3-dimensional Cauchy-Riemann structures and 2nd order ordinary differential equations

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs

Finite type invariants of 3-manifolds

A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for manifolds with large first betti number, encompassing much of the complexity of Ohtsuki's theory for homology spheres. (For example, it is seen that the quantum SO(3) invariants, though not of finite type, are determined by finite type invariants.) The algebraic structure of the set of all finite type invariants is investigated, along with a combinatorial model for the theory in terms of trivalent "Feynman diagrams".Comment: Final version for publication, with figures. The most significant changes from the original posted version are in the exposition of section 3 (on the Conway polynomial) and section 4 (on quantum invariants

Third Order ODEs Systems and Its Characteristic Connections

We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, determine when a system of third order ODEs is trivializable. As application differential invariants of equations on circles in ${\mathbb R}^n$ are computed