## OVERVIEW

Language: English

## AIMS AND CONTENT

LEARNING OUTCOMES

The purpose of the course is to provide an introduction to gauge theories. Specifically, after introducing the necessary notions of differential geometry (the theory of connections on vector and principal fibrations, Hodge theory), we will address some salient aspects of Yang-Mills theory on 4-dimensional Riemannian varieties, studying the structure Of the space module space.

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

PIERRE OLIVIER MARTINETTI (President)

NICOLA PINAMONTI

CLAUDIO BARTOCCI (President Substitute)

MARCO BENINI (President Substitute)

## LESSONS

TEACHING METHODS

Teaching style: In presence

LESSONS START

The class will start according to the academic calendar.

Class schedule

## EXAMS

EXAM DESCRIPTION

Oral.