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Sheaves in Topology (2023/2024: Semester 2)
Cursusdoel
Er is geen informatie over het cursusdoel bekend.
Vakinhoudelijk
Course organiser. Dr. R. van Dobben de Bruyn.
Schedule. Wednesdays 09.00-12.00.
Sheaves are an important organisational tool in topology, geometry, number theory, and logic. At the most basic level, a sheaf captures the ‘local’ behaviour of many constructions in topology. For instance, a continuous function f : X → R on a manifold may be specified by defining maps in local coordinates that glue to a well-defined global function, which shows that C⁰(X,R) (or rather, the association U ↦ C⁰(U,R)) is a sheaf.
This course is an introduction to sheaves, from a topological perspective. We will study their relation with covering spaces, fundamental groups, cohomology, local systems (≈ “Z-module bundles”), and relative versions of these. Along the way, we develop all the necessary tools from homological algebra and category theory, and this course is a good way to get more experience in those topics.
Although we will work entirely in the topological setting, we also recommend this course for students interested in algebraic geometry or number theory, because the material covered in this course inspired the definition and main results of étale cohomology, which has become a cornerstone of arithmetic geometry.
Contents.
Recommended literature. We will not follow any text closely, but you are encouraged to consult one or more of the following standard works (the first one of which is probably closest to the contents of the course):
Schedule. Wednesdays 09.00-12.00.
Sheaves are an important organisational tool in topology, geometry, number theory, and logic. At the most basic level, a sheaf captures the ‘local’ behaviour of many constructions in topology. For instance, a continuous function f : X → R on a manifold may be specified by defining maps in local coordinates that glue to a well-defined global function, which shows that C⁰(X,R) (or rather, the association U ↦ C⁰(U,R)) is a sheaf.
This course is an introduction to sheaves, from a topological perspective. We will study their relation with covering spaces, fundamental groups, cohomology, local systems (≈ “Z-module bundles”), and relative versions of these. Along the way, we develop all the necessary tools from homological algebra and category theory, and this course is a good way to get more experience in those topics.
Although we will work entirely in the topological setting, we also recommend this course for students interested in algebraic geometry or number theory, because the material covered in this course inspired the definition and main results of étale cohomology, which has become a cornerstone of arithmetic geometry.
Contents.
- Presheaves and sheaves on a topological space, sheafification of a presheaf.
- Dictionary between sheaves and local diffeomorphisms (‘espace étalé’).
- Local systems, covering spaces, and review of the monodromy correspondence.
- Homological algebra and sheaf cohomology.
- Čech cohomology and relation to sheaf cohomology.
- Comparison with singular cohomology.
- Sheaf cohomology with compact support.
- Higher pushforwards and the proper base change theorem.
- Constructible sheaves and the exodromy correspondence.
Recommended literature. We will not follow any text closely, but you are encouraged to consult one or more of the following standard works (the first one of which is probably closest to the contents of the course):
- B. Iversen, Cohomology of sheaves. Universitext, Springer-Verlag, 1986.
- G. E. Bredon, Sheaf theory (2nd edition). Graduate texts in mathematics 170, Springer-Verlag, 1997.
- B. R. Tennison, Sheaf theory. London Mathematical Society lecture note series 20, Cambridge University Press, 1975.
- M. Kashiwara and P. Schapira, Sheaves on manifolds. Grundlehren der mathematischen Wissenschaften 292, Springer-Verlag, 1994.
- The Stacks Project, Tag 006A (chapter on sheaves).
- A. Dimca, Sheaves in topology. Universitext, Springer-Verlag, 2004.
- S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic. Universitext, Springer-Verlag, 1994.
- D. Treumann, Exit paths and constructible stacks. Compos. Math. 145.6 (2009), p. 1504–1532.
- J. Curry and A. Patel, Classification of constructible cosheaves. Theory Appl. Categ. 35 (2020), paper no. 27, p. 1012–1047.
- Topology: a solid understanding of point-set topology is required: covering spaces, fundamental groups, and the relation between the two. Some familiarity with algebraic topology (in particular singular (co)homology) will help.
- Category theory: familiarity with the basic definitions and examples of categories and functors.
- Algebra: a working knowledge of module theory over arbitrary rings.
- 40% homework (every two weeks, 9 sets total).
- 60% final exam.
Werkvormen
Hoorcollege
Toetsing
Eindresultaat
Verplicht | Weging 100% | ECTS 7,5
Ingangseisen en voorkennis
Ingangseisen
Je moet een geldige toelatingsbeschikking hebben
Voorkennis
Er is geen informatie over benodigde voorkennis bekend.
Voertalen
- Engels
Cursusmomenten
Gerelateerde studies
Tentamens
Er is geen tentamenrooster beschikbaar voor deze cursus
Verplicht materiaal
Er is geen informatie over de verplichte literatuur bekend
Aanbevolen materiaal
-
SOFTWAREGeen software nodig
Coördinator
| dr. M. Kool | m.kool1@uu.nl |
Docenten
Inschrijving
Deze cursus is open voor bijvakkers. Controleer wel of er aanvullende ingangseisen gelden.
Inschrijving
Van maandag 30 oktober 2023 tot en met vrijdag 24 november 2023
Na-inschrijving
Van maandag 22 januari 2024 tot en met maandag 19 februari 2024
Inschrijving niet geopend
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