Unit GEOMETRY II

Course
Mathematics
Study-unit Code
GP006039
Curriculum
In all curricula
Teacher
Massimo Giulietti
Teachers
  • Massimo Giulietti
  • Marco Timpanella (Codocenza)
Hours
  • 63 ore - Massimo Giulietti
  • 20 ore (Codocenza) - Marco Timpanella
CFU
9
Course Regulation
Coorte 2023
Offered
2023/24
Learning activities
Base
Area
Formazione matematica di base
Academic discipline
MAT/03
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
Bilinear forms. Quadratic forms. Euclidean vector spaces. Euclidean affine spaces. Orthogonal operators, symmetric operators and the spectral theorem. Classification of conics. Projective spaces. Conics in the projective space. Topological spaces. Continuous functions. Connected and compact spaces. Product spaces and quotient spaces.
Reference texts
Marco Abate, Geometria, McGraw-Hill

Edoardo Sernesi, Geometria I, Bollati-Boringhieri

Marco Abate e Chiara De Fabritiis, Esercizi di Geometria, McGraw-Hill

Gianluca Occhetta, Note di Topologia Generale e primi elementi di Topologia Algebrica (online)
Educational objectives
The main goal of the teaching is to provide students with basic
knowledge in the field of bilinear and quadratic forms,
projective geometry and general topology, in order to then be able to
undertake subsequent studies. Particular attention is given to the
understanding of the arguments and to the rigor in the presentation of
concepts and argumentations.

Knowledge and comprehension:
Mathematical comprehension of the proposed topics, knowledge of the
theory developed on bilinear and quadratic forms, euclidean spaces, general topology and fundamental examples
discussed on these topics.
Methods for verifying knowledge:
Written and oral exam.

Capacity:
Being able to independently read and understand basic Geometry texts,
connect arguments, find examples and counterexamples.
Being able to produce simple rigorous proofs of mathematical results and
problem solving of simple problems related to what has been illustrated
in class.
Methods for verifying skills:
Written and oral exam.

Autonomy of judgment:
The display of the contents and arguments will be carried out in order to
improve the ability of the student to recognize rigorous proofs

Communication skills:
The presentation of the topics will be organized to allow the acquisition
of a good ability to communicate problems, ideas and solutions
concerning Geometry, both in written and oral form.
Prerequisites
In order to be able to understand and reach the objectives of the course of Geometria II, it is important that the students have successfully passed the exam of Geometria I.In particular basic topics, such as : vector spaces, linear maps and matrices, affine spaces, parametric and cartesian equations of affine subspaces, are required.
Teaching methods
The course is organized as follows: face-to-face lessons on all the topics of the course and practical training usueful to prepare the students for the written test.
It is planned a tutor teaching activity.
Other information
Attendance:Optional but advised
Learning verification modality
The exam consists of a written test and an oral test. The written test
verifies the ability to produce rigorous demonstrations of problems and
statements related to the topics of the course. The oral exam verifies the
ability to clearly and rigorously explain some of the course contents.
All written tests last
about three hours and consist in solving some problems that can also be
small parts of theory and serve to check the level of understanding of the
topics covered and the ability to connect them. The written test of each
session contains 5 problems, one on endomorphisms of linear spaces, one on euclidean spaces, one on quadratic forms/conics, one on euclidean topology, and one on non-euclidean topology.. The oral exam, lasting about 30 minutes, tends to confirm the
level of understanding of the topics covered and of critical study and
personal re-elaboration.
For information on support services for students with disabilities and / or
SLD, visit the page http://www.unipg.it/disabilita-e-dsa
Extended program
Bilinear forms. Quadratic forms. Euclidean vector spaces. Orthogonal operators, symmetric operators and the spectral theorem. Canonical forms of quadratic forms and conics. Projective spaces and conics in the projective plane. Topological spaces. Continuous functions. Connected and compact spaces. Product spaces and quotient spaces.
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