Introduction to Geometry of Differential Equations and its Applications

Differential equations can be studied from a purely geometric point of view, translating many constructions from finite-dimensional differential geometry into their language. This approach helps to clarify such notions as symmetries, conservation laws, presymplectic structures, and others. However, a number of questions arise in this framework whose answers are either incomplete or currently unknown. In particular, the problem of defining the cotangent equation in terms of the intrinsic geometry of PDEs remains open. This problem is directly related to the Hamiltonian formalism for differential equations.

From an applied perspective, methods for constructing exact solutions of differential equations are of particular interest. One of the most powerful approaches is based on the study of solutions invariant under certain symmetries of the given equation. A question of practical importance in this context is how the systems describing such invariant solutions inherit geometric structures from the original system.

In this talk, I will explain how these two topics are brought together within a reduction mechanism, which in particular clarifies how Hamiltonian operators are inherited by systems describing solutions of a given equation that are invariant under some of its symmetries. To fully implement this mechanism, an interpretation of cotangent equations in intrinsic geometric terms is also required. This can be achieved in the case where the reduced system turns out to be finite-dimensional.