Unit GEOMETRY II
- Course
- Mathematics
- Study-unit Code
- GP006039
- Curriculum
- In all curricula
- Teacher
- Massimo Giulietti
- Teachers
-
- Massimo Giulietti
- Marco Timpanella
- Hours
- 63 ore - Massimo Giulietti
- 20 ore - Marco Timpanella
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- 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
- Eigenvalues and eigenvectors. Diagonalization. Bilinear forms. Quadratic forms. Euclidean vector spaces. Euclidean affine spaces. Orthogonal operators, symmetric operators and the spectral theorem. Classification of conics. Topological and metric 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 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
- Eigenvalues and eigenvectors. Diagonalization. Bilinear forms. Quadratic forms. Euclidean vector spaces. Orthogonal operators, symmetric operators and the spectral theorem. Canonical forms of quadratic forms and conics. Topological and metric spaces. Continuous functions. Connected and compact spaces. Product spaces and quotient spaces.