To quantize a classical system we have to consider the kinematic relation between the classical and quantum case: In the quantum case the states of a system are represented by the rays in a Hilbert space H and the observables by a collection of symmetric operators on H. In the classical case the state space is a symplectic manifold M and the observables are the algebra of smooth functions on M. The kinematic problem is: given M and its symplectic form is it possible to reconstruct the Hilbert space H and the symmetric operators? Geometric quantization gives a well defined procedure to construct the Hilbert space H and the operators corresponding to the classical observables. This procedure also satisfies the Dirac's quantum conditions. In this talk I will discuss these constructions in detail and the three stages of geometric quantization: pre-quantization, polarization and metaplectic correction.

Grothendieck's anabelian conjectures predict that the etale fundamental group is a fully faithful functor from certain anabelian schemes to profinite groups with Galois action, or equivalently that the maps between anabelian schemes are the same as maps between their etale homotopy types. This is analogous to an equivalence between fixed points and homotopy fixed points for Galois actions. We will discuss the anabelian conjectures and their topological analogues, and in the case of maps from the spectrum of a field, relate certain nilpotent obstructions to Massey products. As a corollary one has that the order n Massey product vanishes, where x denotes the image of x in k* under the Kummer map k* -> H^1(Gal(kbar/k), Z_l(1)).

Rigidity theory of circle diffeomorphisms, which concerns smooth conjugacy to a rigid rotation, is a classic problem in dynamical systems initiated by Arnol'd and settled by Herman and Yoccoz. We present complete renormalization and rigidity theory for circle maps with breaks, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity. We prove that renormalizations of any two C^{2+alpha}-smooth (alpha>0) circle maps with breaks, with the same irrational rotation number and the same size of the break, approach each other exponentially fast. As a corollary, we obtain a strong rigidity statement for such maps: for almost all irrational rotation numbers, any two circle maps with breaks, with the same rotation number and the same size of the break, are C^1-smoothly conjugate to each other. As we proved earlier, the latter result cannot be extended to all irrational rotation numbers. (This joint work with Kostya Khanin)