We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model complete and o-minimal structure which is exponentially bounded, and in which the Gamma function on the positive real line is definable. 2000 Mathematics Subject Classification: primary 03C10, 32B05, 32B20; secondary, 26E05.

This paper is devoted to the complete classification of integrable one-component evolution equations whose field variable takes its values in an associative algebra. The proof that the list of non-commutative integrable homogeneous evolution equations is complete relies on the symbolic method. Each equation in the list has infinitely many local symmetries and these can be generated by its recursion (recursive) operator or master symmetry. 1991 Mathematics Subject Classification: 13A50, 16-XX, 22E70, 35A30, 35Q53, 58F07.

We introduce a variant of the usual K\"{a}hler forms on singular free divisors, and show that it enjoys the same depth properties as K\"{a}hler forms on isolated hypersurface singularities. Using these forms it is possible to describe analytically the vanishing cohomology, and the Gauss--Manin connection, in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an isolated complete intersection singularity, due to Brieskorn and Greuel. This applies in particular to the family $\{D(f_\lambda)\}_{\lambda\in \Lambda}$ of discriminants of a versal deformation $\{f_\lambda\}_{\lambda\in\Lambda}$ of a singularity of a mapping. 1991 Mathematics Subject Classification: 14B07, 14D05, 32S40.

This paper investigates the relation between a branching process and a non-linear dynamical system in $\mathbb{C}^2$. This idea has previously been fruitful in many investigations, including that of the FKPP equation by McKean, Neveu, Bramson, and others. Our concerns here are somewhat different from those in other work: we wish to elucidate those features of the dynamical system which correspond to the long-term behaviour of the random process. In particular, we are interested in how the dimension of the global attractor corresponds to that of the tail $\sigma$-algebra of the process. The Poincar\'e--Dulac operator which (locally) intertwines the non-linear system with its linearization may sometimes be exhibited as a Fourier--Laplace transform of tail-measurable random variables; but things change markedly when parameters cross values giving the `primary resonance' in the Poincar\'e--Dulac sense. Probability proves effective in establishing {\it global} properties amongst which is a clear description of the global convergence to the attractor. Several of our probabilistic results are analogues of ones obtained by Kesten and Stigum, and by Athreya and Ney, for discrete branching processes. Our simpler context allows the use of It\^o calculus. Because the paper bridges two subjects, dynamical-system theory and probability theory, we take considerable care with the exposition of both aspects. For probabilist readers, we provide a brief guide to Poincar\'e--Dulac theory; and we take the view that in a paper which we hope will be read by analysts, it would be wrong to fudge any details of rigour in our probabilistic arguments. 1991 Mathematics Subject Classification: 60H30, 60J85, 34A20.

Let $k$ be an algebraically closed field, and let $A$ be a finite-dimensional $k$-algebra. The first aim of classsical representation theory was to classify all indecomposable $A$-modules, but the homomorphisms between modules were not considered. One big step forward, made around 1975, was the invention of the Auslander--Reiten quiver of $A$. The Auslander--Reiten quiver yields a description of the category of $A$-modules modulo its infinite radical ${\rm rad}_A^\omega$. But it contains no information about the homomorphisms in ${\rm rad}_A^\omega$. We denote by mod-$A$ the category of finite-dimensional right modules over $A$. Let ${\rm rad}_A$ be the radical of mod-$A$. Following Prest, we define ${\rm rad}_A^\alpha$ for each ordinal number $\alpha$ as follows: if $J$ is an ideal in mod-$A$ and $n \geq 1$ is a natural number, then $J^n$ is the $n$-fold product of the ideal $J$. For a limit number $\beta$ define ${\rm rad}_A^\beta = \bigcap_{\alpha < \beta} {\rm rad}_A^\alpha$. Finally, let $\alpha$ be an arbitrary infinite ordinal number. Thus $\alpha = \beta + n$ for some limit number $\beta$ and some natural number $n \geq 0$. In this case, define ${\rm rad}_A^\alpha = ({\rm rad}_A^\beta)^{n+1}$. We denote the first limit number by $\omega$, the second by $\omega 2$, and so on. The minimal limit number, which is not of the form $\omega n$ for some natural number $n \geq 1$, is denoted by $\omega^2$. {\sc Theorem 1.} {\it For each ordinal number $\alpha$ with $\omega < \alpha < \omega^2$ there exists a finite-dimensional $k$-algebra $A$ with ${\rm rad}_A^\alpha \not= 0$ and ${\rm rad}_A^{\alpha + 1} = 0$. Let $Q$ be a quiver, and let $I$ be an admissible ideal of the path algebra $kQ$. We call $kQ/I$ {\it special biserial} if the following hold. Any vertex of $Q$ is the starting point of at most two arrows and also the end point of at most two arrows. Given an arrow $\beta$, there is at most one arrow $\alpha$ with $e(\alpha) = s(\beta)$ and $\alpha \beta \notin I$ and at most one arrow $\gamma$ with $e(\beta) = s(\gamma)$ and $\beta \gamma \notin I$. {\sc Theorem 2.} {\it Let $A$ be a special biserial algebra. Then the following are equivalent: \begin{enumerate} \item[{\rm (i)}] $A$ is domestic; \item[{\rm (ii)}] ${\rm rad}_A^{\omega^2} = 0$. \end{enumerate} } 1991 Mathematics Subject Classification: 16G20, 16G60, 16N40

