Geometry II

The aim of the course is to build upon the linear algebra knowledge acquired in the Geometry 1 course by introducing some fundamental concepts of this theory. It also seeks to provide an almost complete treatment of the geometry of affine and Euclidean spaces on one hand, and of projective spaces on the other. The tools introduced are also used to study the geometry of algebraic hypersurfaces in various settings, with particular emphasis on the case of quadratic objects in arbitrary dimensions and on the local and global properties of plane algebraic curves.

Understanding the statements and proofs of fundamental theorems in advanced linear algebra and in the various areas of geometry covered, both from a theoretical perspective (developing a rigorous mathematical language, assimilating definitions, theorems, and the main ideas behind their proofs) and from a practical perspective (solving exercises in written assessments).

Applying knowledge and understanding: Demonstrating known mathematical results using techniques different from those already learned; Constructing rigorous proofs and simple examples. These skills will be developed through interactive teaching: students will constantly test their knowledge by working independently or in small groups on new, simple problems proposed during both lecture-based and support sessions.

Making judgements: Developing informed and independent judgement in evaluating and interpreting the solution of a geometry problem; Being able to construct and develop logical arguments with a clear identification of assumptions and conclusions; Being able to recognize correct proofs and identify flawed reasoning. These objectives will be pursued through practice sessions and supplementary support within the Geometry II course; they will offer students the opportunity to independently develop their decision-making and judgement abilities. The above-mentioned skills will be developed through interactive teaching: Mathematics students will constantly test their understanding by working individually or in small groups on simple new problems proposed during lectures and support sessions.

Communication skills: Communicating information, ideas, problems, solutions, and conclusions clearly and unambiguously; Presenting, both orally and in writing, the most important theorems of linear algebra and analytic geometry in a clear and understandable way; Being able to work in a team and to operate with defined levels of autonomy. To foster the development of communication skills, students will be given various opportunities to submit and discuss written assignments. The final examination will also offer a further opportunity for students to demonstrate their ability to analyse, elaborate, and clearly communicate their work.

Learning skills: Developing the necessary skills to pursue further studies with a high degree of autonomy; Possessing learning abilities and a high level of knowledge and competence that allows access to advanced mathematics courses or graduate programs; Having a flexible mindset and being able to integrate quickly into professional environments, adapting easily to new challenges. Learning skills will be developed throughout the program thanks to the overall organization of the workload, which assigns significant and appropriate importance to individual study time.

If the course is delivered in a blended or remote format, necessary adjustments may be introduced in order to comply with the planned program as outlined in the syllabus.

Geometry 2 total 12 CFU (94 hours)

Teaching Organization FIRST SEMESTER

First semester:

35 hours of frontal lecture
12 hours of exercises

Frontal lectures and  classroom exercise. There is no standard way of lecturing: some lectures will be written exclusively on blackboards or sometimes the student receive printed notes. The method used depends  also on the sort of material that they are covering.

Part of the programme (max 3CFU)  could be done by a visiting professor (italian or not).

Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof.ssa Daniele.

The student should at least know the contents of the Geometry 1 course and it is strongly recommended to have learned the basic concepts of the Algebra course (group, ring, field).

Highly recommended.

 
The detailed program of the course is available on the web page of the course:  https://sites.google.com/view/dmiunictfrusso/geometria-ii
 
Brief description of the contents:

Bilinear forms, generalized inner pooducts. Real and complex inner products, ortogonality, linear maps preserving inner product.
Adjoint endomorphisms, normal matrices, Spectral Theorem for normal operators.
Affine spaces, linear subspaces and their directions. Parallelism. Intersection and linear span of subspaces. Dimension and codimension of subspaces.
Isomorphisms of affine spaces, isometries. Projective spaces, linear subspaces. . Dimension and codimension of linear spaces.
Isomorphisms of projective spaces, projective transformations.
some notions on Bezout's Theorem.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, according to the programme planned and outlined in the syllabus

United Nations Sustainable Development Goals (SDGs) 2030

a) E. Sernesi: Geometria I, Bollati Boringhieri, Torino
b) E. Sernesi: Geometria II, Bollati Boringhieri, Torino.
 
Materiale didattico prof. Russo: Appunti di tutti gli argomenti del corso sono disponibili al seguente link: 
 https://drive.google.com/file/d/1pCkZRyqSCyawPybrHTLgh3gN8IQD4hca/view

The final exam consists of a written test and an oral exam.

The oral exam will require a clear and precise exposition of the theoretical content developed during the course, verifying the maturation of the students' learning, their expository ability and the degree of elaboration of the content achieved.

Students who take a written test with a grade lower than 15/30 are not admitted to take the oral exam and will have to repeat the written exam.

The first term test will be held during the break between semesters. It consists of a written test related to the contents of the first semester.

The students that pass the first term test with a grade of at least 15/30 can complete the final exam with an oral one until the July.

The learning assessment may also be carried out electronically, if conditions require it. In this case, the duration of the written test may be subject to change.
Generally, grades will be assigned according to the following scheme:
- not approved: the student has not acquired the basic concepts and is unable to complete the exercises.
- 18-23: the student demonstrates minimal mastery of the basic concepts, his/her skills in expounding and connecting the contents are modest, he/she can solve simple exercises
- 24-27: the student demonstrates good mastery of the contents of the course, his/her skills in expounding and connecting the contents are good, he/she solves the exercises with few errors
- 28-30 cum laude: the student has acquired all the contents of the course and is able to fully explain them and connect them with a critical spirit; solves the exercises completely and without errors.

Information for students with disabilities and/or DSA:

To ensure equal opportunities and in compliance with current laws, interested students can request a personal interview in order to plan any compensatory and/or dispensatory measures, based on the educational objectives and specific needs.
It is also possible to contact the CInAP (Center for Active and Participated Integration - Services for Disabilities and/or DSA) contact teacher of the Department or the President of the Course of Studies.

On the Microsoft Team of the Course it is possible to download the written exans assigned in previous years.

1) prodotti scalari e loro proprietà

2) teorema spettrale

3) spazi affini

4) Spazi proiettivi