Equivariant K Theory C Algebras And Localization At The Standard Trace

Equivariant K-theory, C*-algebras and Localization at the Standard Trace

This dissertation presents a generalization of Atiyah's L2-index theorem to equivariant elliptic operators on manifolds with a proper, cocompact discrete group action. Instead of demanding that the action be free as Atiyah did, we put a restriction on the symbol classes of the operators. These symbol classes are studied in the framework of C*-algebraic K-theory, and in the topological K-theory of Lück-Oliver. We show that, when the symbol class lies in the "tau-part" of the C*-algebraic equivariant K-theory, the L2-index is equal to the orbifold index, and we explain how this generalize Atiyah's theorem. The "tau-part" of C*-algebraic equivariant K-theory was introduced recently in the works of Antonini, Azzali and Skandalis. We determine a topological counterpart of their construction, and then present an explicit realization using genuine equivariant vector bundles. The central elements to this work are equivariant vector bundles which are appropriately stable under tensor-square operation, and whose isotropy representations are multiples of the regular representations.

Mathematical Reviews

Towards Non-Abelian P-adic Hodge Theory in the Good Reduction Case

The author develops a non-abelian version of $p$-adic Hodge Theory for varieties (possibly open with ``nice compactification'') with good reduction. This theory yields in particular a comparison between smooth $p$-adic sheaves and $F$-isocrystals on the level of certain Tannakian categories, $p$-adic Hodge theory for relative Malcev completions of fundamental groups and their Lie algebras, and gives information about the action of Galois on fundamental groups.