• MATH 569 "Advanced Topics in Geometric and Differential Topology"

  • DESCRIPTION:

    • Some of the most interesting topological spaces are algebraic varieties. Examples include toric varieties, flag varieties, Schubert varieties, and modular varieties. Algebraic varieties admit special topological methods of study such as stratifications, intersection homology, Hodge theory, Lefschetz theorems etc. This course will be an introduction to some of these examples and topological methods.

  • Tentative SYLLABUS:

    • Goresky-MacPherson Intersection (co)homology is a geometric-type homology theory which is well suited for the study of singular spaces. I will 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 (cohomology does not). I will also characterize intersection cohomology in terms of sheaves (description due to P. Deligne). This puts the so-colled perverse sheaves in the picture. I will discuss the formalism behind perverse sheaves (derived categories, the geometric functors etc.) and describe various applications to Singularity theory. If time permits, I will provide the basics of the theory of characteristic classes in the singular setting, including motivic classes that come up in birational geometry.

  • Prereqs:

    • Basic notions of Algebraic Topology and Algebraic Geometry.

  • GRADE:

    • Based on in-class participation. The will be no homework and no final exam.

  • TEXTBOOK:

    • There is no prefered textbook, though you can find the topics covered in this course in one of the following

    Alexandru Dimca: "Sheaves in Topology", Springer, Universitext.
    Markus Banagl: "Topological Invariants of Stratified Spaces" (Springer Monographs in Mathematics).
    Borel et all: "Intersection cohomology".
    F. Kirwan, J. Woolf: "An Introduction to Intersection Homology", 2nd ed.

    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