Geometry and Differential Equations (22/23)

Faculteit Science and Engineering
Jaar 2021/22
Vakcode WMMA017-05
Vaknaam Geometry and Differential Equations (22/23)
Niveau(s) master
Voertaal Engels
Periode semester II a
ECTS 5
Rooster tweejaarlijks, niet in 2019/2020

Uitgebreide vaknaam Geometry and Differential Equations (tweejaarlijks 2022/2023)
Leerdoelen At the end of the course, the student 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.
Omschrijving This course in geometry of differential equations is oriented equally towards mathematicians and physicists.
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).

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.
Uren per week
Onderwijsvorm Hoorcollege (LC), Werkcollege (T)
Toetsvorm Schriftelijk tentamen (WE)
(If Exam grade is at least 4.5, Homeworks count; otherwise not (so that final grade = Exam grade). max (100% final exam, 60% final exam + 40% homeworks))
Vaksoort master
Coördinator A.V. Kiselev
Docent(en) A.V. Kiselev
Verplichte literatuur
Titel Auteur ISBN Prijs
Geometrical methods in the theory of ordinary differential equations. Arnol'd V.I. (1988) 0-387-96649-8
The geometry of Physics: an introduction (CUP, Cambridge, 1997, revised 2001) Frankel Th. (2012) 978-1-107-60260-1
Lecture notes:The twelve lectures in the (non)commutative geometry of differential equations (Part I), on-line IHES/M-12-13, 140 pages. A. V. Kiselev (2012)
Applications of Lie groups to differential equations. 2nd ed. Graduate Texts in Mathematics, 107. Springer-Verlag, NY. xxviii + 513 pp.
Olver P.J. (1993) 0-387-94007-3
Entreevoorwaarden On the math side, knowledge of real analysis, differential geometry (manifolds, vector bundle),.ODE and PDE. Familiarity with group theory (including Lie groups and algebras), de Rham cohomology, and functional analysis would be helpful --- still not compulsory. About PDE, any standard book is enough: e.g., Olver P.J. (2014) Introduction to partial differential equations.
On the physics side, knowledge of classical mechanics (Lagrangian and Hamiltonian formalisms), classical field theory (Maxwell, Yang-Mills and/or Einstein gravity equations) would be advisable but not compulsory.
Opmerkingen This course was registered last year with course code WMMA14002
Opgenomen in
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