TOPICS IN ADVANCED GEOMETRY B
Natural prerequisites are the courses of Linear Algebra and Geometry, Geometry I and II, and Algebra I and II.
The examination will take place on two levels:
1) The assignment of exercises, chosen among those proposed in the exercise sheets and not discussed during the problem sessions. The exercises should be solved at home and their results should be presented during an oral examination. The goal is to verify that the students can apply the theoretical, abstract results to concrete situations.
2) The assignment of a topic, agreed between student and instructor, which will be presented in an oral examination. The goal is to check the ability of the student to develop, in an autonomous way, concepts which are similar to those presented in the theoretical lectures.
3) A traditional oral examination, during which students are required to explain the basic notions of the course and to illustrate the proofs of the main theorems.
The three oral examinations can happen at different moments.
The final mark, to be expressed over 30 points, is a global assessment of each part of the oral examination.
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) understand three versions of homology: simplicial, singular and cellular;
3) apply the computation techniques of homology to the study of homotopic properties of compact surfaces and CW-complexes;
4) understand the fundamental principles of persistent homology.
The contents of the course can be summarized as follows:
1) Classification of compact surfaces
2) Cell complexes
3) Simplicial, singular and cellular homology
4) Classical applications
5) Persistent homology with applications to data analysis
The teaching methods are based on lectures in presence, which could be recorded, if needed.
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
Borrowers
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Degree course in: MATHEMATICS
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Degree course in: MATHEMATICS