GEOMETRY 1

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2017/2018
Year: 
2
Academic year in which the course will be held: 
2018/2019
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
68
Detail of lecture’s hours: 
Lesson (56 hours), Exercise (12 hours)
Requirements: 

It is useful to have followed (and passed the exams of)
Algebra Lineare e Geometria, Algebra 1, Analisi1, Analisi2.

Final Examination: 
Orale

Written and oral examination.
The written examination lasts 2 hours and 30 minutes and tipically consists of 4 or 5 exercises divided in subquestions.
The written examination is graded on a scale from 0 to 30. Students need a grade of 14/30 or more to be admitted to the oral examination.
The oral examination usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous
Passing examination and the final grading depend both on oral and written tests.

Assessment: 
Voto Finale

Acquisition of the basilar notions of General Topology; n particular the student has to understand the concepts of connection, compactness, and numerability properties of topological spaces. Ability to recognize in concrete cases the topological properties of a space, and the continuity of maps between spaces. Understanding of some basic concepts of Algebraic Topology: in particular omotopy, retraction, and basic techniques for the computation of the fundamental group of a space.

General Topology:
Topological spaces and their bases. Remindings of metric spaces. Metrizable topologies.
Hausdorff spaces.
Subspace topology.
Internal part, closure, border of a subspace and their properties.
Continuity between topological spaces.
Product topology.
Separation axioms.
Quoteint topology. Group actions on a topological space. Projective spaces.
Connection and arc connection.
Compactness.
Numerability axioms.
Alexandroff compactification.

Elements of Algebraic Topology:
Omotopy of functions and deformation retraction. Omotopical equivalence of topological spaces.
Fundamental group of a pointed space.
The fundamental group of the circle.
A simplified version of the Theorem of Seifert Van Kampen.
Application: the fundamental group of the spheres.

Textbooks and didactic material

1) M. Manetti, Topologia. Springer, 2008.
2) C. Kosniowski, Introduzione alla topologia algebrica. Zanichelli, 2004 (l'ultima edizione).
Other useful references is are:
Sernesi, Geometria 2
Munkres, Topology (in English).
For exercises and old exams see the personal webpage of the Teacher.

Lessons and exercise sessions

To fix an an appointment send me an email.

Professors

STOPPINO LIDIA