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TOPICS IN ADVANCED GEOMETRY B

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2022/2023
Year: 
1
Academic year in which the course will be held: 
2022/2023
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: 

Natural prerequisites are the courses of Linear Algebra and Geometry, Geometry 1 and 2, and Algebra 1 and 2.

Final Examination: 
Orale

The examination will take place on two levels:

1) A written examination, with a duration of 2 hours, in which students solve some problems, whose difficulty is akin to that of the assigned exercises. The goal of the written examination is to verify that the students can apply the theoretical, abstract results to concrete situations. Passing the written examination requires a mark of 18/30 and is necessary in order to be admitted to the oral examination. The mark of the written examination remains valid for all the subsequent rounds, within the academic year.

Students who obtain a mark of at least 16/30 will also be provisionally admitted to the oral examination. In order to formalize the admission, these students will be required, during a preliminary oral discussion, to defend the written examination.

2) A traditional oral examination, during which the performance of the written examination is discussed; moreover, students are required to explain basic notions, to illustrate the proofs of the main theorems, and to analyze concrete examples.

The final mark, to be expressed over 30 points, is determined by the outcome of the oral examination, which can confirm that of the written examination, or increment it, or cause the failure, in case the exhibited competences should not be considered as sufficient.

Assessment: 
Voto Finale

This course is addressed to Master (Laurea Magistrale) students; it can also be taken by Bachelor (Laurea Triennale) students, as "Fundamentals of Advanced Geometry".

The course aims to introduce Algebraic Topology, a branch of Mathematics whose goal is the study the properties of topological spaces from an algebraic point of view. One of the main motivations is provided by the classification theorem of compact surfaces.

For its contents, for the main ideas which lay at its foundations, and for the power of its results, Algebraic Topology is one of the fundamental areas of modern Mathematics.

At the end of the lectures, students will be able to:

1) understand the classification theorem of compact surfaces;

2) apply the computation techniques of the fundamental group to the study of homotopic properties of CW-complexes;

3) understand the fundamental principles of homology theory.

The contents of the course can be summarized as follows:

1) Classification of compact surfaces

2) The theorem of Seifert-Van Kampen

3) Cell complexes

4) Simplicial, singular and cellular homology

5) Classical applications

The teaching methods will strongly depend on how the Covid-19 emergency will evolve. The most likely scenario is that the lectures will be in presence, and also streamed/recorded.

The teaching method consists of:

1) theoretical lectures, during which I will provide the students with the key notions of the course.

2) Weekly exercise assignments, to apply the theoretical content of the lecture. The exercises can be worked out individually, or in small groups.

3) Problem sessions, during which we will discuss the solutions of the exercises; ideally, the students will play an important part in the problem session.

Office hours: by appointment. Please send an email to giovanni.bazzoni@uninsubria.it

Professors

BAZZONI GIOVANNI

Borrowers