Mathematics

GUIDO PEZZINI
GUIDO PEZZINI

General objectives: to acquire basic knowledge in elementary theory of numbers and finite fields (useful when studying public key cryptography or code theory in other courses or contexts).
 
Specific objectives:
 
Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to the elementary theory of numbers, the resolution of equations to the comparisons (with particular regard to polynomial equations), akke arithmetic functions, the theory of quadratic residues, to the problem of enlargements, extensions and the detailed structure of finite fields.
 
Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of techniques related to equations to congruences, to the most important arithmetic functions; will be able to describe in a concrete way a finite field and his group of Galois.
 
Critical and judgmental skills: the student will have the basics to use the tools that underlie public-key cryptography and the theory of error-correcting codes. The aim of the course, which is purely theoretical, is to allow students interested in the cryptographic applications to easily manipulate the mathematical objects of the course.
 
Communication skills: ability to expose the contents in the oral part of the verification and in any theoretical questions present in the written test.
 
Learning skills: the acquired knowledge will allow a study, individual or given in an LM course, related to standard aspects of number theory and finite fields.

Knowledge of basic theory of matrix Lie groups (the exponential map, Lie algebra of a group, differential of a continuous morphism, etc.), of Lie algebras (nilpotency, solvability, semisimplicity, etc.), and of root systems (bases, Weyl group, etc.). Knowledge of the classification of semisimple Lie algebras using their root systems, and the theoretical tools needed to complete the classification. Being able to analyze concrete examples of such groups, algebras, and root systems, recognizing which of the known properties they satisfy, and being able to prove general statements on these topics, simple or similar to those seen in the course.

Basic knowledge of algebraic structures (groups, rings, fields) introduced in the course Algebra I are required, as well as linear algebra seen in the course Algebra Lineare (vector spaces, linear maps, etc.).

Matrix Lie groups:
- Definitions, examples
- Exponential map and logarithmic coordinates
- Subgroups and their Lie algebras
- Representations of groups

Lie algebras
- Basic definitions (algebras, subalgebras, ideals, etc.)
- Nilpotent and solvable Lie algebras
- Representations of Lie algebras
- Structure of semisimple Lie algebras: Cartan subalgebras, root systems
- Root systems and their classification
- Classification of semisimple Lie algebras

Brian C. Hall, "Lie Groups, Lie Algebras, and Representations"
James Humphreys, "Introduction to Lie algebras and representation theory''.

Brian C. Hall, "Lie Groups, Lie Algebras, and Representations"
James Humphreys, "Introduction to Lie algebras and representation theory''.

Lectures (75%), recitation classes (25%).

Lectures (75%), recitation classes (25%).

In addition to weekly exercise sheets, the exam will have a written part (consisting of exercises similar to those seen during the semester) and an oral part (on the most relevant results seen in the course).
The minimum required note is 18/30.

Study properties of a continuous representation of a matrix Lie group.

Prove that a certain Lie algebra, given by generators and relations or as a set of matrices, is nilpotent, solvable, semisimple, etc.

Find elements in a root system satisfying certain requirements, and studt properties of elements of a Weyl group.

• Gruppi di Lie di matrici
  • Books: Brian C. Hall, "Lie Groups, Lie Algebras, and Representations"


• Algebre di Lie
  • Books: James Humphreys, "Introduction to Lie algebras and representation theory''.


• Sistemi di radici
  • Books: James Humphreys, "Introduction to Lie algebras and representation theory''.