TOPICS IN ALGEBRAIC TOPOLOGY

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SCHEDULE: TR 2:30PM - 3:45PM, Room Van Vleck B129.

CLASS HOMEPAGE: http://www.math.wisc.edu/~maxim/853f16.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. If time permits, I plan to discuss the Kahler package for the intersection (co)homology of complex algebraic varieties, including a basic introduction to Saito's theory of mixed Hodge modules.

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.
Greg Friedman: Singular Intersection Homology.

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.

LECTURE NOTES: this is a draft and it shall be updated regularly.