We construct an elliptic analogue of Sklyanin‘s separation of variables for the sl(2) Gaudin system, using an adaptation of Drinfeld‘s Radon transformations.

We consider the moduli space $M$ of stable principal $G$-bundles over a compact Riemann surface $C$ of genus $g ⩾ 2, G$ being any reductive algebraic group and give an explicit description of the generic fibre of the Hitchin map $H:T^*M→ K$. If $T ⊂ G$ is a fixed maximal torus with Weyl group $W$, for each given generic element $&phis; ∈ K$ one may construct a $W$-Galois covering $&Ctilde;$ of $C$ and consider the generalized Prym variety $P=Hom_W(X(T),J(&Ctilde;))$, where $X(T)$ denotes the group of characters on $T$ and $J(&Ctilde;)$ the Jacobian. The connected component $P_0 ⊂ P$ which contains the trivial element is an abelian variety. In the present paper we use the classical theory of representations of finite groups to compute dim $P = dim M$. Next, by means of mostly elementary techniques, we explicitly construct a finite map $F$ from each connected component $H^-1(&phis;)_c$ of the Hitchin fibre to $P_0$ and study its degree. In case $G=PGl(2)$ one has that the generic fibre of $F:H^-1$()$_c → P_0$ is a principal homogeneous space with respect to a product of $(2d-2)$ copies of $Z/2Z$ where $d$ is the degree of the canonical bundle over $C$. However if the Dynkin diagram of $G$ does not contain components of type $B_l, l⩾ 1$ or when the commutator subgroup $(G,G)$ is simply connected the map $F$ is injective.

Let A and x be d × d and d × 1 real matrices. We study the asymptotic distribution of the sequence ((A$^n$x) mod.Z$^d$) in the torus $T^d$. We prove that for any A, for almost all x, this distribution exists; we characterize the case where this distribution is uniform; we give a description of the non uniform case. Finally we ask a question on the asymptotic distribution modulo 1 of the coefficients of the sequence of powers (A$^n$).

For manifolds M,M‘ of the form S$^2$ ∪ e$^4$ ∪ e$^6$ we compute the homomorphisms H$_*$M → H$_*$M‘ between homology groups which are realizable by a map F: M → M‘.

We develop a ’higher genus‘ analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor $F$ on the category of modular operads, the Feynman transform, which generalizes Kontsevich‘s graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick‘s theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.