A) Loring W. Tu, An Introduction to Manifolds, Universitext, 2011
Springer-Verlag New York
Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups
Graduate Texts in Mathematics, 94, 1983
Springer-Verlag New York
B) James Munkres, Topology, 2nd Edition, Pearson, 2000 (o mettiamo la prima edizione del
75)
C) Raoul Bott, Loring W. Tu, Differential Forms in Algebraic Topology,
Graduate Texts in Mathematics, 82, 1982, Springer-Verlag New York
D) I.M. Singer, J.A. Thorpe,Lecture Notes on Elementary Topology and Geometry,
Undergraduate Texts in Mathematics, 1967, Springer-Verlag New York
E) Loring W. Tu, Differential Geometry
Connections, Curvature, and Characteristic Classes
Graduate Texts in Mathematics, 275, 2017
Springer International Publishing
Learning Objectives
We aim at giving some advanced notions in differential topology and geometry; these concepts might be then useful in approaching and developing further more specific arguments in theoretical and mathematical physics.
Prerequisites
Linear algebra, some notion of general topology, calculus.
Teaching Methods
Lecturing using the blackboard with the students active participation.
Type of Assessment
The final exam is oral. The aim is to assess the knowledge of the basic notions given in the course, as well as the ability of the student in applying these to the solution of simple geometric problems.
Course program
Introduction to diferentiable manifolds and Lie groups.
Multilinear algebra.
Partitions of unity.
Algebra of differential forms and differntial of a form.
DeRham complex of a differentiable manifold, inclusive of compact support forms.
The Mayer-Vietoris sequence.
Orientation, integation and Stokes theorem.
Homotopy.
Poincaré Lemma and homotopy invariance.
Simplicial homology and de Rham Theorem.
Principal fibre bundles: fundamental examples and definition, transition functions, sections and local triviality. Classification of principal fibre bundles with group U(2), SU(2) and SO(3) over n-dimensional spheres; sketch of the classification of G-principal bundles over a manifold M via H^1(M,G).
Associated bundles: definition and basic examples, vector bundles, gauge group.
Kunneth formula and Leray-Hirsh theorem (without proof).
Connections on bundles: defintions and basic examples, connection forms, horizontal lifts and parallel transport.
Curvature form and its geometric meaning.
Holonomy. Characteristic classes.
The Young-Mills functional.