Corso di laurea in matematica per la Finanza e l'Assicurazione

Students who have taken the classes of "Algebra Lineare e Geometria" and "Analisi Matematica 1" already have the first two prerequisites. To cover the third prerequisite, students can use any basic textbook in algebra, for instance the book "Algebra e matematica discreta. Per studenti di informatica, ingegneria, fisica e matematica, con numerosi esempi ed esercizi svolti", Alberto Facchini, Zanichelli, 2000.

This course will provide students in Matematica per la Finanza e l'Assicurazione the mathematical background essential to those who wish to enroll in the Laurea Magistrale in Matematica (Master Degree in Mathematics).

The theoretical structure of the course consists in a series of theorems and their proofs, the study of which will enable the student to autonomously produce rigorous proofs of mathematical results not identical to those already known but inspired to them in a relevant manner and to solve problems of moderate difficulty in the field of topology.

 In particular, the course will provide:

• theoretical training objectives: development of a rigorous mathematical language; assimilation of abstract concepts, theorems and their proofs related to general topology and algebraic topology
  
• applied training objectives: the student will learn computing techniques to solve problems; the student will be able to solve standard exercises and new problems, in which it will be necessary to develop new strategies and apply the concepts learned or develop simple proofs similar to those seen in class.

• have acquired the fundamental concepts of general topology and know some aspects of algebraic topology;
• be able to communicate and express problems pertaining to the topics of the course: to be able to state and prove theorems, but also to discuss problems concerning the statement of a theorem and its applications;
• be able to apply the notions and the techniques learnt in the course both to standard exercises and to new problems, which require the autonomous elaboration of a strategy, or of a small rigorous proofs, not identical to the ones seen at the lectures but similar.

Class schedule is here (Orario 2021/2022 - Secondo anno)

https://www.matematica.unito.it/do/home.pl/View?doc=Orario_LT.html

Lectures and exercise sessions will be held in the classroom. You will need the Green Pass to enter Palazzo Campana.

Lectures and exercise sessions will be also available on live streaming. You can find the links to the streaming on the Moodle page (see the Moodle button at the bottom of this page).

Lectures and exercise sessions will not be recorded.

Recordings of last year lectures are available on the Moodle page relative to last year class.

https://math.i-learn.unito.it/course/view.php?id=1303

The written examination consists in exercises to solve, and usually lasts 2 hours. The students can consult their own books and notes during the exam, but not in electronic form; a basic calculator is allowed.

For admission to the oral examination, it is necessary to get a grade of at least 18/30 at the written examination. The oral examination must be taken in the same exam session of the written examination.  If a student fails the oral examination, s/he must repeat also the written examination.

The oral examination consists of questions on the theory and the proofs treated in the course, and ofter includes a discussion of the written examination.

For more details, and for the written examinations of the previous years, please see the web page of the course on moodle.

Because of the health emergency COVID-19, the exams may take place online. Please look at moodle for more details.

1. General topology (4.5 CFU): definition of topological space, open and closed sets, neighborhoods. Topologies induced by a metric. Basis of a topology. Continuous functions, homeomorphisms. Subspaces, product topology and quotient topology. Axioms of separation. Connectedness. Compactness. Axioms of countability. Sequences, convergence.

2. Homotopy and fundamental group (1.5 CFU): Homotopy between functions. Omotopically equivalent spaces. Retractions. Paths and homotopy between paths. The fundamental group. Properly discontinuous actions and quotients. The fundamental group of the circle. Coverings.

C. Kosniowski, Introduzione alla topologia algebrica, Zanichelli, for parts 1 and 2.

G. Occhetta, Geometria III, freely downloadable, for parts 2 and 3.

N. Hitchin, Geometry of surfaces,  freely downloadable, for part 3.

There are notes by Professor Albano for part 4, available on Moodle.