We investigate definability in $\mathcal{R}$, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMAs) as a tool. First we show that an SMA can be interpreted in $\mathcal{R}$ without parameters. Building on this, we prove that the recursively enumerable $T$-degrees satisfy a weak form of the bi-interpretability conjecture which implies that all jump classes $\mathrm{Low}_n$ and $\mathrm{High}_{n-1}$$(n\ge 2)$ are definable in $\mathcal{R}$ without parameters and, more generally, that all relations on $\mathcal{R}$ that are definable in arithmetic and invariant under the double jump are actually definable in $\mathcal{R}$. This partially answers Soare's Question 3.7 (R. Soare, {\emRecursively enumerable sets and degrees} (Springer, Berlin, 1987), Chapter XVI).

1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.

Bushnell and Kutzko gave a complete and effective classification of the smooth dual of $\mbox{GL}(N,F)$, where $F$ is a non-archimedean local field. Similarly, Zink gave a classification of the smooth dual of $D^{\times}$, where $D$ is a division algebra with centre $F$, in terms of non-canonical objects and under the restrictive hypothesis that $F$ has characteristic $0$. In this paper, we extend part of Bushnell and Kutzko's formalism to $D^{\times}$ and obtain a complete classification of the smooth dual working for any characteristic. The crucial point of this work is to define a good way of splitting the algebra $D$ so that the important notion of {\it simple stratum}, and its properties, can be translated to $D^{\times}$ by some descent arguments.

1991 Mathematics Subject Classification: 12E15, 20G05, 20G25, 22E50.

In this paper we use the Hecke algebra of type $\bf B$ to define a new algebra $\mathcal S$ which is an analogue of the $q$-Schur algebra. We show that $\mathcal S$ has ‘generic’ basis which is independent of the choice of ring and the parameters $q$ and $Q$. We then construct Weyl modules for $\mathcal S$ and obtain, as factor modules, a family of irreducible $\mathcal S$-modules defined over any field.

1991 Mathematics Subject Classification: 16G99, 20C20, 20G05.

The ground field $F$ is an algebraically closed field of zero characteristic, and ${\Bbb N}$ is the set of natural numbers. Let $L$ be a locally finite-dimensional (or {\em locally finite}, for brevity) Lie algebra. Assume for simplicity that $L$ has countable dimension. Then one can choose finite-dimensional subalgebras $L_i$ ($i\in{\Bbb N}$) of $L$ in such a way that $L_{i}\subset L_{i+1}$ for all $i$ and $L=\bigcup_{i\in{\Bbb N}}L_i$.The set $\{L_i\mid i\in{\Bbb N}\}$ is called a{\em local system} of $L$. Assume that $L$ is simple. Then all $L_i$ can be chosen to be perfect (that is, $[L_i,L_i]=L_i$). We shall call such local systems {\em perfect}. Let $S_i=S_i^1\oplus\ldots\oplus S_i^{n_i}$ be a Levi subalgebra of $L_i$, where $S_i^1,\ldots,S_i^{n_i}$ are simple components of $S_i$, and let $V_i^k$ be the standard $S_i^k$-module. Since $L_i$ is perfect, for every $k$ there exists a unique irreducible $L_i$-module${\cal V}_i^k$ such that the restriction ${\cal V}_i^k{\downarrow} S_i^k$ is isomorphic to $V_i^k$. An embedding $L_i\to L_{i+1}$ is called {\em diagonal} if, for each $l$, any non-trivial composition factor of the restriction ${\cal V}_{i+1}^l{\downarrow} L_i$ is isomorphic to ${\cal V}_i^k$ or its dual ${\cal V}_i^{k^*}$ for some $k$. We prove that a simple locally finite Lie algebra $L$ can be embedded into a locally finite associative algebra if and only if there exists a perfect local system $\{L_i\}_{i\in{\Bbb N}}$ of $L$ such that all embeddings $L_i\to L_{i+1}$ are diagonal. Note that this result can be considered as a version of Ado's theorem for simple locally finite Lie algebras. Let $V$ be a vector space. An element $x\in \frak{gl}(V)$ is called {\em finitary}if $\dim xV<\infty$. The finitary transformations of $V$ form an ideal $\frak{fgl}(V)$ of $\frak{gl}(V)$, and any subalgebra of $\frak{fgl}(V)$ is called a {\em finitary Lie algebra}. We classify all finitary simple Lie algebras of countable dimension. It turns out that there are only three: $\frak{sl}_\infty(F)$, $\frak{so}_\infty(F)$, and $\frak{sp}_\infty(F)$. The author has classified all finitary simple Lie algebras and will describe this in a subsequent paper.The analogous problem in group theory has been recently solved by J.I. Hall. He classified simple locally finite groups of finitary linear transformations.

1991 Mathematics Subject Classification: 17B35, 17B65.

In this paper we study the notion of joint functional calculus associated with a couple of resolvent commuting sectorial operators on a Banach space $X$. We present some positive results when $X$ is, for example, a Banach lattice or a quotient of subspaces of a $B$-convex Banach lattice. Furthermore, we develop a notion of a generalized $H^\infty$-functional calculus associated with the extension to $\Lambda(H)$ of a sectorial operator on a $B$-convex Banach lattice $\Lambda$, where $H$ is a Hilbert space. We apply our results to a new construction of operators with a bounded $H^\infty$-functional calculus and to the maximal regularity problem.

1991 Mathematics Subject Classification: 47A60, 47D06, 46C15.

In this paper we investigate how the volume of a hyperbolic manifold increases under the process of removing a curve, that is, Dehn drilling. If the curve we remove is a geodesic, we show that for a certain family of manifolds the volume increase is bounded above by $\pi \cdot l$ where $l$ is the length of the geodesic drilled. We further construct examples to show that there is no lower bound to the volume increase in terms of a monotonic unbounded function of length. In particular, this shows that volume increase is not bounded below linearly in length.

1991 Mathematics Subject Classification: primary 51M10, 51M20; secondary 51M25, 52C25.

Let $M$ be a compact, smooth, $2$-connected, $2m$-dimensional manifold with $\partial M$ simply connected. If $M$ has the homotopytype of an $m$-dimensional $CW$-complex, then it supports a smooth, self-indexed function, maximal, constant and regular on $\partial M$ with at most $\mbox{cat}(M)+2$ critical points, all of which are of a certain ‘reasonable’ type. To such a critical point there corresponds, homotopically, the attachment of a cone. Conversely, to a cone attachment we may associate, under certain dimensionality and connectivity conditions, a ‘reasonable’ critical point.

1991 Mathematics Subject Classification: primary 55M30, 57R70; secondary 58E05, 55P50, 55P62.

The functional Ito formula, in the form $df({\bf \Lambda})=f({\bf \Lambda} +d{\bf \Lambda})-f({\bf \Lambda} )$, is formulated and proved in the context of a Lie algebra $\mathcal{L}$ associated with a quantum (non-commutative) stochastic calculus. Here $f$ is an element of the universal enveloping algebra $\mathcal{U}$ of $\mathcal{L}$, and $f({\bf \Lambda} + d{\bf \Lambda})-f({\bf \Lambda} )$ is given a meaning using the coproduct structure of $\mathcal{U}$ even though the individual terms of this expression have no meaning. The Ito formula is equivalent to a chaotic expansion formula for $f({\bf \Lambda} )$ which is found explicitly.

1991 Mathematics Subject Classification: primary 81S25; secondary 60H05; tertiary 18B25.