TOPICS IN ALGEBRAIC TOPOLOGY
INSTRUCTOR: Laurentiu Maxim.
office: 713 Van Vleck
e-mail: maxim@math.wisc.edu
office hours: TBD
SCHEDULE: TR 1:00-2:15, Room VAN VLECK B129.
CLASS HOMEPAGE:
http://www.math.wisc.edu/~maxim/853f12.html
TOPICS: Intersection (co)homology, perverse sheaves, applications to Singularity theory.
DESCRIPTION:
The intersection homology of Goresky-MacPherson is a
homology theory well-suited for the study of singular spaces.
I will first introduce intersection homology in the geometric way,
i.e. using chains that meet the strata of a singular space in a controlled way,
and I will prove the basic properties of this theory, e.g. that it
satisfies Poincare Duality (while the usual homology does not).
I will also characterize the intersection (co)homology groups in terms of
sheaves (using a description due to P. Deligne).
This brings the perverse sheaves in the picture. I will discuss the
formalism behind perverse sheaves and describe various applications to
Singularity theory.
READING SOURCES:
- Alexandru Dimca: Sheaves in Topology, Springer, Universitext, ISBN: 3-540-20665-5.
- Markus Banagl: Topological Invariants of Stratified Spaces, Springer Monographs in Mathematics, ISBN: 978-3-540-38585-1; 3-540-38585-1.
- Borel et all: Intersection cohomology, ISBN: 978-0-8176-4764-3.
- F. Kirwan, J. Woolf: An Introduction to Intersection Homology, 2nd ed., ISBN: 978-1-58488-184-1; 1-58488-184-4.
The following article explains the history of intersection homology and its connections with various problems in mathematics:
- Steven L. Kleiman: The Development of Intersection Homology Theory, math.HO/0701462, Pure Appl. Math. Q. 3 (2007), no. 1, 225-282.
GRADE:
Based on in-class participation. There is no exam scheduled for this class.
As part of class participation, each student is expected to type up a week worth of lecture notes.
These notes will be posted on the class page.