Lectures are held in English or Italian, at the students' choice. The course is addressed to students in mathematics. but it can also be also attended by students in physics.
The purpose of the teaching is to provide an introduction to gauge theories. In particular, after having introduced the necessary notions of differential geometry (theory of connections on vector and principal bundles, Hodge theory), we will deal with some salient aspects of Yang-Mills theories on 4-dimensional Riemannian manifolds, arriving at the study of the structure local of the instanton module space.
The course aims at making students familiar with some differential-geometric methods and techniques which lie at the basis of physical "gauge theories".
Basics of differential geometry.
The course follows a traditional approach.
1. Vector bundles
2. Connections on vector bundles; curvature
3. Chern-Weil homomorphism; characteristic classes
4. Lie groups (overview)
5. Principal bundles
6. Connections on principal bundles; curvature; associated bundles
7. Differential operators on vector bundles elliptic and Fredholm operators
8. The Yang-Mills functional; gauge group; istantons
9. Local structure of the instanton moduli space
10. Kähler manifolds (overview); Hitchin-Kobayashi's correspondence
Detailed notes will be made available to students on the Aulaweb site.
Ricevimento: By appointment (email address: bartocci@dima.unige.it)
According to the academic calendar.
The examination consists of the preparation and presentation of a "seminar" on a specific topic chosen by the candidate in agreement with the teacher (the latter will provide all the necessary study materials).
The evaluation will take into account the overall quality of the presentation along with the student's ability to answer 2 questions related to her/his seminar topic (30-point grading scale; highest result: 30 e lode; pass result: 18)
For any further information please write an email to me at the address bartocci@dima.unige.it
