Geometry & Differential Equations (14/15)

Faculteit Science and Engineering
Jaar 2014/15
Vakcode WMMA14002
Vaknaam Geometry & Differential Equations (14/15)
Niveau(s) master
Voertaal Engels
Periode semester II a
ECTS 5
Rooster rooster.rug.nl

Uitgebreide vaknaam Geometry & Differential Equations (tweejaarlijks 2014/2015)
Leerdoelen The competences of a graduate in this course include:
1. ability to 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. ability to calculate the classical and higher symmetries of a given PDE system and find its invariant solutions;
3. ability to find the generating sections of conservation laws and reconstruct the conserved currents by using the homotopy;
4. ability to derive the equations of motion from a given action functional , inspect whether a given PDE system is manifestly Euler-Lagrange (and reconstruct its action functional), and find Noether symmetries of a given Euler-Lagrange equation;
5. ability to calculate the generations of Noether identities for the equations of motion (e.g., for the Yang-Mill models or Einstein's gravity equations) and construct the respective classes of gauge symmetries.
Omschrijving This is an introduction to the geometry of non-linear partial differential equations that arise in many models of mathematical physics. The course is designed equally for mathematicians and physicists. Its topic is closely related to (but not only and not viewing those as pre-requisites), e.g., "Symmetry in Physics", "Analysis on Manifolds", "Riemannian geometry", "General relativity", "Group theory", or "Representations of Lie groups". In particular, the content of this course is independent from the material of "QU Geometry & topology 2013/14" (in fact, the two modules will alternate).

The aim of this course is to introduce the geometry of jet spaces, in which differential equations are realised as surfaces, and to explain the notions of (i) symmetries that take solutions to solutions, (ii) conservation laws that can effectively be calculated via advanced algebraic techniques, (iii) First Noether's Theorem that associates conserved quantities with symmetries of the action functional in a given model, and (iv) Second Noether's Theorem that produces gauge symmetries from differential identities satisfied by the equations of motion. Of course, there remains much to be discovered in this vast domain of science and its applications in pure mathematics and theoretical physics.
Uren per week
Onderwijsvorm Hoorcollege (LC), Werkcollege (T)
Toetsvorm ~mondelinge presentatie
(Students will report on one or several topics typically, a small research question that will have been assigned during lectures; a student can also report on his/her individual supervised research if it pertains to this course and only if approved by the lecturer)
Vaksoort master
Docent(en) A.V. Kiselev
Verplichte literatuur
Titel Auteur ISBN Prijs
"The twelve lectures in the (non)commutative geometry of differential equations" (Part I, 2012), on-line IHES/M-12-13, 140 pages. A.V.Kiselev
Entreevoorwaarden
Opmerkingen
Opgenomen in
Opleiding Jaar Periode Type
MSc Applied mathematics  (Courses offered locally at the university of Groningen) - semester II a keuze
MSc Astronomy  (Optional Courses in Theoretical and Observational Astronomy) - semester II a keuze
MSc Mathematics  ( Courses offered locally at the university of Groningen) - semester II a keuze
MSc Physics  (Optional Courses in Quantum Universe) - semester II a keuze
Tweejaarlijkse vakken  (Even jaren) - semester II a -