We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory; the generalised group algebras are called ‘host’ algebras. The main property of a host algebra is that its representation theory should be isomorphic (in the sense of the Gelfand–Raikov theorem) to a specified subset of representations of the algebraic object. Here we obtain both existence and uniqueness theorems for host algebras as well as general structure theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. To make contact with harmonic analysis, we consider general convolution algebras associated to representation sets, and consider conditions for a convolution algebra to be a host algebra.

The non-degeneracy of the canonical $p$-adic height pairing defined by Perrin-Riou and Schneider on an elliptic curve over a number field with good, ordinary reduction is still unknown.

Following the work done for the real-valued pairing, the behaviour of the $p$-adic height is analysed as a point varies on a section of a family of elliptic curves, and so new information is obtained about this pairing. In particular, the variation is $p$-adically continuous and the non-degeneracy of a set of sections can be checked simultaneously for almost all elements of the family. The paper contains some conjectures about the valuation of the $p$-adic regulator and its consequences for the growth of the Mordell–Weil group in cyclotomic $\mathbb{Z}_p$-extensions.

A finite rewriting system is presented that does not satisfy the homotopical finiteness condition FDT, although it satisfies the homological finiteness condition FHT. This system is obtained from a group $G$ and a finitely generated subgroup $H$ of $G$ through a monoid extension that is almost an HNN extension. The FHT property of the extension is closely related to the ${\rm FP}_2$ property for the subgroup $H$, while the FDT property of the extension is related to the finite presentability of $H$. The example system separating the FDT property from the FHT property is then obtained by applying this construction to an example group.

Let $H$ be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field $k$ of characteristic 0. If $H$ has no nontrivial skew-primitive elements, some bounds are found for the dimension of $H_1$, the second term in the coradical filtration of $H$. Using these results, it is shown that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. Also a Hopf algebra of dimension $pq$ where $p$ and $q$ are odd primes with $p < q \leq 1 + 3p$ and $q \leq 13$ is semisimple and thus a group algebra or the dual of a group algebra. Some partial results in the classification problem for dimension 16 are also given.

In this paper, we are mainly concerned with $n$ -dimensional simplices in hyperbolic space ${\bb H}^n$ . We will also consider simplices with ideal vertices, and we suggest that the reader keeps the Poincaré unit ball model of hyperbolic space in mind, in which the sphere at infinity ${\bb H}^n(\infty)$ corresponds to the bounding sphere of radius 1. It is known that all hyperbolic simplices (even the ideal ones) have finite volume. However, explicit calculation of their volume is generally a very difficult problem (see, for example, [1] or [16]). Our first theorem states that, amongst all simplices in a closed geodesic ball, the simplex of maximal volume is regular. We call a simplex regular if every permutation of its vertices can be realized by an isometry of ${\bb H}^n$ . A corresponding result for simplices in the sphere has been proved by Böröczky [4].

Let T be an ergodic and free ℤdrotation on the d-dimensional torus [ ]d given by

formula here

where (m1, …, md) ∈ ℤd, (z1, …, zd) ∈ [ ]d and [αjk]j,k=1 …, d ∈ Md(ℝ). For a continuous circle cocycle ϕ[ratio ]ℤd × [ ]d → [ ](ϕm+n(z) = ϕm(Tnz)ϕn(z) for any m, n ∈ ℤd), the winding matrix W(ϕ) of a cocycle ϕ, which is a generalization of the topological degree, is defined. Spectral properties of extensions given by

formula here

are studied. It is shown that if ϕ is smooth (for example ϕ is of class C1) and det W(ϕ) ≠ 0, then Tϕ is mixing on the orthocomplement of the eigenfunctions of T. For d = 2 it is shown that if ϕ is smooth (for example ϕ is of class C4), det W(ϕ) ≠ 0 and T is a ℤ2-rotation of finite type, then Tϕ has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(ϕ) = 1, then Tϕ has singular spectrum.

Given a sequence of integers aj, $j\ge 1$, a multiset is a combinatorial object composed of unordered components, such that there are exactly aj one-component multisets of size j. When $a_j\asymp j^{r-1} y^j$ for some r > 0, $y\geq 1$, then the multiset is called expansive. Let cn be the number of multisets of total size n. Using a probabilistic approach, we prove for expansive multisets that $c_n/c_{n+1}\to 1$ and that cn/cn+1 < 1 for large enough n. This allows us to prove monadic second-order limit laws for expansive multisets. The above results are extended to a class of expansive multisets with oscillation.

Moreover, under the condition $a_j=Kj^{r-1}y^j + O(y^{\nu j})$, where K > 0, r > 0, y > 1, $\nu\in (0,1)$, we find an explicit asymptotic formula for cn. In a similar way we study the asymptotic behavior of selections, which are defined as combinatorial objects composed of unordered components of distinct sizes.