OVERVIEW

Language: English

AIMS AND CONTENT

TEACHING METHODS

Teaching style: In presence

SYLLABUS/CONTENT

Geometric Methods in Mathematical Physics

Academic year 2013-2014; first semester

1. FIBRE BUNDLES, CONNECTIONS AND HOLONOMY GROUPS 

•Vector bundles and their operations; vector bundles with metric structure.

• Linear connections on vector bundles; curvature 2-form; Cartan’s strucure equations; Bianchi’s identity; generalized Levi-Civita connection.
• Principle bundles; fundamental vector fields.
• Connections on principal bundles; from vector bundles to principle bundle and back; group of gauge transformations
• Holonomy group; intrnsic torsion
• Classification of Riemannian holonomy gropus (statement of Berger's theorem and examples)
2. TOPICS IN RIEMANNIAN GEOMETRY
• Geodesics and parallel transport 
• Surfaces; "theorema egregium"; the Gauss-Bonnet theorem
• Hopf-Rinow's theorem
• Symmetric spaces
3. INTRODUCTION TO KÄHLER MANIFOLDS
• Introduction to complex manifolds
• Kähler manifold; the complex projective space
• Riemann surfaces; algebraic curves
4. INTRODUCTION TO HODGE THEORY 
• Differential operators on Riemannian manifolds
• The de Rham cohomology
• The Hodge theorem
• The Hodge decomposition theorem on compact Kähler manifolds
• ASD equations; instantons on S4.

TEACHERS AND EXAM BOARD

Exam Board

CLAUDIO BARTOCCI (President)

PIERRE OLIVIER MARTINETTI (President)

NICOLA PINAMONTI (President)

LESSONS

LESSONS START

September 26, 2016

Class schedule

TOPICS IN DIFFERENTIAL GEOMETRY

EXAMS

Exam schedule