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TOPICS IN DIFFERENTIAL GEOMETRY

## OVERVIEW

## AIMS AND CONTENT

### LEARNING OUTCOMES

### TEACHING METHODS

### SYLLABUS/CONTENT

## TEACHERS AND EXAM BOARD

### Exam Board

## LESSONS

### TEACHING METHODS

### LESSONS START

### Class schedule

## EXAMS

### EXAM DESCRIPTION

CODE | 44142 |
---|---|

ACADEMIC YEAR | 2020/2021 |

CREDITS | 5 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA |

SCIENTIFIC DISCIPLINARY SECTOR | MAT/07 |

LANGUAGE | Italian (English on demand) |

TEACHING LOCATION | GENOVA (Mathematics) |

SEMESTER | 2° Semester |

MODULES | This unit is a module of: |

TEACHING MATERIALS | AULAWEB |

Language: English

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 style: In presence

**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.

PIERRE OLIVIER MARTINETTI (President)

NICOLA PINAMONTI

CLAUDIO BARTOCCI (President Substitute)

MARCO BENINI (President Substitute)

Teaching style: In presence

The class will start according to the academic calendar.

Oral.