Lie groups and pseudogroups actions: classical and differential invariants

Valentin Lychagin

University of Tromsø

Abstract:

The main purpose of these lectures is to present a short introduction to the world of classical and differential invariants. The focus will be on the symmetry groups and pseudo-groups of PDEs and algebras of their differential invariants. The main theorem in the differential invariants theory - Lie-Tresse theorem - will be discussed in details. We will explain the importance of certain kind of algebraicity in PDEs theory and outline algebro-differential methods for finding differential invariants as well as their syzygies and construction of factor equations. A number of concrete applications to various problems in algebra, geometry and differential equations shall be considered.

Prerequisites include the basics of the theory of Lie groups, their actions on manifolds and also familiarity with the geometry of the jet spaces. These are discussed in Section 1 of B.Kruglikov and V.Lychagin, " Geometry of differential equations " (Chapter 14 of Elsevier Handbook of Global Analysis, ed.by D.Krupka and D.Saunders, 2008). Familiarity with initial chapters of the book A.Kushner, V.Lychagin and V.Rubtsov, "Contact Geometry and Nonlinear Differential Equations" (Cambridge University Press, 2007) will be useful, but the topics from the geometry of jets will be reviewed.