Generalized Symplectic Geometries and the Index of Families of Elliptic Problems.

Intro -- Contents -- 1 Algebraic preliminaries -- 1.1 Clifford algebras -- 1.2 Clifford modules -- 2 Topological preliminaries -- 2.1 Karoubi's K[Sup(p,q)]-theory -- 2.2 Some elementary examples -- 2.3 Classifying spaces -- 3 (p,q)-lagrangians and classifying spaces for K-theory -- 3.1 (p,q)-lagrangian subspaces -- 3.2 Graphs of linear operators and lagrangians -- 3.3 The components of FL[Sup(p,q)] -- 3.4 The homotopy type of FL[Sup(p,q)] -- 4 Symplectic reductions -- 4.1 Generalized symplectic geometries -- 4.2 Homotopic properties of the symplectic reduction process -- 4.3 The generalized Maslov index -- 4.4 Comparison with the traditional Maslov index -- 5 Clifford symmetric Fredholm operators -- 5.1 The space F[Sup(p,q)] -- 5.2 Hamiltonian (p,q)-modules -- 5.3 The graph map and generalized Floer operators -- 5.4 Examples -- 6 Families of boundary value problems for Dirac operators -- 6.1 (p,q)-Dirac operators and their Calderon projections -- 6.2 The index of families of boundary value problems -- 6.3 The cobordism in variance of the families index -- 6.4 Gluing formulæ -- A. Gap convergence of linear operators -- B. Gap continuity of families of BVP's for Dirac operators -- C. Pseudodifferential Grassmanians and BVP's for Dirac operators -- D. The proof of Proposition 6.1 -- References.

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