Prerequisites
The course is designed equally for mathematicians and physicists.
On the mathematics side, bachelor-level knowledge of real analysis, differential geometry (manifolds, basic understanding of the notion of vector bundle), ordinary and partial differential equations. Familiarity with group theory (including Lie groups and algebras), rings and fields, de Rham cohomology, and functional analysis would be helpful --- still not compulsory if a student prefers to try examples before doing the abstract theory.
About PDE, any standard / recommended undergraduate textbook is enough: e.g., without loss of generality,
Olver P.J. (2014) Introduction to partial differential equations. Undergraduate Texts in Mathematics, Springer, Cham. xxvi + 635 pp. ISBN: 978-3-319-02098-3
Almost all of the compulsory math background is found or referred to in Chapter 1 in the book:
Olver P.J. (1993) Applications of Lie groups to differential equations. 2nd ed. Graduate Texts in Mathematics, 107. Springer-Verlag, NY. xxviii + 513 pp. ISBN: 0-387-94007-3.
On the physics side, knowledge of classical mechanics (Lagrangian and Hamiltonian formalisms), classical field theory (Maxwell, Yang-Mills and/or Einstein gravity equations), and acquaintance with the various (non)linear equations of mathematical physics (KdV, Boussinesq, Burgers, KP, Monge-Ampere, minimal surface equation) would suggest what is actually studied in this course. This `physics' background is advisable but not compulsory.
Preliminary knowledge of categories and functors, finite- and infinite-dimensional integrable systems, Lie algebroids and groupoids, quantum mechanics, QFT, and Batalin-Vilkovisky quantisation, as well as computer programming skills, are not presumed.
Aim of the course
By understanding the geometry of jet bundles in which differential equations are submanifolds, we shall employ the Lie theory to solve and classify nonlinear ODE and PDE systems, propagate known exact solutions to families, effectively find conservation laws that constrain every solution of a given PDE, and relate symmetries of the action to conserved currents (1st Noether Theorem) and gauge symmetries to differential relations between the equations (2nd Noether Theorem).
A graduate in this course is able to:
1. calculate the prolongations of a given PDE system, inspect its formal (non)integrability, and operate "on-shell" by virtue of the equation and its differential consequences;
2. calculate the classical and higher symmetries of a given PDE system and find its invariant solutions;
3. find the generating sections of conservation laws and reconstruct conserved currents by using the homotopy;
4. derive the equations of motion from a given action functional, inspect whether a given PDE system is manifestly Euler-Lagrange (and then reconstruct its action functional), and find Noether symmetries of a given Euler-Lagrange equation;
5. calculate generations of the Noether identities for equations of motion (e.g., for the Yang-Mill models or Einstein gravity equations) and construct the respective classes of gauge symmetries.
Remark
Getting familiar with the geometry of differential equations offers ample opportunities for development and implementation of software algorithms and packages for symbolic calculations, allowing an effective search for symmetries, conserved currents, recursions, Backlund transformations, analytic solutions, and classifications of PDE.
- Docent: Arthemy Kiselev