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:
- Recognise and understand a basic foundation of mathematics.He will have made progress in:
-- Recognise the fundamental concepts of some important current mathematical theories.
-- Establish the main connections between these theories.
- Show evidence of abstract thinking and of a critical spirit. He will have made progress in:
-- Identify the unifying aspects of different situations and experiences.
-- Argue within the context of the axiomatic method.
-- Construct and draw up a proof independently, clearly and rigorously.
- Communicate in a scientific manner. He will have made progress in:
-- Structure an oral presentation and adapt it to the listeners' level of understanding.
- Show evidence of independent learning. He will have made progress in:
-- Correctly locate an advanced mathematical text in relation to knowledge acquired.
- 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.
Learning outcomes specific to the course. By the end of this activity, students will be able to:
- Understand the notion of commutator from a categorical point of view, and use it to compute group homology.
- Characterise Galois coverings corresponding to different Galois structures, making the link with coverings in algebraic topology and Galois extensions in algebra.
- Analyse properties of reflective subcategories and of (semi)localisations using factorization systems and closure operators.
- 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.
- Use sheaf theory and topos theory to study the passage from local to global. Make the link between intuitionistic logic and topos theory.
- Understand some constructions in homological algebra and ring theory using categorical groups.
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