Elements of invariant nonlinear analysis II
7404-MAT-ENANII
1. Action of group G on a set.
2. Properties of the group's action on a set (orbit, isotropy group, set of fixed points of the group's action).
3. G-equivariant and G-invariant maps.
4. Compact Lie group.
5. Compact Lie group representations and their classification.
Type of assessment: oral examination
Total student workload
30 hours — lecture
30 hours — studying literature and materials indicated by the instructors and consultations with the instructor
30 hours — individual work, exam preparation
Total 90 hours.
3 ECTS points
Learning outcomes - knowledge
After completing the course, the PhD student:
W1: understands the action of group G on a set. (P8S_WG)
W2: knows the definition of the orbit, the isotropy group and the set of fixed points of the G group. (P8S_WG)
W3: knows the definitions of the G-equivariant and G-invariant maps. (P8S_WG)
W4: can determine the orbital types of the indicated groups. (P8S_WG)
W5: knows the definition of a compact Lie group. (P8S_WG)
W6: can introduce the operation of a compact Lie group on manifolds. (P8S_WG)
W7: understands the concept of representation of a compact Lie group. (P8S_WG)
W8: understands the concept of the conjugacy classes of subgroups of the Lie group. (P8S_WG)
W9: can classify representations of the group S^1. (P8S_WG)
Learning outcomes - skills
After completing the course, the PhD student:
U1: can introduce the action of the group on a set. (P8S_UW)
U2: can determine the orbit, isotropy group and set of fixed points for the indicated groups. (P8S_UW)
U3: can determine the properties of mapping in relation to the group's operation. (P8S_UW)
U4: can check whether the given group is a compact Lie group. (P8S_UW)
U5: can indicate the representations of compact Lie groups. (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
- practical
- classic problem-solving
Prerequisites
Completed course in mathematical analysis, algebra and differential equations and knowledge of the content of the course Elements of invariant nonlinear analysis I.
Course coordinators
Assessment criteria
Oral examination: W1, W2, W3, W4, W5, W6, W7, W8, W9, U1, U2, U3, U4, U5, K1, K2, K3
Bibliography
1. T. Bröcker, T. Dieck, Representations of compact Lie groups, Springer-Verlag, New York, 1995.
2. J.J. Duistermaat, J.A.C. Kolk, Lie groups, Springer-Verlag, Berlin, 2000.
3. K. Kawakubo, The theory of transformation groups, The Clarendon Press, Oxford University Press, New York, 1991.
Additional information
Additional information (registration calendar, class conductors,
localization and schedules of classes), might be available in the USOSweb system: