Contemporary Mathematics, vol. 219, American Mathematical Society, 1998, 287 pages, ISBN: 0-8218-0828-1, ISSN: 0271-4132.

**M. Asorey, F. Falceto,**and**G. Luzón**-- Unstable bundles in quantum field theory.The relation between connections on 2-dimensional manifolds and holomorphic bundles provides a new perspective on the role of classical gauge fields in quantum field theory in two, three and four dimensions. In particular we show that there is a close relation between unstable bundles and monopoles, sphalerons and instantons. Some of these classical configurations emerge as nodes of quantum vacuum states in nonconfining phases of quantum field theory which suggests a relevant role for those configurations in the mechanism of quark confinement in QCD.

**G. Barnich**-- Brackets in the jet-bundle approach to field theory.In the first part, the sh Lie structure of brackets in field theory, described in the jet bundle context along the lines suggested by Gel'fand, Dickey and Dorfman, is analyzed. In the second part, we discuss how this description allows us to find a natural relation between the Batalin - Vilkovisky antibracket and the Poisson bracket.

**C. Becchi, S. Giusto,**and**C. Imbimbo**-- The BRST structure of twisted N = 2 algebra.The BRST structure of twisted N = 2 superconformal matter coupled to topological gravity is derived by gauging the rigid N = 2 superconformal algebra. This construction provides BRST transformations laws for which holomorphic factorization on the world-sheet is manifest.

**L. Bonora, C. S. Chu,**and**M. Rinaldi**-- Anomalies and locality in field theories and M-theory.We review some basic notions on anomalies in field theories and superstring theories, with particular emphasis on the concept of locality. The aim is to prepare the ground for a discussion on anomalies in theories with branes. In this light we review the problem of chiral anomaly cancellation in M-theory with a 5-brane.

**F. Brandt**-- Gauge covariant algebras and local BRST cohomology.The formulation of the local BRST cohomology on infinite jet bundles and its relation and reduction to gauge covariant algebras are reviewed. As an illustration, we compute the local BRST cohomology for geodesic motion in (pseudo-) Riemannian manifolds and discuss briefly the result (symmetries, constants of the motion, consistent deformations).

**M. Dubois-Violette**-- Generalized homologies for d^{N}= 0 and graded q-differential algebras.We discuss the generalized homology associated with a nilpotent endomorphism d such that d

^{N}= 0. We construct such d on simplicial modules and rely the corresponding generalized homologies to the usual simplicial ones. We also investigate the generalization of graded differential algebras in this context.**M. Fliess, J. Lévine, P. Martin,**and**P. Rouchon**-- Nonlinear control and diffieties, with an application to physics.Problems of nonlinear control theory are considered in the context of diffieties. As an application, the Dirac gauge theory is discussed.

**M. Henneaux**-- Consistent interactions between gauge fields: the cohomological approach.The cohomological approach to the problem of consistent interactions between fields with a gauge freedom is reviewed. The role played by the BRST symmetry is explained. Applications to massless vector fields and 2-form gauge fields are surveyed.

**N. Kamran**and**T. Robart**-- A compatible analytic manifold structure for Lie pseudogroups of infinite type.We show that any local analytic Lie pseudogroup of infinite type can be endowed with a compatible Silva analytic manifold structure. The compatibility condition means that the associated maximal isotropy Lie group endowed with the induced topology becomes an analytic Lie group in Milnor sense, i.e. an analytic manifold such that the group operations are analytic. In that context the second fundamental theorem of Lie is extended for the class of closed Lie subalgebras of the maximal isotropy Lie algebra. So any closed connected subgroup of the isotropy group is a Silva analytic Lie group. Moreover we prove that the group of local analytic paths starting at the identity transformation in any Lie pseudogroup inherits naturally of a Silva analytic Lie group structure in the previous sense.

Our approach treats the transitive and intransitive cases on the same footing and our results are shown to be valid in the wider classes of quasi-analytic transformations of Denjoy's or Gevrey's type. The Cartan - Kähler theorem is notably shown to be valid in the quasi-analytic setting.

**J. Krasil'shchik**-- Cohomology background in geometry of PDE.**N. Maggiore**-- Algebraic renormalization of massive supersymmetric theories.The algebraic BRS method of renormalization is applied to the supersymmetric Yang - Mills theories. The most general invariant counterterms and the supersymmetric extension of the Adler - Bell - Jackiw anomaly are explicitily found. In addition, masses are added in a simple way, preserving gauge invariance to all orders of perturbation theory while breaking supersymmetry ``softly'', in the sense of Girardello and Grisaru.

**G. Marmo**and**A. Ibort**-- A new look at completely integrable systems and double Lie groups.Completely integrable systems are discussed in the realm of the general conjugacy problem from the perspective offered by Lie - Scheffers theorem. Hamiltonian and non-Hamiltonian standard completely integrable systems are briefly reviewed as well as a natural generalization to the non-Abelian setting suggested by the theory of double Lie groups.

**V. Penna, M. Rasetti,**and**M. Spera**-- Quantum dynamics of 3-D vortices.**J. D. Stasheff**-- The (secret?) homological algebra of the Batalin-Vilkovisky approach.After a brief history of ``cohomological physics'', the Batalin - Vilkovisky complex is given a revisionist presentation as homological algebra, in part classical, in part novel. Interpretation of the higher order terms in the extended Lagrangian is given as higher homotopy Lie algebra and via deformation theory. Examples are given for higher spin particles and closed string field theory.

**A. Verbovetsky**-- Notes on the horizontal cohomology.This paper is devoted to the horizontal (``characteristic'') cohomology of systems of differential equations. Recent results on computing the horizontal cohomology via the compatibility complex are generalized. New results on the Vinogradov

*C*-spectral sequenceand Krasil ¢shchik's*C*-cohomologyare obtained. As an application of general theory, the examples of an evolution equation and a p-form gauge theory are explicitly worked out.**C. M. Viallet**-- Invariants of rational transformations and algebraic entropy.We introduce an index associated to any birational transformation of projective spaces. This index, which we call ``algebraic entropy'', is conjectured to measure the obstruction to the existence invariants of the map.

**A. Vinogradov**-- Introdution to secondary calculus.First we exhibit some basic notions and constructions of modern geometry of partial differential equations that lead to the concept of diffiety, an analogue of affine algebraic varieties for partial differential equations. Then it is shown how the differential calculus on diffieties which respects the underlying infinite order contact structure is self-organized into Secondary Calculus in such a way that higher symmetries of PDE's become secondary vector fields and the first term of the

*C*-spectral sequence becomes the algebra of secondary differential forms. Then the general secondarization problem is formulated. Its solution for modules and multi-vector-valued differential forms is proposed and the relevant homological algebra is discussed. Eventually, relations with gauge theories are briefly outlined at the end.**A. Vinogradov**and**M. Vinogradov**-- On multiple generalizations of Lie algebras and Poisson manifolds.The notion of (n,k,r)-Lie algebra (n > k ³ r ³ 0), an n-ary generalization of that of Lie algebra, is introduced and studied. The standard Lie algebras turn out to be (2,1,0)-Lie algebras. Two types of n-ary Lie structures studied in recent few years in the context of the Nambu and ``non-Nambu'' generalizations of dynamics correspond to (n,n-1,0)- and (n,1,0)- Lie algebras, respectively.

Using techniques of Frölicher - Nijenhuis brackets, we associate to any formally integrable equationEa cohomology theory H_{C}^{*}(E) (based on aC-complex) related to deformations of the equation structure on the infinite prolongationE^{¥}. A subgroup in H_{C}^{1}(E) is identified with recursion operators acting on the Lie algebra symEof symmetries. On the other hand, another subgroup of H_{C}^{*}(E) can be understood as the algebra of supersymmetries of the ``superization'' of the equationE. This passing to superequations makes it possible to obtain a well-defined action of recursion operators in a nonlocal setting. Relations to Poisson structures onE^{¥}are briefly discussed.

We give an extensive review of both the methods of approach and the available solutions to the problem of providing a complete quantum description of 3-D vortex dynamics. The leading technique is an appropriate form of geometric quantization based on current algebra, implemented in the framework of the Clebsch fluid description combined with the coadjoint orbit picture. We show how, in the ensuing quantum field theory for the vortex gas, the dynamical constants of motion identify with the topological invariants of the vortex considered as an unknotted link.