2022/2023

2022/2023

DM270

MAT/03 (GEOMETRY)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

2nd semester (01/03/2023 - 09/06/2023)

9

84 lesson hours

Italian

WRITTEN AND ORAL TEST

PIROLA GIAN PIETRO (titolare) - 9 ECTS

A course in Calculus and a course in Linear Algebra

The aim of the course is to introduce the students to the basic notions of general topology and of affine and projective geometry. The students are expected to understand the main structures and properties of general topology (open and closed sets, continuity, product topology, quotient topology conectedness, compactness, numerability axioms, sequences and compactness in metric spaces) and of affine, euclidean and projective geometry; moreover, the students are expected to learn how to solve exercises to verify these properties in concrete cases.

Affine, euclidean and projective geometry:
Affine spaces and affine maps. Affine subspaces and their "giacitura".
Theorems of Talete, Pappo and Desargues.
Affine properties. Grassmann Formula.
Affine geometry in dimension 2 and 3.
Euclidean geometry. Isometries. Euclidean properties.
Projections. Chasles's classification theorem. Theorem of Cartan-Dieudonné.
Introduction to projective geometry. Historical motivation.
Projective space associated to a vector space (particularly over the real numbers)
Projective subspaces; Grassmann formula; homogeneous coordinates.
Affine charts. Pappus theorem in the projective space.
Projection from a point.
Some ideas about duality. Desargues Theorem.
Projectivities. Projective properties.
Algebraic curves, affine and projective.
Affine, euclidean and projective classification of conics.
Some ideas about quadrics.

General topology.
Metric spaces and contiunuity. Equivalent metrics. Properties of open sets.
Topological spaces; open and closed sets, neighbourhoods and related notions. Topological space associated to a metric space: metrizable topology.
Basis of a topological space. Base lemma. Fundamental system of neighbourhoods. Numerability axioms. Sequences and limits in a topological space. Classification of points. Continuous functions between topological spaces. Separation axioms: T0,..., T4.
Subspace topology. Immersions. Product topology, canonical basis. Quotient topology. Quotient of a topological space modulo an equivalence relation. Regular and normal spaces and their properties. Uryson Lemma and metrizabilty Theorem. Compact spaces; compactness and continuous functions. Cauchy sequences. Completeness; extension of Heine-Borel. Some topics on completion of a metric space, and on the construction of the real numbers as a completion of the rationals. Connected spaces; connectedness and continuous maps. Arc connectedness.
Connected and arc connected components.

Lectures, exercise sessions and tutorato.

For 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


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
- L. Steen and J. A. Seebach, Counterexamples in Topology (1970, 2nd ed. 1978) (la bibbia dei controesempi topolgici, con esempi di spazi con le più bizzarre topologie possibili)
- J. Munkres, Topology, 2nd edition, Pearson (in inglese)

The exam consists of a written and oral part.
The written test is divided into two parts. The first consists of the exercise, the second short one is theoretical and will be held just before the oral. The student cannot consult books or any other sources in this second part. The oral exam usually arises from a discussion of the written exam, followed by questions and / or theoretical exercises. To be admitted to the oral exam, the student must obtain a score of at least 15/30 in the written exercise test. Oral exams are public.

More information can be found on the webpage: https://mate.unipv.it/pirola/