MA613 Graduate course/Symposium
McKay correspondence and an introduction to derived categories
The McKay correspondence relates the representation theory of a
finite subgroup G in SL(2,CC) or SL(3,CC) with the geometry of
crepant resolutions of CC^n/G. The course will give a hand-on
introduction to a circle of calculations and theory surrounding
these ideas. One of the recent formulations is in the language of
derived categories, which is the main topic of this year's Warwick
EPSRC symposium, and the course will include a gentle non-dogmatic
introduction to categorical ideas.
The lecture course is scheduled for
Mon 9:00 in B3.03
Tue 16:00-18:00 in B3.01
Thu 9:00 in B3.02
during Term 1, and may continue in Term 2 if there is any
interest. The longer session on Tue will include talks by
symposium visitors and by volunteers or victims from the audience.
Approximate contents:
-> Introductory examples with easy calculations
-> How to list the finite subgroups G in SL(2,CC), GL(2,CC),
SL(3,CC)
-> Invariants of finite group actions on affine varieties
-> Klein's calculation of invariants
-> Invariants, the Du Val surface singularities by equations and
their resolutions in algebraic geometry
-> Representation theory of finite groups
-> G-Hilb and G-Cons and calculations for Abelian groups
-> Moduli problems and correspondences
-> Introduction to DCat and Fourier-Mukai transforms
-> More general theory of DCat and the BKR proof
-> Hilb^n CC^2 and BKR following Haiman
-> Reid's recipe for Abelian groups
-> Theta stability, constellations
-> Other groups: some easy solvable groups, the terminal group
1/r(1,a,r-a) in GL(3,CC), some Abelian subgroups of SL(4,CC) and
SL(n,CC)
-> etc.: motivic integration, topology, relations with string
theory, Calabi-Yau 3-folds, CY3-algebras.
Literature: I will put up a website in the course of time.