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.

More information on this seminar series can be found at the following link:

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

Details

Sheaf Cohomology, Derived and Triangulated Categories

by Arthur Huang | University of Toronto
Time: 09:30  (Tuesday, Feb. 09, 2010)
Location: BA6180, Bahen Center, 40 St. George St.
Abstract:
The cohomology of sheaves is indispensable in the studying of intersection homology theory. In this talk, I will first define Cech cohomology of sheaves and show that it is equivalent to the usual simplicial cohomology for triangulable spaces. Then I will go to the category of sheaves over a topological space X (algebraic varieties, pseudo-maniflods, etc.), which is an abelian category with enough injections, define its derived category and sheaf cohomology in terms of derived functors. After showing the equivalence of this definition with the Cech cohomology, I will give basic constructions of the triangulated categories and show that distinguished triangles play a similar role as exact triples in abelian categories: they induce long exact sequences. With these setting-ups, I'll briefly talk about constructable sheaves, the triangulated categories related to them, and duality.

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)