These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in that we have

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We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesski i's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups.

Let G be a reductive complex linear algebraic group, which in this paper for simplicity we shall always assume to be simple and simply connected, and let Lie(G) = g. This

Fix a prime number ℓ. In this paper we prove a conjecture [16, p. 300], which Ihara attributes to Deligne, about the action of the absolute Galois group on the pro-ℓ completion of the fundamental group of the thrice punctured projective line. It is stated below. Similar techniques are also used to prove part of a conjecture

We prove a condition for the existence of flat bundles on the punctured two-sphere with prescribed holonomies around the punctures, involving Gromov-Witten invariants of generalized flag varieties. This generalizes the case of special unitary connections described by Agnihotri and the second author [1] and Belkale [5].

We prove that for a simple simply connected quasi-split group of type 2, 3 the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant. It follows immediately from the result above that if cd F 2 (resp. vcd F 2) then Serre's Conjecture II (resp. the Hasse principle) holds for such a group. For a (C 2 )-field, in particular C(x, y), we prove the stronger result that Serre's Conjecture II holds for all (not necessary quasi-split) exceptional groups of type D 4 , E 6 , E 7 .

Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X, G); we will single out three of them for discussion here. First of all, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of G-valued singular cochains. Alternatively, one may regard H n (•, G) as a representable functor on the homotopy category of topological spaces, and thereby define H n rep(X, G) to be the set of homotopy classes of maps from X into an Eilenberg-MacLane space K(G, n). A third possibility is to use the sheaf cohomology H n sheaf (X, G) of X with coefficients in the constant sheaf G on X. If X is a sufficiently nice space (for example, a CW complex), then all three of these definitions agree. In general, however, all three give different answers. The singular cohomology of X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect H n sing (X, G) to tell us very much about X. Similarly, the cohomology group H n rep(X, G) is defined using maps from X into a simplicial complex, which (ultimately) relies on the existence of continuous real-valued functions on X. If X does not admit many real-valued functions, we should not expect H n rep (X, G) to be a useful invariant. However, the sheaf cohomology of X seems to be a good invariant for arbitrary spaces: it has excellent formal properties in general and sometimes yields

Abstract. Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two results of perhaps wider interest: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space.

We study the relationship between the period and the index of a principal homogeneous space over an abelian variety, obtaining results which, in particular, generalize work of Cassels and Lichtenbaum on curves of genus one. In addition, we show that the p-torsion in the Shafarevich-Tate group of a fixed abelian variety over a number field k grows arbitarily large when considered over field extensions l/k of bounded degree. Essential use is made of an abelian variety version of O’Neil’s period-index obstruction.