## Topology of Algebraic Varieties

This is a learning seminar on the topology of algebraic varieties.  We will cover Lefschetz theory, intersection homology, perverse sheaves, and applications to representation theory.

http://wiki.math.toronto.edu/TorontoMathWiki/index.php/Topology_of_Algebraic_Varieties_Learning_Seminar

### Intersection Homology Theory

by Mikhail Mazin  (www) | University of Toronto
Time: 09:30  (Tuesday, Jan. 26, 2010)
Location: BA6180, Bahen Center, 40 St. George Street
Abstract:
Classical homology of compact oriented manifolds enjoy many nice properties. In particular, one can define the intersection product $H_i(X)\times H_j(X) -> H_i+j-n(X)$. Moreover, in the case of complementary dimensions $i+j=n$ one has the Poincare duality: $H_i(X)\times H_j(X) -> H_0(X)=Z$ is nondegenerate when tensored with rational numbers. Unfortunately, in the case of singular algebraic varieties the intersection product is not defined. And even if one replaces the homology by cohomology and intersection product by cup product, still the Poincare duality is false. Even the Betti numbers in complementary dimensions could be different. Intersection homology fixes this problem. One can define it for any pseudomanifold X, in particular complex algebraic variety, and it enjoys an analog of Poincare duality. In the case of a smooth manifold, intersection holomogy coincide with classical homology. In this talk I will follow the original Mark Goresky and Robert MacPherson paper "Intersection Homology Theory". I will define the Intersection Homology for a pseudomanifold, define the intersection product, and sketch the proof of the Poincare duality. I'm planning to say few words about stratifications and triangulations of algebraic varieties in the beginning.

#### Dates in this series

· Tuesday, Jan. 19, 2010: Cohomology of families of smooth projective varieties (Stephen Morgan)
· Tuesday, Jan. 26, 2010: Intersection Homology Theory (Mikhail Mazin)
· Tuesday, Feb. 02, 2010: Intersection homology (continued) (Mikhail Mazin)
· Tuesday, Feb. 09, 2010: Sheaf Cohomology, Derived and Triangulated Categories (Arthur Huang)
· Tuesday, Feb. 23, 2010: Sheaf Cohomology, Derived and Triangulated Categories (Continued) and IC-Sheaves (Arthur Huang/Omar Antolín Camarena)
· Tuesday, Mar. 02, 2010: IC-sheaves and Perverse Sheaves (Omar Antolín Camarena)
· Tuesday, Mar. 09, 2010: Perverse Sheaves and Decomposition theorem I (Omar Antolín Camarena and Chris Brav)
· Tuesday, Mar. 16, 2010: Applications of the decomposition theorem (Chris Brav)
· Tuesday, Mar. 23, 2010: Semismall maps and Springer theory (Sergey Arkhipov)
· Tuesday, Mar. 30, 2010: Semismall maps and Springer theory - Part II (Sergey Arkhipov)
· Tuesday, Apr. 13, 2010: Kazhdan-Lusztig polynomials (Brad Hannigan-Daley)
· Thursday, Apr. 15, 2010: The Geometric Satake Correspondence (Bruce Fontaine)
· Tuesday, Apr. 20, 2010: The Riemann-Hilbert Correspondence (Daniel Rowe)