Please note that, starting from the next academic year 2022/23, some courses will have new names which are more adherent to the actual content of the course. In order to avoid confusion, we summarize below the algebra courses offered within the Master’s Degree in Mathematics.
Homological Algebra (6 CFU) is the new name for the course that was called “Representation Theory” in the past. The course provides a first introduction to homological algebra and representation theory of quivers, an important branch of modern algebra with connections to geometry, topology, theoretical physics and data science. The first part of the course introduces to quivers and representations and covers fundamental notions and results from module theory. The second part provides some basic knowledge on categories and functors and introduces to homological algebra: homological functors, complexes, homology. The course will be offered in the first semester of the academic year 2022/23.
Methods of Representation Theory (Seminar Course, 6 CFU) is the new name for the Seminar course that was called “Homological Algebra” in the past. This is a reading course which is devoted to some more advanced topics in homological algebra and representation theory, including applications to other areas such as persistence theory. It will be offered in the academic year 2023/24.
Computational Algebra (6 CFU). The course provides an introduction to coding theory, presenting the main notions and techniques for error detection and correction. In particular, linear and cyclic codes are studied. The topics are presented both from a teorical and computational point of view. In the first part of the course, basic concept from algebra arereviewed, and finite fields are studied. At the end of the course the students will know the main terminology and main results in coding theory, the more relevant linear and cyclic codes, their decoding algorithms. They will be able to produce rigorous arguments and proofs on these topics and they will be able to read articles and advanced texts. The course will be offered in the first semester of the academic year 2023/24.
Algebraic Geometry (6 CFU). The goal of the course is to introduce the basic notions and techniques of algebraic geometry including the relevant parts of commutative algebra, and create a platform from which the students can take off towards more advanced topics, both theoretical and applied, also in view of a master's thesis project. The fist part of the course provides some basic concepts in commutative algebra, such as localization, Noetherian property and prime ideals. The second part covers fundamental notions and results about algebraic and projective varieties over algebraically closed fields and develops the theory of algebraic curves from the viewpoint of modern algebraic geometry. Finally, the student will be able to deal with some applications, as for instance Gröbner basis or cryptosystems on elliptic curves over finite fields. The course is offered in the second semester of the academic year 2022/23.
According to the study plan, students in the Master’s Degree in Mathematics have to attend at least one of the courses “Homological Algebra” and “Computational Algebra”.
The course “Algebraic Geometry” is offered every academic year and can be also attended by students in the Bachleor’s Degree in Applied Mathematics.