Let $\Gamma$ be a countable abelian semigroup and $\mathcal{A}$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times \mathcal{A})$-graded Lie superalgebra $\mathfrak{L} =\bigoplus_{(\alpha, a) \Gamma \times \mathcal{A}} \mathfrak{L}_{(\alpha, a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma \times \mathcal{A})$-gradation. In this paper, we show that the Euler--Poincar\'e principle yields the generalized denominator identity for $\mathfrak{L}$ and derive a closed form formula for the supertraces $\text{str}(g| \mathcal{L}_{(\alpha, a)})$ for all $g\in G$, where $(\alpha, a) \in \Gamma \times \mathcal{A}$. We discuss the applications of our supertrace formula to various classes of infinite-dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac--Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $\text{GL}(n) \times \text{GL}(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and irreducible highest-weight modules over a generalized Kac--Moody superalgebra $\mathfrak{g}$ corresponding to the Dynkin diagram automorphism $\sigma$ are the same as the usual characters of Verma modules and irreducible highest-weight modules over the orbit Lie superalgebra $\breve{\mathfrak{g}} = \mathfrak{g}(\sigma)$ determined by $\sigma$. 1991 Mathematics Subject Classification: 17A70, 17B01, 17B65, 17B70, 11F22.

A semi-algebra of continuous functions is a cone $A$ of continuous real functions on a compact Hausdorff space $X$ such that $A$ contains the products of its elements. A cone $A$ is said to be of type $n$ if $f\in A$ implies $f^n(1 + f)^{-1} \in A$. Uniformly closed semi-algebras of types 0 and 1 have long been characterized in a manner analogous to the Stone--Weierstrass theorem, but, except for the case when $A$ is generated by a single function, little has been known about type 2. Here, progress is reported on two problems. The first is the characterization of those continuous linear functionals on $C(X)$ that determine semi-algebras of type 2. The second is the determination of the type of the tensor product of two type 1 semi-algebras. 1991 Mathematics Subject Classification: 46J10.

For $K$ a connected finite complex and $G$ a compact connected Lie group, a finiteness result is proved for gauge groups ${\mathcal G}(P)$ of principal $G$-bundles $P$ over $K$: as $P$ ranges over all principal $G$-bundles with base $K$, the number of homotopy types of ${\mathcal G}(P)$ is finite; indeed this remains true when these gauge groups are classified by $H$-equivalence, that is, homotopy equivalences which respect multiplication up to homotopy. A case study is given for $K = S^4$, $G = \text{SU}(2)$: there are eighteen $H$-equivalence classes of gauge group in this case. These questions are studied via fibre homotopy theory of bundles of groups; the calculations in the case study involve $K$-theories and $e$-invariants. 1991 Mathematics Subject Classification: 54C35, 55P15, 55R10.