1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2023/2024
4. Mathematics - drugiego stopnia z tyt. magistra
5. Topology

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Topology - opis przedmiotu

Cel przedmiotu

The basic notions of algebraic and geometric topology.

Wymagania wstępne

Zapisz zmiany

General topology, group theory.

Zakres tematyczny

Lecture
The Fundamental group

1. Induced, quotient and compact-open topology (1 hour)
2. Homotopy (2 h)
3. Retractions (1 h)
4. Construction of the fundamental group (3 h)
5. The Fundamental group of the Cartesian product (1 h)
6. Symplices and symplicial complexes (2 h)
7. Calculating of the fundamental groups (2 h)
8. The fundamental group of the circle, the torus, the sphere, the projective plane (2 h)

The Jordan theorem (proof), the Schoenfliesa theorem (3 h)

Topology in art – Alexander’s sphere, Wady’s leaks, art of M.C. Escher (2 h)

Classification Theorem for Surfaces

1. Surfaces (1\2 h)
2. Polytopes (1\2 h )
3. Triangulation of surfaces (1 h)
4. The proof of Classification Theorem for Surfaces (2 h )

The Borsuka-Ulama theorem

1. The various forms of the Borsuk-Ulam theorem (2 h)
2. The Tucker lemma and the proof of the Borsuk-Ulam theorem (2 h)
3. Sandwich theorem (with proof) (2 h)
4. Fair distribution theorems (1 h)
5. Proof of Brouwer's fixed point theorem based on the Borsuk-Ulam theorem (1 h)
6. Information about Sperner's lemma (1/2 h)

Degree of mappings. (1 h)

Class
Topologies

1. Basic exercises on topologies (1 h)
2. Examples of topologies (1 h)

Homotopy

1. Exercises on homotopy and equivalence relations (1 h)
2. Exercises on homotopy abstraction classes. (2 h)
3. Exercises dealing with the construction of fundamental group (3 h)
4. Exercises on retractions (1 h)
5. Exercises on the fundamental group (3 h)

Classification Theorem for Surfaces

1. Exercises on classification of surfaces based on the proof of the classification theorem. (2 h)
2. Exercises on triangulations of surfaces. (1 h)

The Borsuka-Ulama theorem

1. Proofs of various versions of The Borsuka-Ulama theorem (3 h)
2. The low-dimensional sandwich theorem (1h)
3. Exercises which use The Borsuka-Ulama theorem (2 h)
4. Proof of the Sperner lemma (2 h)
5. Exercises on fair distribution (2 h)

Even-dimensional sphere combing theorem (1 hour)

Presentations and class tests (4 h)

Metody kształcenia

Conventional lecture with emphasis on joint discussion of the problems discussed. During classes, students solve tasks together (usually given a week in advance). Blackboard discussions with multiple students are preferred. Constant access to the Internet is assumed (all examples, especially graphics, animations).

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. The condition for passing the exercises is a positive grade in the test. It is allowed to present a paper on topology. The topic is to be chosen independently by the student. Papers can be prepared by a group of two or three students. The topic of the paper must be approved by all students and the instructor.
2. The exam is in written form with the possibility of discussion of solutions between the examiner and the student being examined. The grade for the course consists of the grade for the exercises (40%) and the grade for the exam (60%). The condition for taking the exam is a positive grade from the exercises. The condition for passing the course is a positive grade in the exam.

Literatura podstawowa

1.. Allen Hatcher, Algebraic Topology, www.math.cornell.edu/~hatcher/

2. Jiri Matousek, Using the Borsuk-Ulam theorem, Springer, 2003

Literatura uzupełniająca

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 10-04-2024 15:46)