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.
* Localization 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.
Course coordinators
Type of course
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.
* Recognizes the interaction between algebraic and topological methods.
* Can search for, read, and analyze mathematical literature and prepare a seminar 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: