Conducted in
term:
2024/25
ECTS credits:
unknown
Language:
Polish
Organized by:
Doctoral School of Exact and Natural Sciences
(in Polish) Elementy niezmienniczej analizy nieliniowej IV 7404-MAT-ENANIV
1. Admissible function.
2. Morse function and special Morse function.
3. Degree for gradient mappings invariant to the action of a given group.
Type of assessment: oral exam
Total student workload
30 hours - lecture
30 hours - studying literature and materials indicated by the lecturers and consultations with the lecturer
30 hours - individual work, exam preparation
Total 90 hours.
3 ECTS points
Learning outcomes - knowledge
After completing the course, the PhD student:
W1: knows the definition of an Euler ring. (P8S_WG)
W2: knows the concept of an admissible function on an open set. (P8S_WG)
W3: knows the concept of a Morse function and a special Morse function.
(P8S_WG)
W4: knows the definition of degree for gradient invariant mappings. (P8S_WG)
W5: knows the properties of degree for gradient invariant mappings. (P8S_WG)
Learning outcomes - skills
After completing the course, the doctoral student:
U1: is able to perform operations of adding and multiplying elements in the Euler ring. (P8S_UW)
U2: is able to check whether a function is admissible on a certain set. (P8S_UW)
U3: is able to check whether a given function is a Morse function and a special Morse function. (P8S_UW)
U4: is able to calculate the degree of a special Morse function under the action of the S^1 group. (P8S_UW)
Learning outcomes - social competencies
After completing the course, the PhD student achieves the following results:
K1: understands the appropriate formulation of questions and problems, uses professional terminology correctly (P8S_KK)
K2: analyzes the problem correctly using the principles of logic (P8S_KK)
K3: conveys the acquired knowledge in an understandable way (P8S_KK)
Observation/demonstration teaching methods
- display
Expository teaching methods
- informative (conventional) lecture
Exploratory teaching methods
- classic problem-solving
- practical
- practical
Prerequisites
Completed course in mathematical analysis, topology and knowledge of the content of the course Elements of invariant nonlinear analysis I, II, III.
Course coordinators
Assessment criteria
Oral examination: W1, W2, W3, W4, W5, U1, U2, U3, U4, K1, K2, K3
Bibliography
1. T. Dieck, Transformation Groups and Representation Theory, Springer-Verlag, Nowy Jork, 1979.
2. G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972.
3. K. H. Mayer, G-invariante Morse-Funktionen. (German), Manuscripta Math. 63 (1989).
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: