Lefschetz Formulas for Singular Flows on Foliated Manifolds

In the talk, we discuss the Lefschetz formulas for flows on compact manifolds that preserve a foliation of codimension one. We start with the case when the orbits of the flow are everywhere transversal to the fibers of the foliation. Then we will dwell in more detail on the case when the flow has fixed points. An approach to the Lefschetz formulas for such flows will be described, based on the pseudo-differential b-calculus on manifolds with boundary developed by R. Melrose. The resulting Lefschetz formula contains eta-invariant-type terms associated with compact fibers of the foliation containing fixed points of the flow. We discuss some properties of such eta-invariants. The report is based on joint works with J. Alvarez Lopez and E. Leuchtnam.