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| MATH0465-1 | Algebraic Topology
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| Duration : | 30h Th, 10h Pr, 20h Mon. WS |
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| Number of credits : |
| Master in Mathematical Sciences, in-depth approach, 1st year | | 8 |
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| Master in Mathematical Sciences, didactic approach, 1st year | | 8 |
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| Master in Mathematical Sciences, professional focus in management, 1st year | | 8 |
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| Master in Mathematical Sciences, professional focus in computer science, 1st year | | 8 |
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| Master en sciences mathématiques, à finalité spécialisée en statistique, 1st year | | 8 |
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| Master in Mathematical Sciences, specialized approach, 1st year | | 8 |
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| Master in Mathematical Sciences | | 8 |
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| Lecturer : | Jean-Pierre Schneiders |
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Language(s) of instruction :
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| French language |
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Organisation and examination :
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| Teaching in the first semester, review in January |
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Course contents :
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| This course constitutes an introduction to algebraic topology and homological algebra.
One starts by studying the singular homology of topological spaces in general and by treating explicitly a few significant examples (sphere, torus, sphere with handles, projective spaces, ...). To illustrate the usefulness of the preceding theory, we conclude this first part by establishing Jordan's theorem in arbitrary dimension.
In the second part, the study of product spaces leads us to define and study tensor products of complexes and to introduce the "Tor" functors. We conclude this part with Künneth's theorem.
The third part of the course is devoted to the duality between cohomology and homology and motivates the introduction and study of the "Ext" functors.
The course ends with a brief study of the homology and cohomology of topological manifolds and of Poincaré duality. |
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Learning outcomes of the course :
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| At the end of the course, the students should have a good idea of what algebraic topology is and of how its study leads naturally to homological algebra and derived functors. |
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Prerequisites and co-requisites/ Recommended optional programme components :
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| A good knowledge of basic algebra and topology is essential. |
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Planned learning activities and teaching methods :
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| The course consists of blackboard lessons, exercises sessions and a personal work.
During the lessons, the main theoretical results are introduced, established and illustrated with examples.
During the exercises sessions, the students are trained to solve by themselves various problems using the results considered in the lessons
The personal work consists of the preparation of a short paper presenting and establishing a result related to the course but not considered during the lessons. |
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Mode of delivery (face-to-face ; distance-learning) :
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| Face-to-face course. |
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Recommended or required readings :
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| Lecture notes are in preparation and a list of reference works is available. |
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Assessment methods and criteria :
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| An examination comprising an oral part on the theory and a presentation of the personal work is organized during the first session. A similar examination is organized during the second session. |
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Work placement(s) :
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Organizational remarks :
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| The course follows the official schedule handed out to the students at the beginning of the academic year. |
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Contacts :
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| Jean-Pierre Schneiders
Département de Mathématique (Bât. B37, Bureau 1/60)
Grande Traverse 12 - 4000 Liège (Sart-Tilman)
Phone: (04) 366.94.01 - E-Mail: jpschneiders@ulg.ac.be
Web page: http://www.analg.ulg.ac.be/jps/ |
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| Items online : |
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| Course web page |
| Web page giving access to various informations on the course and to the electronic version of the notes. |
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