Unit GEOMETRY II
- Course
- Mathematics
- Study-unit Code
- GP006039
- Curriculum
- In all curricula
- Teacher
- Federico Alberto Rossi
- Teachers
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- Federico Alberto Rossi
- Giovanni Giuseppe Grimaldi
- Hours
- 63 ore - Federico Alberto Rossi
- 20 ore - Giovanni Giuseppe Grimaldi
- CFU
- 9
- Course Regulation
- Coorte 2025
- Offered
- 2025/26
- 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. 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
- Recommended Textbooks:
1. Edoardo Sernesi “Geometria 1” Bollati Boringhieri
2. Edoardo Sernesi “Geometria 2” Bollati Boringhieri
Other Textbooks:
1. Marco Abate "Geometria" McGraw-Hill
2. Marco Abate, Chiara de Fabritiis "Esercizi di Geometria" McGraw-Hill
3. Marco Manetti “Minitopologia” (dispense online)
4. Gianluca Occhetta “Note di Topologia Generale e primi elementi di Topologia Algebrica” (dispense online)
5. Marius Stoka “Corso di geometria per matematici” CEDAM
Students are encouraged to obtain one of the listed textbooks or a textbook that covers the course material. The professor will not provide any textbooks or handouts.
Students are responsible for obtaining any assistive devices they may need. - 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.
Skills:
Being able to independently read and understand basic Geometry and Topology texts. Connect similar 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 a way that enhances the student's ability to recognize rigorous demonstrations, to identify fallacious reasoning and to adopt optimal strategies for solving problems and exercises.
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 and Topology, 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, diagonalizability, complex numbers.
- Teaching methods
- The course is organized in classroom lectures on all the topics of the course and practical training useful to prepare the students for the written test.
- Other information
- Attendance is strongly recommended.
Independent and dedicated personal study is essential and indispensable.
The University of Perugia has organised a student listening and support service whose FOCUS is aimed at the prevention and management of psychological problems, study and learning difficulties for students during their university career. Special attention is paid to students with disabilities and DSA.
More information can be found on the following University page:
https://www.unipg.it/servizi/counseling-psicologico-e-pedagogico-didattico
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For information on support services for students with disabilities and/or learning disability ("DSA") visit the university web page https://www.unipg.it/disabilita-e-dsa .
Students with special needs (e.g., disabled students, learning disabilities students, working students, non-attending students) are kindly requested to contact the professor at least 15 days before the exam.
Students are responsible for obtaining any assistive devices they may need. - Learning verification modality
- The examination consists of a written and an oral test. The written test tests the ability to produce rigorous demonstrations of problems and statements related to the course topics. The oral examination tests the ability to explain some of the course content clearly and rigorously.
The written test consists of solving problems that can also be small parts of theory and serves to check the level of understanding of the topics covered and the ability to connect them.
The oral test tends to confirm the level of understanding of the topics covered and of critical study and personal reworking. A score of not less than 16/30 in the written test must be obtained to be admitted to the oral test.
For information on support services for students with disabilities and/or learning disability ("DSA") visit the university page: https://www.unipg.it/disabilita-e-dsa .
Students with special needs (e.g., disabled students, learning disabilities students, working students, non-attending students) are kindly requested to contact the professor at least 15 days before the exam.
Students are responsible for obtaining any assistive devices they may need. - 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. Classification of conics. Topological spaces. Continuous functions. Connected and compact spaces. Product spaces and quotient spaces.
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- Quality Education: a university course.
Decent Work and Economic Growth, Reduced Inequalities: more knowledge, education and culture leads to greater economic benefit and levelling out inequalities.