Course Unit Page
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Teacher Roberto Pagaria
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Credits 6
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SSD MAT/03
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Teaching Mode Traditional lectures
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Language Italian
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Campus of Bologna
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Degree Programme Second cycle degree programme (LM) in Mathematics (cod. 8208)
Also valid for Second cycle degree programme (LM) in Mathematics (cod. 5827) -
Course Timetable from Sep 20, 2021 to Dec 15, 2021
SDGs
This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.
Academic Year 2021/2022
Learning outcomes
At the end of the course, the student reaches advanced knowledge in the field of algebraic topology, necessary for the understanding of various mathematical problems. He is able to compute cohomological and homotopic invariants of spaces of ordered and non-ordered configurations, including the group of braids.
Course contents
The aim of this course is to provide tools such as sheaves and spectral sequences for the computation of cohomology groups. They are used in various fields of mathematics: in complex geometry, differential, commutative algebra, algebraic geometry and algebraic topology.
The course is divided into two parts: in the first one we will briefly recall the definitions of homology and cohomology of a topological space and their main properties. We will see some main theorems (e.g. Kunneth, Lefschetz fixed point, Poincaré duality) and their applications to the computation of (co)homology groups.
In the second part we introduce the two important tools: sheaf theory and spectral sequences. Finally, we will apply these tools in algebraic topology by making some examples concerning configuration spaces.
Readings/Bibliography
Hatcher - Algebraic topology
Hatcher - Spectral sequences
Iversen - Cohomology of sheaves
Teaching methods
Frontal lesson with examples and exercises. Some exercises will be left as homework to become familiar with the tools introduced.
Assessment methods
Oral exam or seminar.
Office hours
See the website of Roberto Pagaria
Alumni