Stanislav Dubrovskiy

Abstract:   In this thesis we investigate the moduli spaces of two geometric structures: the symmetric connection and Fedosov structure.

We consider the action of the group of origin-preserving diffeomorphisms on the space of germs of generic symmetric connections at a point. The resulting moduli space gives rise to a Poincaré series. We calculate the series using information from the corresponding moduli spaces of k-jets of symmetric connections.

We then apply this technique to Fedosov structure. Fedosov manifold is a natural generalization of Kähler manifold. There is a canonical deformation quantization for Fedosov manifolds.

Poincaré series of both structures are shown to be rational functions, just as the ones given by a finite number of functional invariants.

This confirms the long-standing "finiteness" assertion, that algebras of invariants of "natural" differential-geometric structures are finitely generated.