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## TOPICS IN DIFFERENTIAL GEOMETRY

CODE 44142 2017/2018 5 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA MAT/07 Italian (English on demand) GENOVA (Mathematics) 2° Semester This unit is a module of: AULAWEB

## 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

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

### TEACHING METHODS

Teaching style: In presence

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

TOPICS IN DIFFERENTIAL GEOMETRY

## EXAMS

Oral.

### Exam schedule

Date Time Location Type Notes
17/01/2018 10:00 GENOVA Orale
13/02/2018 10:00 GENOVA Orale
08/06/2018 10:00 GENOVA Orale
26/06/2018 14:00 GENOVA Orale
27/07/2018 10:00 GENOVA Orale
11/09/2018 14:00 GENOVA Orale