Mathematical Methods in Classical Mechanics and Supergeometry

Chapters

1- Fiber bundles,

2- Poisson, symplectic and twisted Poisson structures,

3- Marsden-Weinstein reduction, generalizations, and applications,

4- Lie algebroids, Dirac algebroids,

5- Lie groups and Poisson--Lie groups,

6- Lie bialgebras,

7- Integrable systems and dressing transformations,

8- Supermanifolds, supertangent bundles, supercotangent bundles, supertangent morphisms, superdifferential forms,

9- Super mechanics, super-Hamilton equations, supersymmetric models.

Evaluation

Students are evaluated against two criteria:

I) Participation during lectures and fleshing out, at least, one of the chapters of the script of the subject (60%). In particular, each chapter will contain a series of topics/examples/extra proofs to be developed. The student has to work on five of them, which accounts for the 50% of the final mark. The remainig 10% concerns the corrections or small improvements in the whole script, or questions made during the course. In particular, students are highly recommended to propose possible topics of their interest that will extend themes treated in the subject. Students must declare the topics to be developed on May 25th and works must be sent by June, 25th. It is expected that answers will last up to 5 pages long.

II) Presentation of one of the topics of the subject at the end of the course (40%). Due to coronovirus, it is not longer possible to accomplish this part of the subject by giving a lecture in a student conference, as initially proposed. The student shall present a chapter of the script at the end of the subject instead. In case this will be still impossible due to coronovirus, a conversation on Skype (or anything of the short) concerning the topics of one chapter of the script is expected.

Lectures

Due to coronovirus, the lecturer will make available the topics to study on each Wednesday morning.

I am looking for solutions to restore our lectures in a remote way. I expect to have news soon.

Literature

P.M. Gadea, J.M. Masque, Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers, Springer.

J. Gancarzewicz, Zarys współczesnej geometrii różniczkowej, Script.

V. Guillemin, S. Sternberg, Symplectic methods in physics, Cambridge University Press, New York, 1990.

J.M. Lee, Manifolds and differential geometry, Graduate Studies in Mathematics 107, American Mathematical Society.

R. Abraham, J.E. Marsden, T. Ratiu, Foundations on Mechanics, AMS, Providence, 2008.