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: tolopological surfaces and their classification, 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:
- classify a topological surface from a given polygon representing it;
- 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).

TEACHING METHODS

Theoretical lectures and exercises sessions

SYLLABUS/CONTENT

The following topics will be dealt with from both a theorical and operative point of view.
Classifications of topological surfaces.
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.

Gauss-Bonnet Theorem might also be mentioned.

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

LESSONS

LESSONS START

February 23rd 2026

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written exam followed by oral exam. The result of the written exam contributes to the final grade and can be adjusted based on the outcome of the oral exam, either positively or negatively. A midterm written exam is scheduled during the semester.

Students with disabilities or specific learning disorders (DSA) are reminded that, in order to request exam accommodations, they must first upload their certification to the University’s online services portal at servizionline.unige.it under the “Students” section. The documentation will be verified by the University’s Inclusion Services Office for students with disabilities and DSA.

Afterwards, well in advance (at least 7 days) before the exam date, students must send an email to the professor responsible for the exam, copying both the School’s Inclusion Contact Professor for students with disabilities and DSA (sergio.didomizio@unige.it) and the aforementioned Inclusion Services Office. The email must include the following information:

• Course name
• Date of the exam session
• Student’s last name, first name, and student ID number
• The compensatory tools and dispensatory measures deemed useful and being requested

The Inclusion Contact Professor will confirm to the exam instructor that the student is entitled to request accommodations and that such accommodations must be agreed upon with the instructor. The instructor will then reply, indicating whether the requested accommodations can be granted.

Requests must be submitted at least 7 days before the exam date to allow the instructor sufficient time to review them. In particular, if the student intends to use concept maps during the exam (which must be much more concise than those used for studying), failure to meet the submission deadline will result in insufficient time to make any necessary revisions.

For more information on requesting services and accommodations, please refer to the document: Linee guida per la richiesta di servizi, di strumenti compensativi e/o di misure dispensative e di ausili specifici.

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.

FURTHER INFORMATION

Ask the professor for other information not included in the teaching schedule