Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 853: Topics in Algebraic Topology

Not offered in 2017-2018

Laurentiu Maxim, Fall 2016:

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.

 
For a more detailed pdf description, see
 
As for the reading sources, I typically recommend the following (though I will post my own notes, so students should not buy all these books):
 
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.

 

credits: 
3 (N-A)
semester: 
Fall

UW-Madison Department of Mathematics
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