Let $X$ be a regular separated scheme of finite Krull dimension and let $U^{n}_{X} \subset A^{n}_{X}$ be the punctured affine $n$-space over $X$. We show that the total graded Witt ring of $U^{n}_{X}$ is a free graded module over the total graded Witt ring of $X$ with two generators $1$ and $\epsilon$. The second generator satisfies the equation $\epsilon^{2} = 1$ when $n = 1$ and $\epsilon^{2} = 0$ when $n \geq 2$.

We study the global root number of the complex $L$-function of twists of elliptic curves over $\mathbb{Q}$ by real Artin representations. We obtain examples of elliptic curves over $\mathbb{Q}$ which, while not having any rational points of infinite order, conjecturally must have points of infinite order over the fields $\mathbb{Q}( \sqrt[3] {m} )$ for every cube-free $m > 1$. We describe analogous phenomena for elliptic curves over the fields $\mathbb{Q}( \sqrt[r] {m} )$, and in the towers $(\mathbb{Q}( \sqrt[r^n] {m})_{n \ge 1} )$ and $(\mathbb{Q}( \sqrt[r^n] {m}, \mu_{r^n})_{n \ge 1})$, where $r \ge 3$ is prime.

A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode ‘conjugacy’ relations among subgroups of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as $p$-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system $\mathcal{F}$ over a finite $p$-group $S$ is saturated can be determined by just looking at smaller classes of subgroups of $S$. We also prove that the homotopy type of the classifying space of a given $p$-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of $\mathcal{F}$-centric $\mathcal{F}$-radical subgroups (at a minimum) to the set of $\mathcal{F}$-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to $p$-constrained finite groups, and prove that they in fact all arise from groups.

The purpose of this article is to determine the representation type for all of the Hecke algebras of classical type. To do this, we combine methods from our previous work, which is used to obtain information on their Gabriel quivers, and recent advances in the theory of finite-dimensional algebras. The principal computation is for Hecke algebras of type B with two parameters. Then, we show that the representation type of Hecke algebras is governed by their Poincaré polynomials.

We study the rigidity problem for periodic orbits of (continuous) graph maps belonging to the same homotopy equivalence class. Since the underlying spaces are not necessarily homeomorphic, we define a new notion of pattern which enables us to compare periodic orbits of self-maps of homotopy-equivalent spaces. This definition unifies the known notions of pattern for other spaces. The two main results of the paper are as follows: given a free group endomorphism, we study the persistence under homotopy of the periodic orbits of its topological representatives, and in the irreducible case, we prove the minimality (within the homotopy class) of the set of periodic orbits of its efficient representatives.

We give an example of an affinoid curve without analytic continuation. We use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow-ups. Thus the global rigid analogue of Hironaka's complex analytic flattening theorem given by T. Gardener and H. Schoutens, in Theorem 2.3 of ‘Flattening and subanalytic sets in rigid analytic geometry’, Proc. London Math. Soc (3) 83 (2001) 681–707, is not true. Since this is a key step in the proof of the affinoid elimination theorem (loc. cit. Theorem 3.12), that proof contains a serious gap. We also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map. This clarifies the relationship among several natural categories of rigid subanalytic sets.

Let $\Omega$ be a smooth bounded domain in ${R}^N$. We prove general uniqueness results for equations of the form $- \Delta u = au - b(x) f(u)$ in $\Omega$, subject to $u = \infty$ on $\partial \Omega$. Our uniqueness theorem is established in a setting involving Karamata's theory on regularly varying functions, which is used to relate the blow-up behavior of $u(x)$ with $f(u)$ and $b(x)$, where $b \equiv 0$ on $\partial \Omega$ and a certain ratio involving $b$ is bounded near $\partial \Omega$. A key step in our proof of uniqueness uses a modification of an iteration technique due to Safonov.

New criteria and Banach spaces are presented (for example, $GL$-spaces and Banach spaces with property $(\alpha)$) that ensure that the Boolean algebra generated by a pair of bounded, commuting Boolean algebras of projections is itself bounded. The notion of $R$-boundedness plays a fundamental role. It is shown that the strong operator closure of any $R$-bounded Boolean algebra of projections is necessarily Bade complete. Also, for a Dedekind $\sigma <formula form="inline" disc="math" id="frm006"><formtex notation="AMSTeX">$-complete Banach lattice $E$, the Boolean algebra consisting of all band projections in $E$ is $R$-bounded if and only if $E$ has finite cotype. In this situation, every bounded Boolean algebra of projections in $E$ is $R$-bounded and has a Bade complete strong closure.

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant $K > 0$ such that if $\mathcal{M}$ is a semi-finite von Neumann algebra and $( \mathcal{M}_n )^{ \infty }_{n = 1}$ is an increasing filtration of von Neumann subalgebras of $\mathcal{M}$ then for any given martingale $x = ( x_n )^{\infty}_{n = 1}$ that is bounded in $L^2 ( \mathcal{M} ) \cap L^1 ( \mathcal{M} )$, adapted to $( \mathcal{M}_n )^{\infty}_{n = 1}$, there exist two martingale difference sequences, $a = ( a_n )_{n = 1}^\infty$ and $b = ( b_n )_{n = 1}^\infty$, with $dx_n = a_n + b_n$ for every $n \geq 1$,

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{2} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{2} \leq 2 \| x \|_2,

and

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{1, \infty} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{1, \infty} \leq K \| x \|_1.

As an application, we obtain the optimal orders of growth for the constants involved in the Pisier–Xu non-commutative analogue of the classical Burkholder–Gundy inequalities.