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## TOPICS IN DIFFERENTIAL GEOMETRY

CODE 44142 2022/2023 6 cfu during the 2nd year of 9011 MATEMATICA(LM-40) - GENOVA 6 cfu during the 1st year of 9011 MATEMATICA(LM-40) - GENOVA MAT/07 GENOVA 2° Semester 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

MARCO BENINI (President Substitute)

## LESSONS

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

TOPICS IN DIFFERENTIAL GEOMETRY

Oral.