Organizers: Dominic Joyce, Pierrick Bousseau, Hulya Arguz,
Chenjing Bu
Enquiries: e-mail dominic.joyce@maths.ox.ac.uk.
Let A be a geometric C-linear abelian category, such as the category coh(X) of coherent sheaves on a smooth projective complex manifold X, or the category mod-CQ of representations of a quiver Q. Let M be the moduli stack of objects in A. A Cohomological Hall Algebra (CoHA) is the structure of an associative algebra on the cohomology H*(M,Q). These are a part of Geometric Representation Theory.
In
good cases, people can identify the CoHA as some interesting
infinite-dimensional algebra studied in Representation Theory,
such as a Heisenberg algebra, a W-algebra, … CoHAs may also have
interesting representations, for example, the CoHA of
0-dimensional sheaves on a complex projective surface X
has a representation on the cohomology of the moduli scheme of
torsion-free rank one sheaves on X (and so on the
cohomology of Hilbert schemes of points on X ).
This is a large subject. We are open to suggestions from the
audience, but provisionally we plan to focus on the following
areas:
Prerequisites:
We will be assuming background in Algebraic Geometry at the
level of Hartshorne: projective varieties, schemes, vector
bundles, coherent sheaves, and so on. There will be some Artin
stacks as well, but if you are prepared to believe that a stack is
some kind of space which looks locally like the quotient of a
scheme by a group, and don't ask too many questions, you can
probably cope without having studied stacks. Having some idea of
what an abelian category is would be helpful.
Provisional
programme:
Week 1:
Dominic: introduction to Hall algebras and CoHAs.
Week 3: Hulya:
CoHAs of coherent sheaves on surfaces,
Mellit-Minets-Schiffmann-Vasserot.
Week 4: No
meeting
Week 5: Pierrick:
Kontsevich-Soibelman critical CoHAs.
Week 6: Chenjing
Week 7: TBA
Week 8: TBA
Notes for week 1
by Dominic
Notes for week 2
by Dominic
Notes for week
3 lecture by Hulya, notes taken by Dominic