2018/2019

2018/2019

DM270

MAT/03 (GEOMETRY)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

2nd semester (04/03/2019 - 14/06/2019)

9

72 lesson hours

Italian

WRITTEN AND ORAL TEST

STOPPINO LIDIA (titolare) - 9 ECTS

A course in Calculus and a course in Linear Algebra

The main part of the course is an introduction to general topology. The second part is an introduction to projective geometry.

Topological spaces and continuous functions. Compactness, connectedness, separation and numerability properties. Subspaces, products, quotients of topological spaces. Metric spaces: completeness, compactness, Baire's theorem, Ascoli's theorem. Homotopy.
Introduction to projective geometry. Projective space associated to a vector space; subspaces;homogeneous coordinates. Embedding of the euclidean plane in the real projective plane. Projectivities. Conics: projective and affine classification; polarity. Quadrics (outline). The Erlangen programme (outline).

Extended summary

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.
Outline of the "Erlangen program".

Lectures and problem sessions

For the topology:
E. Sernesi, Geometria 2, seconda edizione, Bollati Boringhieri, 2000
- M. Manetti, Topologia, seconda edizione, Springer, Milano 2014.
- C. Kosniowski, Introduzione alla topologia algebrica, Zanichelli, Bologna 1988

For projective geometry:
- E. Sernesi, Geometria 1, seconda edizione, Bollati Boringhieri, Torino 2000,
E. Fortuna, R. Frigerio, R. Pardini, Geometria Proiettiva, Esercizi e richiami di teoria, Springer Milano, 2011

Written and oral exam

Written and oral exam