Abstract: The invariant which I will discuss is a number associated with a closed Riemannian manifold $(M,g)$, a closed one form $\omega,$ and a tangent vector field with isolated singularities on $M$. A number of very interesting geometric situations create such data as I will show in this lecture.

It is "strange" because is defined by a (most of the time) divergent integral which, fortunately, can be regularized using geometry.

It is important in topology because it expresses the difference between some invariants defined analytically (with a Riemannian metric) and combinatorially (with a triangulation) like for example Ray Singer torsion and Reidemeister torsion.

It also permits to differentiate between the (less familiar but important) topological structures known as Euler structures.

The invariant is based on a characteristic form defined by Chern and rediscovered by Mathai Quillen, is implicit in the work of Bismut - Zhang on torsion and plays an important role in my recent work with S. Haller (on closed trajectories of some class of vector fields).