In this paper exact asymptotic formulae are found for singular values of the Cauchy operator and the logarithmic potential type operator (on a bounded domain), as well as their products with Bergman's projection. It is shown that these spectral characteristics detect geometric properties of a domain $\Omega$ (area and the length of the boundary). The hypothesis “can we hear the shape of a drum”, from a paper by J.M. Anderson, D. Khavinson, and V. Lomonosov [‘Spectral propertiesof some integral operators arising in potential theory’, {\em Quart.\ J. Math.\ Oxford} (2) 43 (1992) 387-407], is correct in the above sense.

1991 Mathematics Subject Classification: 47B10.

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.

We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.

For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.

2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.

We study finite and infinite entangled graphsin the bond percolation process in three dimensionswith density $p$.After a discussion of the relevant definitions,the entanglement critical probabilities are defined.The size of the maximal entangled graph at the origin is studied for small $p$, and it is shown that this graph has radius whose tail decays at least as fast as $\exp(-\alpha n/\log n)$; indeed, the logarithm may be replaced by any iterate of logarithmfor an appropriate positive constant $\alpha$. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when $p$ is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation. http://www.statslab.cam.ac.uk/$\sim$grg/ 1991 Mathematics Subject Classification: primary 60K35; secondary 05C10, 57M25, 82B41, 82B43, 82D60.

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex $X$ and $k \geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, which is obtained by taking a generalization of the Spivak bundle on $X$ (which however is not a stable sphere bundle unless $X$ is a Poincaré space), pulling back to $Maps(S^k, X)$ and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the $(k + 1)$-dimensional unframed little disk operad $\mathcal{C}_{k + 1}$. I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based $\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad $C_{\ast}\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual $C_{\ast}\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and the chain complex of $\Omega^k X$ are Koszul dual to each other as $C_{\ast}\mathcal{C}_k$-algebras, and that the chain complex of $Maps(S^k, X)^{S(X)}$ is naturally equivalent to their (equivalent) Hochschild cohomology in the category of $C_{\ast}\mathcal{C}_k$-algebras.

We consider a Hamiltonian setup $(\mathcal M,\omega,H,\mathfrak L,\Gamma,\mathcal P)$, where $(\mathcal M,\omega)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ is a Lagrangian submanifold of $\mathcal M$, $H$ is a smooth time-dependent Hamiltonian function on $\mathcal M$, and $\Gamma:[a,b]\to\mathcal M$ is an integral curve of the Hamiltonian flow $\vec H$ starting at $\mathcal P$. We do not require any convexity property of the Hamiltonian function $H$. Under the assumption that $\Gamma(b)$ is not $\mathcal P$-focal, we introduce the Maslov index $\mathrm i_{\mathrm{maslov}}(\Gamma)$ of $\Gamma$ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances $\mathrm i_{\mathrm{maslov}}(\Gamma)$ is computed as a sort of algebraic count of the $\mathcal P$-focal points along $\Gamma$. We prove the following version of the Index Theorem: under suitable hypotheses, the Morse index of the Lagrangian action functional restricted to suitable variations of $\Gamma$ is equal to the sum of $\mathrm i_{\mathrm{maslov}}(\Gamma)$ and a convexity term of the Hamiltonian $H$ relative to the submanifold $\mathcal P$. When the result is applied to the case of the cotangent bundle $\mathcal M=TM^*$ of a semi-Riemannian manifold $(M,g)$ and to the geodesic Hamiltonian $H(q,p)=\frac12 g^{-1}(p,p)$, we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.

2000 Mathematical Subject Classification: 37J05, 53C22, 53C50, 53D12, 70H05.

We introduce a notion of a category with full projective functors. It encodes certain common properties of categories appearing in representation theory of Lie groups, Lie algebras and quantum groups. We describe the left or right exact functors which naturally commute with projective functors and provide a unified approach to the verification of relations between such functors.

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.

This paper continues the study of quantised function algebras $\mathcal{O}_{\epsilon}[G]$ of a semisimple group $G$ at an $\ell$th root of unity $\epsilon$. These algebras were introduced by De Concini and Lyubashenko in 1994, and studied further by De Concini and Procesi and by Gordon, amongst others. Our main purpose here is to increase understanding of the finite-dimensional factor algebras $\mathcal{O}_{\epsilon}[G](g)$, for $g \in G$. We determine the representation type and block structure of these factors, and (for many $g$) describe them up to isomorphism. A series of parallel results is obtained for the quantised Borel algebras $U_{\epsilon}^{\geq 0}$ and $U_{\epsilon}^{\leq 0}$.

2000 Mathematical Subject Classification: 16W35, 17B37.

We discuss which part of the rationalized algebraic $K$-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.

The definition of Thom polynomials for Lagrange, Legendre and critical point function singularities is given. The approach is based on the notion of classifying space of singularities. This approach provides a universal method for computing Thom polynomials. Characteristic classes of complex Lagrange and Legendre singularities of small codimension are computed. Two independent methods for these computations are presented. The first is the direct one based on the resolution of singularities, the Gysin homomorphism, and the notion of adjacency exponents. The second uses the existence theorem for Thom polynomials and is based on Rimányi's idea of using symmetries. The expressions for the resulting polynomials reduced modulo 2 agree with those obtained by Vassiliev for the real case.

We give a complete classification (up to unitary equivalence) of extensions of $C(X)$ by a separable simple AF-algebra $A$ with a unique trace (up to scalar multiples), where $X$ is a compact subset of the plane. In particular, we show that there are non-trivial extensions $\tau$ such that $[\tau]=0$ in ${\rm Ext}(C(X),A).$ A new index is introduced to determine when an extension is trivial. Extensions of $C(S^2)$ and other algebras are also studied. Our results work for a larger class of C*-algebras of real rank zero.

1991 Mathematics Subject Classification: primary 46L05,46L35; secondary 46L80.

It is shown that the system of two additive equations a_1 x_1^k + \ldots + a_s x_s^k = b_1 x_1^k + \ldots + b_s x_s^k =0 where $k \ge 2$ and $a_j$, $b_j$ are any given integers, has non-trivial solutions in all $p$-adic fields provided only that $s > 8k^2$. The constant 8 can be reduced when $k$ is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound $8k^2$ can be replaced by $2k^2$.

2000 Mathematical Subject Classification:11D72.

This paper establishes recognition theorems for the finite Lie-type groups defined in odd characteristic $p$. The hypotheses of these theorems are couched in terms of certain local data in the form of an amalgam which is called a weak $BN$-pair of rank 2. The groups to be recognized are completions of these amalgams which satisfy the condition of being of local characteristic $p$ (a condition satisfied by the finite Lie-type groups defined in characteristic $p$). The results presented will find application in one of the ongoing revisions of the finite simple group classification.

The purpose of this paper is to give a detailed study of the basic theory of C*-categories. The study includes some examples of C*-categories that occur naturally in geometric applications, such as groupoid C*-categories, and C*-categories associated to structures in coarse geometry. We conclude the paper with a brief survey of Hilbert modules over C*-categories.

2000 Mathematical Subject Classification: 18D99, 46L05, 46L08.

To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function $$ F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '), $$ where $\alpha$ is real and $T\ge 2$, $\gamma$ and $\gamma '$ denote the imaginary parts of zeros of the Riemann zeta-function, and $w(u) = 4/(4 + u^2)$. Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for $F(\alpha)$ when $|\alpha|\le 1$, and made the conjecture that $F(\alpha) = 1 + o(1)$ as $T\to \infty$ for any bounded $\alpha$ with $|\alpha |\ge 1$. In this paper we use an approximation for the prime indicator function together with a new mean value theorem for long Dirichlet polynomials and tails of Dirichlet series to prove that, assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions, then for any $\epsilon >0$ we have $$ F(\alpha) \ge {3\over 2} - |\alpha| - \epsilon ,$$ uniformly for $1\le |\alpha| \le \frac32 -2\epsilon $ and all $T \ge T_0(\epsilon)$.

1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.

