Geometry 3

A.Y. 2020/2021
Overall hours
Learning objectives
The course aimt at providing the basic notions on topological structures, also useful in the study of Geometry and Analysis.
Expected learning outcomes
Improving of the logical-deductive skills, of the ability to abstract and of the flexibility in facing mathematical problems. Ability to solve specific exercises on the topics presented during the class.
Course syllabus and organization

Single session

Lesson period
First semester
Teaching methods
Teaching activity will be held following current regulations.
All the lectures and exercise sessions will take place on Zoom and the students will be able both to attend in real time following the schedule of the first semester, and/or to watch the video later, since all the sessions will be recorded and made available for the students on Ariel. Moreover, tutoring activities will be organized for groups of students, also, if allowed, in place at the university.

Program and references
The program and references for the course will not change.

If circumstances allow, the written examination will be held at the university, as described in Section "Examination". Alternatively, the written examination will be held on an online platform, it will include exercises solving and will require to upload the scan of a written resolution by the student.
The oral part of the examination will be at the university or on Zoom platform depending on current regulations.
Course syllabus
Topological spaces: definitions, examples and properties.
Continuous functions; omeomorphisms.
Induced topology and product topology.
Separation axioms. Quotient topology and group actions.
Connectedness; path connectedness.
Complements of general topology.
Introduction to homotopy theory.
Prerequisites for admission
Basic knowledge of Algebra, Linear Algebra and Analysis (suggested courses: Geometria 1, Geometria 2, Analysis 1)
Teaching methods
Classroom lessons (36 hours lessons, 24 hours exercises sessions; a tutoring is also provided).
Teaching Resources
Suggested textbooks:
M.Manetti, Topologia (seconda edizione), Springer,
E.Sernesi, Geometria 2, Bollati Boringhieri,
J. Munkres, Topology (second edition), Pearson
Assessment methods and Criteria
The final examination consists of two parts, a written exam and an oral exam.
During the written exam, the student must solve some exercises analogous to those presented during the exercises sessions, with the aim of assessing the student's ability to solve problems in topology.
During the semester there will be a partial written examination and a second one will take place on the day of the first complete written examination.
The oral exam can be taken only if the written component has been successfully passed. The oral exam will be an interview on the topics of the course.
MAT/03 - GEOMETRY - University credits: 6
Practicals: 24 hours
Lessons: 36 hours
Educational website(s)
Office2101, second floor, via C. Saldini 50