# TOPICS IN ADVANCED GEOMETRY B

Degree course:
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree:
2018/2019
Year:
1
Academic year in which the course will be held:
2018/2019
Course type:
Compulsory subjects, characteristic of the class
Credits:
8
Period:
Second semester
Standard lectures hours:
64
Detail of lecture’s hours:
Lesson (64 hours)
Requirements:

The calculus, and some analysis, in several real variables is needed from the very beginning of the course. It is also required that students have already had a thorough knowledge of the point-set topology and of the fundamental group of a topological space.

Final Examination:
Orale

Verification of learning will consist of two parts:

1) A 2 hours written exam where the student is asked to solve some of the exercises assigned during the course. These exercises have a sufficiently hight degree of complexity to verify if the student has acquired both the ability to analyze concrete situations and the ability to make autonomous reasoning towards the deduction of more abstract properties;

2) A traditional oral exam, during which the student will have to show that she/he has acquired the basic notions and the proofs of the main theorems.

Assessment:
Voto Finale

The purpose of the teaching is to let the student acquire, via both abstract results and many concrete examples, the basic concepts from the theory of n-dimensional differentiable manifolds. These are the natural spaces where the notion of differentiability of maps can be introduced and where tools from Analysis can be extended and developed.
The course is intended as preparatory for different areas of mathematics such as Differential Topology, Riemannian Geometry, Analytic Mechanics and Smooth Dynamical Systems. At the end of the course we expect that:

1) The student has acquired the main notions and the fundamental theorems of the theory of differentiable manifolds, of the smooth maps between these spaces and of the vector bundles and theirs sections, with a special emphasis on the tangent bundle and the dynamics of vector fields'

2) On the base of the proofs illustrated during the lectures, the student is able to make by him/herself reasonings of medium complexity that lead him/her to deduce abstract properties of the above mentioned objects;

3) The student is able to investigate the main properties of the objects alluded to above in concrete situations.

Schematically, the main topics of the course can be described as follows:

1) Topological manifolds
2) Differentiable structures on topological manifolds
3) Smooth functions and partition of the unit
4) Tangent space and differential of a smooth map
5) Tangent bundle and vector fields
6) Smooth coverings and smooth actions of discrete groups
7) The inverse function theorem, the implicit function theorem and the rank theorem
8) Immersions, regular submanifolds and embeddings
9) Zero measure and Sard theorems
10) The Whitney embedding theorem

Textbooks

1) J. M. Lee. Introduction to Smooth Manifolds. Graduate Texts in Mathematics, 218. Springer.

2) W. M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry. Volume 120, Pure and Applied Mathematics. Academic Press.

Further teaching support

1) Lecture notes written by the teacher on the main topics of the course

2) Exercises of medium-hight complexity to be solved at home.

Final examination

Written examination based on homeworks and oral examination.

The teaching method will consist in frontal lectures. Exercises both of abstract nature and on concrete examples will be assigned regularly

Office hours: by appointment

PIGOLA STEFANO