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Course, academic year 2018/2019
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Topological Methods in Combinatorics - NDMI014
Title in English: Topologické metody v kombinatorice
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2018
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. RNDr. Martin Tancer, Ph.D.
Class: DS, diskrétní modely a algoritmy
Classification: Informatics > Discrete Mathematics
Annotation -
Last update: T_KAM (07.05.2001)
One of the important proof techniques in discrete mathematics is the application of theorems from algebraic topology. The course covers the necessary topological preliminaries and establishes several combinatorial and geometric results by topological methods, mainly using the Borsuk-Ulam theorem.
Course completion requirements -
Last update: Mgr. Jan Kynčl, Ph.D. (23.05.2019)

For getting the credit from tutorials, the students are required to get at least 20 points from homework. The total number of available points will be at least 80. There is no provision for repeated attempts for the credit. Credit from tutorials is a necessary condition for an attempt to pass an exam.

Literature - Czech
Last update: doc. Mgr. Milan Hladík, Ph.D. (06.05.2014)

J. Matousek, Using the Borsuk-Ulam Theorem

V. V. Prasolov, Elements of Combinatorial and Differential Topology

J. R. Munkres, Elements of Algebraic Topology

Requirements to the exam -
Last update: Mgr. Jan Kynčl, Ph.D. (23.05.2019)

The exam will be oral based on the contents of the lectures. Extra points gained by students by solving problems for tutorials will be considered in favor of the students.

Syllabus -
Last update: doc. RNDr. Martin Tancer, Ph.D. (25.10.2018)

Simplicial complexes, connectedness of a space.

Borsuk-Elam theorem, equivalent versions.

Ham-sandwich theorem, Necklace theorem.

Theorems on non-embeddability and colorings (chromatic number of Kneser graphs, Radon theorem).

Additional (possible) topics: homology, degree of a map, colorful Tverberg theorem, Z_2 index.

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