B005492 - GEOMETRY II

Italian

Topological space. Continuous application. Subspaces, products and quotients of topological spaces. Homeomorphisms. Hausdorff spaces. Connected spaces. Compact spaces. Complete metric spaces. Differential geometry of curves and surfaces.

V. Checcucci, A. Tognoli, E. Vesentini "Lezioni di topologia generale" Feltrinelli
J. R. Munkres "Topology" Springer Verlag
J. L. Kelley "General Topology" Van Nostrand
I. Singer, J. Thorpr "Lezioni di topologia elementare e geometria" Boringhieri
Allan Hatcher "Algebraic Topology"
Manfredo P. Do Carmo "Differential Geometry of Curves and Surfaces" Dover Publ. Inc (2016)
Material provided by the teachers

Knowledge acquired:
The course is focused on topology, metric space, Non-differentiable curve and surface. Exercises and applications will be illustrated.

Competence acquired: Basic notions of General Topology, Metric Spaces and classical Differential Geometry

Skills acquired (at the end of the course): Ability to use the foundamental notions of General Topology and Differential Geometry

Courses required: Geometry I, Mathematical Analysis I.

Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 300

Hours reserved to private study and other indivual formative activities: 180

Contact hours for: Lectures (hours): 120

Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understunding of the subject.

Training section: training of the students to modelling and solving a wide selection of problems in Topology and in Differential Geometry. The training sections are conducted so to help the students develop communication skills and apply the theoretical knowledge and to encourage independence of students.

Moodle learning platform: online teacher-student interaction, posting of additional notes, weekly exercise sheets, copy of testes.

Frequency of lectures, practice and lab: Recommended
Teaching Tools

The final exam consists of a written and oral part. The aim is to assess the knowledge of the basic notions that have been discussed in the course, as well as the ability of the student in applying these to the solution of simple geometric problems.
In the final written examination a selection of exercises is proposed. In the evaluation special attention is paid to the correctness of the solution procedure as well as to the originality and effectiveness of the adopted methods.
In the oral examination a number of questions are posed. In the valuation special attention is paid to communication skills and appropriate use of mathematical language.

PART 1 - TOPOLOGY: Metric spaces. Topological spaces and continuos maps. Subspaces. Products. Quotiens. Separation axioms. Compact Spaces. Connected spaces. Complete metric spaces.

PART 2 - DIFFERENTIAL GEOMETRY: Submanifolds of R^n. Regular curves , curvature, torsion, fundamental theorem of local theory. Surfaces in R^3, first and second fundamental form, curvature and local theory of surfaces. Teorema Egregium. Geodetics. First notions of global theory of surfaces.