OVERVIEW

The course is addressed at second year students and aims at providing some basic notions of Algebraic Topology and Differential Geometry.

AIMS AND CONTENT

LEARNING OUTCOMES

The first part of the course is an introduction to Algebraic Topology. The first elements of homotopy theory are introduced, with the aim of defining the fundamental group of a topological space. In the second part the students will face some basic notions of Differential Geometry, by studying curves and surfaces in the real three-dimensional space.

AIMS AND LEARNING OUTCOMES

The course aims at providing the necessary theoretical and operative knowledge for the understanding and solving of the problems in the following topics: notions of cathegory theory and functors, homotopy classes of functions and topological spaces, fundamental groups of topological spaces, study and characterisation of a parametrised curve in R^n, study and characterisation of a surface in R^3.
At the end of the course the student will be able to recall and present the theoretical notions she will have faced during the lectures. Moreover, by applying the algorithms and the solving technique introducted in the course and justifying every step, she will be able to:
- compute the fundamental group of a given topological space;
- distinguish different topological spaces by means of suitable invariants;
- study and characterise a parametrised curve in R^n, with particular focus on curves in R^3;
- study and characterise a regular surface in R^3, also by means of gaussian curvature, sectional curvatura, geodesics.

PREREQUISITES

Notions of linear algebra and analytic geometry. Notions of general topology. Notions of differential calculus (partial derivatives, integrals).

SYLLABUS/CONTENT

The following topics will be dealt with from both a theorical and operative point of view.
Notions of cathegory theory and functors.
Homotopy classes of functions and topological spaces.
Fundamental groups.
Seifert-Van Kampen Theorem.
Curves in R^n and R^3: curvature, torsion, Frenet-Serret frame.
Regular surfaces in R^3: fundamental forms, Gauss map, curvatures, Theorema Egregium.

The following additional topics may be introduced.
Classifications of topological surfaces.
Gauss-Bonnet Theorem.

RECOMMENDED READING/BIBLIOGRAPHY

1. Manetti, Topologia.
2. do Carmo, Differential Geometry of Curves and Surfaces.
3. Kosniowski, Introduzione alla topologia algebrica.
4. Massey: A basic Course in Algebraic Topology.
5. Abate, Tovena, Curve e superfici.
Other books or notes may be added during the semester.

TEACHERS AND EXAM BOARD

Exam Board

FABIO TANTURRI (President)

MATTEO PENEGINI

ELEONORA ANNA ROMANO (President Substitute)

EXAMS

EXAM DESCRIPTION

Written exam followed by an oral exam

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools

ASSESSMENT METHODS

The written part will consist in the resolution of exercises on the topics of the teaching programme (see also the Aims and Learning Outcomes section). Through the resolution of the exercises the student will be evaluted in:
- her skills in identifying the theoretical and practical results which are necessary to approach and solve the problems, and general knowledge of the very same results;
- her skills in applying the suitable algorithms and procedures to solve the exercises;
- her skills in providing the right arguments and justifications for the involved steps she follows.

During the oral part the teacher will evaluate the student's argumentation skills. General knowledge which has not positively emerged during the written part will be also assessed.

Exam schedule