**This biannual learning unit is being organized in 2022-2023**

Teacher(s)

Language

English

Prerequisites

Depending on the subject, mathematics skills at the level of the end of the Bachelor in Mathematics or first year Master in Mathematics.

Main themes

The topic considered varies from year to year depending on the research interests of the course instructor.

Learning outcomes

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1 | Contribution of the course to learning outcomes in the Master in Mathematics programme. By the end of this activity, students will have made progress in: - Show evidence of independent learning.
- Analyse a mathematical problem and suggest appropriate tools for studying it in depth.
- Begin a research project thanks to a deeper knowledge of one or more fields and their problematic issues in current mathematics. He will have made progress in:
- Develop in an independent way his mathematical intuition by anticipating the expected results (formulating conjectures) and by verifying their consistency with already existing results.
- Ask relevant and lucid questions on an advanced mathematical topic in an independent manner.
The course aims to initiate research in the field under consideration. Specific learning outcomes vary depending on the field. |

Content

The course will provide an introduction to topos theory from a geometric perspective. The following topics will be covered during the course:

- Presheaves and sheaves on topological spaces and locales
- Localic topoi
- Grothendieck topologies and sites
- Presheaves and sheaves on a Grothendieck site
- The sheafification functor
- Grothendieck topoi and their properties
- Characterization of Grothendieck topoi
- Morphisms of sites and geometric morphisms

Teaching methods

The course will consist of lectures and exercise sessions. During the lectures the theoretical foundations of the subject will be provided, while the exercise sessions will permit students to work on examples and problems to assimilate and apply the material covered in the lectures.

Evaluation methods

The final mark consists of two parts:

- continuous evaluation in the form of tasks/exercises/small projects (40%)
- a written exam at the end of the quadrimester (60%)

Other information

It is recommended that the student is familiar with the basic concepts of category theory (LMAT2150 or a similar course).

Online resources

MoodleUCLouvain

Bibliography

- Artin, Michael & Grothendieck, Alexandre & Verdier, Jean-Louis.
*Théorie des Topos et Cohomologie Étale des Schémas*(Seminaire de Geometrie Algebrique du Bois-Marie, SGA4) - Borceux, Francis.
*Handbook of categorical algebra 3: Categories of sheaves.* - Caramello, Olivia.
*Theories, Sites, Toposes.**Relating and studying mathematical theories through topos-theoretic bridges.* - Jonhstone, Peter T.
*Sketches of an elephant: A topos theory compendium*, volumes 1 and 2. - Johnstone, Peter T.
*Topos theory*. - Maclane, Saunders & Moerdijk, Ieke.
*Sheaves in Geometry and Logic. A First Introduction to Topos Theory*. *The Stacks Project*, Chapter 34 Topologies on Schemes.

Faculty or entity

#### Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme

Sigle

Credits

Prerequisites

Learning outcomes

Master [120] in Mathematics