ELEMENTS OF ADVANCED GEOMETRY

The student will learn some basic facts of algebraic topology.

Training in the use of formal language in abstract mathematics. Parent topic is the fundamentals of Algebraic Topology.

Knowledge and understanding: students, at the end of the training course, must: - understand statements and proofs of fundamental theorems in the context of Algebraic Topology; demonstrate mathematical skills in reasoning, manipulation and calculation; - solve mathematical problems which, although not common, are of a similar nature to others already known by students.

Applying knowledge and understanding students must be able to: - demonstrate known mathematical results with techniques other than those known; - build rigorous proofs; - build simple examples. The skills listed above will be achieved through interactive teaching: students will constantly verify their knowledge, working independently or in collaboration in the context of small work groups, on simple new problems, proposed during the exercises.

Making judgments: the student must: be able to construct and develop logical arguments with a clear identification of assumptions and conclusions; - be able to recognize correct proofs, and to identify fallacious reasoning. Students must autonomously develop their decision-making and judgment skills.

Communication skills: the student must be able to communicate information, ideas, problems, solutions and their conclusions in a clear and unambiguous way; - be able to present, orally or in writing, in a clear and understandable way, the most important theorems of Algebraic Topology; - be able to work in a team and operate with defined degrees of autonomy. The final exam will also offer the student a further opportunity to study and verify the analysis, processing and communication skills of the work done.

Learning skills: the student must have developed the skills necessary to undertake subsequent studies with a high degree of autonomy; - possess learning skills and a high standard of knowledge and competence, such as to allow access to the lectures or programs of the master's degree courses in Mathematics; - have a flexible mentality, and be able to readily enter the workplace, easily adapting to new problems. Learning skills will be acquired during the course of study thanks to the subdivision of the total working hours, which gives an important and adequate emphasis to those dedicated to personal study.

Lectures with slides. Discussion and correction of exercises.

If the teaching is given in a mixed or remote mode, the necessary changes may be introduced with respect to what was previously stated, in order to respect the program envisaged and reported in the syllabus.

Homotopy of functions and paths. The construction of the fundamental group. Applications. Fixed point theorem in dimenion 2. The Jordan's curve theorem.

1. Notes of the lecturer.

2. Topologia by M. Manetti, Springer