In this paper a completely integrable Hamiltonian system on $T^*M$ is constructed for every Riemannian symmetric space $M$. We show that the solutions of this system correspond to the solutions of Nahm's equations for suitably chosen maps. Nahm's equations were introduced by Nahm as a rewriting of Bogomolny equations for magnetic monopoles.

We represent our system as a degeneration of a certain case of Hitchin's algebraically integrable system. We prove the complete integrability of our system by means of this representation.

Some concrete examples of our Hamiltonian system on $T^*M$ are described. When $M= S^n$, we obtain the classical system of C. Neumann. If the configuration space $M$ of our system is the $n$-dimensional hyperbolic space, we get the Minkowskian analogue of the C. Neumann system. Other examples that we describe are a many-body C. Neumann system, a spherical pendulum, and a spherical pendulum with an additional magnetic force.

Let $(W,S)$ be a finite Coxeter system and $B=B(W)$ the corresponding Artin-Tits braid group. The natural map $B \rightarrow W$ has a canonical section $r:W \rightarrow B$ defined by the condition that if $w \in W$ is written as a reduced expression in the generators in ~$S$ then $r(w)$ is the corresponding product taken in $B$. The main result of the present paper is as follows. Let $C$ be a conjugacy class in $W$ whose elements have order $d$ say. Then there exists an element $w \in C$ of minimal length in $C$ such that $r(w)^d$ is a product of terms of the form $r(w_I)^{d_I}$ where the following hold: $d_I$ is a non-negative even integer, $I$ runs over a sequence of subsets of $S$ which decreases (which implies that the terms commute), and $w_I$ is the longest element in the corresponding parabolic subgroup of $W$. Such an element will be called a ‘good’ element in $C$. The result is proved case by case, using the classification of irreducible finite Coxeter groups and the knowledge of representatives of minimal length from the article by Geck and Pfeiffer in Advances in Math. 102 (1993) 79-94.The main application of this result concerns the problem of calculating character values of Iwahori-Hecke algebras. The generic Iwahori-Hecke algebra $H$ associated with $(W,S)$ is a quotient of the group algebra of $B$ by the ideal generated by quadratic relations of the form $(s-q)(s+1)$ where $s \in S$ and $q$ is an indeterminate. Thus, $H$ is an algebra over a suitable field of rational functions in the variable ~$q$. The above result implies that if $w$ is a good element in the class $C$ of $W$, then the eigen values of the standard basis element $T_w$ of $H$ in an irreducible representation of $H$ are roots of unity times fractional powers of ~$q$, and the fractional powers occurring can be explicitly determined from the ordinary character table of $W$. This result is used to compute the character table of the Iwahori-Hecke algebra of type $E_8$. To determine the roots of unity, we use additional relations coming from the modular representation theory of ~$H$. This completes the program of determining the character tables of Iwahori-Hecke algebras.

1991 Mathematics Subject Classification: 20C20, 20F36.

Let $(X_n,d_n)$ be a sequence of finite metric spaces of uniformly bounded diameter. An equivalence relation $D$ on the product $\prod_n X_n$ defined by $\vec x\, D\,\vec y$ if and only if $\limsup_n d_n(x_n,y_n)=0$ is a {\em $c_0$-equality\/}. A systematic study is made of $c_0$-equalities and Borel reductions between them. Necessary and sufficient conditions, expressed in terms of combinatorial properties of metrics $d_n$, are obtained for a $c_0$-equality to be effectively reducible to the isomorphism relation of countable structures. It is proved that every Borel equivalence relation $E$ reducible to a $c_0$-equality $D$ either reduces a $c_0$-equality $D'$ additively reducible to $D$, or is Borel-reducible to the equality relation on countable sets of reals. An appropriately defined sequence of metrics provides a $c_0$-equality which is a turbulent orbit equivalence relation with no minimum turbulent equivalence relation reducible to it. This answers a question of Hjorth. It is also shown that, whenever $E$ is an $F_\sigma$-equivalence relation and $D$ is a $c_0$-equality, every Borel equivalence relation reducible to both $D$ and to $E$ has to be essentially countable. 2000 Mathematics Subject Classification: 03E15.

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:

(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);

(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).

Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.

2000 Mathematical Subject Classification: 14E15.