Real and complex differential geometry. Analysis on manifolds. Lie groups, Lie algebras and linear representations. Action of Lie groups on manifolds. Classification and geometry of Riemann surfaces.
The text books will be suggested according to the different subjects to treat or to be deepen.
Learning Objectives
Knowledge acquired: All what is specified in the course contents (see below)
Competence acquired: Notions of manifold and symmetry as fundamental concepts in the development of modern mathematics and its applications.
Skills acquired (at the end of the course):
Possibility of establishing a rigorous framework and of understanding the foundations of physical theories defined on curved spaces acted upon by symmetry transformations.
Prerequisites
Courses to be used as requirements (required and/or recommended)
Courses required: All the fundamental teachings of the three year laurea course.
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...):
150
Hours reserved to private study and other individual formative activities:
Contact hours for: Lectures (hours): 50
Further information
Office hours:
Upon appointment with the students
Type of Assessment
The final test is an oral test upon the arguments treated in the course.
Course program
Elements of real and complex differential geometry: differentiable and holomorphic manifolds, vector fields, differential forms and calculus of differential forms. Elements of real and complex analysis on manifolds. Fibered structures and vector bundles. Derivatives of vector fields. Lie groups, Lie algebras and their linear representations. Representations of finite and compact groups. Geometry of Lie groups and their action on differentiable manifolds. Introduction to the theory of Riemann surfaces, their classification and their geometry.