Geometry 2

A.Y. 2020/2021
Overall hours
Learning objectives
The aim of this course is to introduce some basic concpets from differential geometry, including Lie groups and principal fibrations, which are essential in some parts of theoretical physics. In the last part of the course we introduce De Rham cohomology and complex manifolds.
Expected learning outcomes
At the end of the course, the student will have acquired the following skills:

1) knows how to use the concepts and methods of differential geometry
2) will be able to understand and use vector fields and differential forms
3) will be able to understand and use properties of Lie groups (left invariant vector fields, the Lie algebra and the canonical one form)
4) will be able to understand principal fibrations and connections
5) will be able to understand and use De Rham cohomology (Poincare' Lemma, the Mayer-Vietoris sequence, cohomology groups of spheres and projective spaces)
Course syllabus and organization

Single session

Lesson period
Second semester
The lectures will be by video (not live), and they will be put on "Ariel" shortly before the scheduled lectures. Occasionally there will be meetings on Skype or Zoom or Teams for further information and to discuss any questions the students may have.
Course syllabus
Differentiable manifolds: smooth maps and functions, tangent space and fibration, Lie groups, Lie algebras, differential forms, the cotangent bundle of a Lie group and the Maurer-Cartan equations, principal fibrations and connections with applications to Gauge theory.
De Rham cohomology: the mayer-Vietoris sequence, the Poincare' lemma examples.
Integration on manifolds: orientation, Stokes' theorem. Complex manifolds: complex structures and basics of Kähler manifolds.
Prerequisites for admission
Bachelor degree in physics.
Teaching methods
Traditional lectures at the black board.
Teaching Resources
M. Abate, F. Tovena, Geometria Differenziale. Springer Verlag 2011.
D. Huybrechts, Complex geometry, an introduction. Berlin Springer-Verlag 2005.
G.L. Naber, Topology, Geometry, and Gauge Fields. Interactions. Springer Verlag 2000.
G.L. Naber, Topology, Geometry, and Gauge Fields. Foundations. Springer Verlag 1997.
C. Taubes, Differential Geometry. Oxford University Press 2011.
L.W. Tu, An introduction to manifolds. Springer 2011.
Assessment methods and Criteria
The final examination consists of an oral exam on appointment. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding the matter covered in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 6
Practicals: 24 hours
Lessons: 28 hours