Symmetries in topology and algebra 1000-1S26STA
Fundamental constructions in equivariant topology.
Equivariant (co)homology theories, including those of Borel and
Bredon (equivariant singular (co)homology).
Principal bundles and universal (classifying) spaces for groups and families of subgroups; various constructions.
Linear representations of compact Lie groups and the linearization of group actions on manifolds.
Symmetries of spheres.
Homotopy orbit spaces and homotopy fixed-point sets.
Localisation theorems in equivariant (co)homology theories.
P. A. Smith fixed-point theorems for finite group actions; various proofs.
"Smith Theory Revisited" via Steenrod algebra methods.
Categorification of P. A. Smith theory.
The detailed seminar program will take into account the interests and background of participants.
Main fields of studies for MISMaP
mathematics
physics
Course coordinators
Type of course
Prerequisites
Prerequisites (description)
Learning outcomes
LEARNING OUTCOMES
Upon completing the seminar, the student:
* Appreciates the importance of transformation groups as a tool for studying geometric objects.
* Recognises the interaction between algebraic and topological methods.
* Can search for, read, and analyse mathematical literature and prepare a presentation based on it.
* Can prepare an outline and slide presentation for a mathematical talk.
* Can present mathematical content in a manner appropriate to the intended audience.
Bibliography
LITERATURE
* Allday, C., Puppe, V., "Cohomological Methods in Transformation
Groups". Cambridge University Press, 2009.
* Bredon, G. E., "Introduction to Compact Transformation Groups"
Academic Press, 1972.
* tom Dieck, T., "Transformation Groups and Representation Theory"
Lecture Notes in Mathematics 766, Springer.
* tom Dieck, T., _Transformation Groups_. De Gruyter Studies in
Mathematics, 1987.
* Dwyer, W. G., Wilkerson, C. W., “Smith Theory Revisited.” Annals
of Mathematics 127 (1988), 191–196.
* Eilenberg, S., “Sur les transformations périodiques de la surface
de sphère.” Fundamenta Mathematicae 22 (1934), 28–41.
* Quillen, D. G., “Spectrum of an Equivariant Cohomology Ring I,
II.” _Annals of Mathematics_ 94(3) (1971), 549–602.
* Treumann, D., “Smith Theory and Geometric Hecke Algebras.”
Mathematische Annalen 375 (2019), 595–628.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes: