Geometry and Differential Equations (18/19)
Faculteit  Science and Engineering 
Jaar  2017/18 
Vakcode  WMMA14002 
Vaknaam  Geometry and Differential Equations (18/19) 
Niveau(s)  master 
Voertaal  Engels 
Periode  hele jaar 
ECTS  5 
Rooster  tweejaarlijks, niet in 2017/2018 
Uitgebreide vaknaam  Geometry and Differential Equations (tweejaarlijks 2018/2019)  
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 "onshell" 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 EulerLagrange (and reconstruct its action functional), and find Noether symmetries of a given EulerLagrange equation; 5. ability to calculate the generations of Noether identities for the equations of motion (e.g., for the YangMill models or Einstein's gravity equations) and construct the respective classes of gauge symmetries. 

Omschrijving  This is an introduction to the geometry of nonlinear 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 prerequisites), 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 
Presentatie (P)
(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  
Coördinator  prof. dr. H. Waalkens  
Docent(en)  prof. dr. H. Waalkens  
Verplichte literatuur 


Entreevoorwaarden  
Opmerkingen  
Opgenomen in 
