Universal algebra

Teacher(s)

Vitale Enrico;

Language

French
> English-friendly

Prerequisites

Knowledge of mathematics bachelor level algebra.
It is recommended that the student be familiar with abstract mathematical structures such as vector spaces as covered in LMAT1131 or LINFO1112 or LEPL1101, Euclidean or affine spaces as covered in LMAT1131 or LMAT1141, groups as covered in LMAT1231 or LPHYS2211 or topological spaces as covered in LMAT1323.

Main themes

Three approaches to universal algebra will be introduced, compared and developed (at different levels) :
the approach in terms of finitary operations and equations, the approach in terms of Lawvere's algebraic theories and the approach in terms of finitary monads on the category of sets.

Learning outcomes

At the end of this learning unit, the student is able to :
1	Contribution of the course to the learning outcomes of the master's program in mathematics. At the end of this activity, the student will have progressed in his ability to : - Know and understand a fundamental base of mathematics. In particular, he will have developed his ability to : -- Recognize the fundamental concepts of important current mathematical theories. -- Establish the main links between these theories. - Demonstrate evidence of reasoning, abstraction and critical thinking. In particular, he will have developed his ability to : -- To identify the unifying aspects of different situations and experiences. -- Reasoning in the framework of the axiomatic method. -- Build and write a demonstration independently, clearly and rigorously. - Communicate in a scientific way. In particular, he will have developed his ability to: -- Structuring an oral presentation by adapting it to the level of expertise of the public. - Be autonomous in learning. In particular, he will have developed his ability to: -- Correctly locate an advanced mathematical text in relation to the acquired knowledge. - Start a search through a deeper knowledge of a field of current mathematics. In particular, he will have developed his ability to : -- Develop an autonomous mathematical intuition by anticipating the expected results (formulating conjectures) and checking the consistency with already existing results. -- To autonomously ask relevant questions on an advanced subject of mathematics. Learning outcomes specific to the course (depending on the topics covered). At the end of this activity, the student will be able to : To find, in his general mathematical knowledge, several significant examples of algebraic structures and to situate them in relation to the new concepts introduced in the course. To concretely illustrate the different notions and the abstract results in the categories of sets, groups, abelian groups, modules, etcetera. Recognize and demonstrate important exactness properties of algebraic categories. Use the point of view of algebraic theories and the point of view of monads to understand the structures of general algebra and their fundamental properties.

Content

This activity consists of introducing the basic language and some fundamental results of universal algebra to explain situations encountered in other courses of the bachelor's and master's program in mathematics.
The following contents are covered in the course :

• signatures, Sigma-algebras and equational categories,
• algebraic theories, algebraic categories and algebraic functors,
• monads and algebras over a monad, finitary monads.

Teaching methods

Volume 1 of the course is given as a lecture. During the sessions, students are called upon to ask questions, give suggestions and formulate ideas to move the course forward based on their prior knowledge. Volume 2 will be devoted to supervising students' work of reflection and research, work which will require a certain degree of autonomy and which will lead to the presentation on the part of the students of certain course additions previously set by the teacher.

Evaluation methods

The assessment aims to test knowledge and understanding of concepts, examples and fundamental results, the ability to build a coherent reasoning, mastery of demonstration techniques introduced during the course. The evaluation can take different forms which will be set by the teacher at the start of the activity. It can be based only on the presentations made by the student during the course, but it can also be supplemented by an assignment to be handed in after the end of the course or by a more traditional oral exam. The student can choose the language of the final assessment (English, French or Italian), and of the presentations made during the course (English or French).

Other information

The course is biennial and will not be activated in 2022-2023.

Online resources

Moodle site in preparation.

Bibliography

Teaching materials

• J. Adamek, J. Rosicky, E.M. Vitale : Algebraic Theories (Cambridge University Press, 2010)

Faculty or entity

MATH