Event № 651
We consider ordinary differential equations of arbitrary order up to differentiable changes of variables. It turns out that starting from 2ndorder ODEs there exist continuous differential invariants that are preserved under arbitrary changes of variables. This was first discovered by Sophus Lie and explored in detail by A. Tresse for 2nd order ODEs. However, its was E. Cartan who first understood the geometric meaning of these invariants and related them to the projective differential geometry. We outline further advances in the equivalence theory of ODEs due to S.-S. Chern (3rd order ODEs) and R. Bryant (4th order ODEs) and present the general solution for arbitrary (systems of) ODEs of any order. It is based on the techniques of so-called nilpotent differential geometry and cohomology theory of finite-dimensional Lie algebras. It is surprising that a part of the invariants can be understood in purely elementary way via the theory of linear ODEs and leads to classical works of E.J.Wilczynsky back to the beginning of 20th century.