Prospectus

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Topics in Algebraic Topology (BM)

Course
2022-2023

Admission Requirements

The course Inleiding in de Algebraïsche Topologie is required.

Description

This course is a sequel to the course Inleiding in de Algebraische Topologie (IAT). The course will include a gentle introduction into homological algebra and the language of categories and functors. We will upgrade the Galois correspondence found in IAT into an equivalence of certain categories. We will apply that result to the study of Riemann surfaces (which we introduce and study first). We will obtain an interesting application in the context of the so-called "abc-theorem" for polynomials. In the second part of the course we introduce sheaves and cohomology, discuss comparison theorems between singular, sheaf and Cech cohomology, and finish with applications.

Course Objectives

The aim is to become acquainted with a number of important results and techniques in algebraic topology. After completing the course, the student will know how to apply these results and techniques in various situations in geometry.

Timetable

The schedule for the course can be found on MyTimeTable

Mode of Instruction

Weekly lectures. Homework on a regular basis.

Assessment method

The final grade is based on homework (25%) and a final oral exam or retake (75%). In order to pass the course the grade for the final exam or retake needs to be at least 5, and the weighted average of homework and final exam or retake needs to be at least 5.5. No mimimum grade for the homework is required in order to be allowed for the exam or in order to pass the course. The homework counts as a practical and there is no retake for it. In the event of a retake, the homework again counts for 25% for the final grade.

Reading list

W. Fulton, "Algebraic Topology, A first course", Springer Graduate Texts in Mathematics 153.
T. Szamuely, "Galois groups and fundamental groups", Cambridge Studies in Advanced Mathematics 117, Cambridge University Press.

Registration

From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.
Extensive FAQ's on MyStudymap can be found here.

Contact

By e-mail: rdejong[at]math.leidenuniv.nl
By phone: +31 (0) 71 5 27 71 40