SITO NON PIU' AGGIORNATO - UNIVERSITÀ DI PAVIA

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Geometry 1

Professors:
Cornalba Maurizio
Year:
2015/2016
Course code:
500316
ECTS:
9
SSD:
MAT/03
DM:
270/04
Lessons:
84
Period:
II semester
Language:
Italian

Objectives

The main section of the course is an introduction to general topology and to the first notions of algebraic topology. The second section is an introduction to projective geometry.

Teaching methods

Lectures and problem sessions

Examination

Written and oral exam

Prerequisites

A course in Calculus and a course in Linear Algebra

Syllabus

Topological spaces; open sets, closed sets, neighborhoods and related notions. Continuous functions. Connected spaces; connectivity and continuous functions. Compact spaces; compactness and continuous functions. Hausdorff spaces; T3 and T4 spaces. Continuous maps between Hausdorff and/or compact spaces. Construction of topological spaces: subspaces, quotient of a topological space modulo an equivalence relation, products of topological spaces. Metric spaces; continuous functions between metric spaces. Completeness; completion of a metric space. Characterization of compactness for metric spaces. Uniformly continuous functions between metric spaces. Baire's theorem. Ascoli's theorem. Homotopy of continuous functions. Simply connected spaces. Coverings; lifting of homotopies. The fundamental group of a topological space. The fundamental group of the circle and of the spheres. Van Kampen's theorem (outline). Review of isometries of the euclidean plane. Introduction to projective geometry; historical motivations. Projective space associated to a vector space (over any field, but particularly over the real field); projective subspaces; homogeneous coordinates. Immersion of the Euclidean plane in the real projective plane. Projectivities; projective properties. Conics; affine and projective classifications; polarity. Outline of quadrics.

Bibliography

For the topology section:
- C. Kosniowski, Introduzione alla topologia algebrica, Zanichelli, Bologna 1988
- M. Manetti, Topologia, seconda edizione, Springer, Milano 2014
- Notes provided by the instructor

For the projective geometry section:
- E. Sernesi, Geometria 1, second edition, Bollati Boringhieri, Torino 2000
- Notes provided by the instructor

Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602