TOPICS IN ALGEBRAIC TOPOLOGY

INSTRUCTOR: Laurentiu Maxim.

office:

e-mail:

office hours:

SCHEDULE: TR 9:30AM - 10:45AM (online, syncronous, on BBCollaborate).

CLASS HOMEPAGE: http://www.math.wisc.edu/~maxim/853S21.html

CANVAS PAGE: https://canvas.wisc.edu/courses/235504

SYLLABUS

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.

RECORDED LECTURES

READING SOURCES:

Laurentiu Maxim: Intersection Homology & Perverse Sheaves, with Applications to Singularities, Graduate Texts in Mathematics, Vol. 281, Springer, 2019; ISBN: 978-3030276430.
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.

The following is a beautiful survey about perverse sheaves, the decomposition theorem and its many applications:

M.A. de Cataldo, L. Migliorini: The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bull. Amer. Math. Soc. 46 (2009), 535-633.

ADVANCED READING:

Masaki Kashiwara, Pierre Schapira: Sheaves on manifolds, Springer-Verlag, Berlin, 1994. ISBN: 3-540-51861-4.
Mark Goresky, Robert MacPherson: Stratified Morse theory, Springer-Verlag, Berlin, 1988. ISBN: 3-540-17300-5.
Jörg Schürmann: Topology of singular spaces and constructible sheaves, Birkhäuser Verlag, Basel, 2003. ISBN: 3-7643-2189-X.

GRADE:

Based on in-class participation. There is no exam scheduled for this class.