Course Details

Name:

Istituzioni Di Geometria Superiore I / Foundations of Advanced Geometry I

Type:

Course

The course profile, written following the Tuning international methodology, is available here.

Basic information

Code:

F1195

Sector:

MAT/03

Credits:

Bachelor Degree in Mathematics: 9 Ects (b)

Term:

2^nd semester

Degree(s):

Compulsory 3^rd year Bachelor Degree in Mathematics curriculum Generale

Language:

Italian

Teacher(s):

Margherita Lelli Chiesa

Course Objectives

The goal is to introduce the basic concepts of projective geometry, algebraic geometry and algebraic topology

Course Content

Projective geometry: affine spaces and subspaces, affinities, projective spaces and subspaces, Grassmann formula, homogeneous coordinates, projective duality, projectivities, projective line and cross-ratio, projective classification of real and complex quadrics.
Algebraic geometry: Zariski topology on the affine space and on the projective space, affine varieties and projective varieties, Hilbert Nullstellensatz, regular functions, dimension of an affine variety, morphisms and rational maps, smooth and singular points.
Algebraic topology: path connected components of a topological space, homotopies, retractions and deformation retractions, homotopy of paths and fundamental group, local homeomorphisms and covering maps, lifitings, homotopy lifting property, the Borsuk Theorem and the Brouwer fixed point theorem, Van Kampen Theorem, fundamental groups of real and complex spheres and projective spheres, the universal covering space.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

have knowledge and understanding of basic projective geometry and basic algebraic geometry;
understand the fundamental concepts of algebraic topology;
demonstrate skill in mathematical reasoning and ability to conceive a proof;
understand and explain the meaning of complex statements using mathematical notation and language.

Prerequisites and Learning Activities

The student must master the linear algebra, the general topology and the differential geometry taught in the courses Geometria A and Geometria B. He should also know the basic concepts of algebra.

Teaching Methods

Language: Italian
Lectures and exercises

Assessment Methods and Criteria

Written and oral exam

Textbooks

M. C. Beltrametti, E. Carletti, D. Gllarati, G. M. Bragadin, Lezioni di geometria analitica e geometria proiettiva. Bollati Boringhieri.
E. Sernesi, Geometria 1. Bollati Boringhieri.
M. Manetti, Geometria Algebrica.
F. Bottacin, Introduzione alla geometria algebrica.
R. Hartshorne, Algebriac Geometry. Springer.

Course page updates

This course page is available (with possible updates) also for the following academic years:

Course information last updated on: 21 gennaio 2019, 12:17