We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation F of a compact Riemannian manifold M with coefficients in a leafwise U(p, q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated.

In this paper we study the problem of recovering the reflecting surface in a reflector system which consists of a point light source, a reflecting surface, and an object to be illuminated. This problem involves a fully nonlinear partial differential equation of MongeAmp`ere type, subject to a nonlinear second boundary condition. A weak solution can be obtained by approximation by piecewise ellipsoidal surfaces. The regularity is a very complicated issue but we found precise conditions for it.

We show that the first Johnson subgroup of the mapping class group of a surface  of genus greater than 1 acts ergodically on the moduli space of representations of π1() in SU2. Our proof relies on a local description of the latter space around the trivial representation and on the Taylor expansion of trace functions.

We define the quantum correction of the Teichmller space \mathcal {T} of Calabi-Yau manifolds. Under the assumption of no weak quantum correction, we prove that the Teichmller space \mathcal {T} is a locally symmetric space with the Weil-Petersson metric. For Calabi-Yau threefolds, we show that no strong quantum correction is equivalent to that, with the Hodge metric, the image \mathcal {T} of the Teichmller space \mathcal {T} under the period map \mathcal {T} is an open submanifold of a globally Hermitian symmetric space \mathcal {T} of the same dimension as \mathcal {T} . Finally, for Hyperkhler manifold of dimension \mathcal {T} , we find both locally and globally defined families of \mathcal {T} and \mathcal {T} -classes over the Teichmller space of polarized Hyperkhler manifolds.

We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.