Ordinary Hilbert schemes; Hilbert scheme of points on a surface; G-Hilb scheme as the G-equivariant part of Hilb scheme; G-Hilb scheme as the moduli space of G-clusters; G-Hilb for subgroups of SL2: the minimal resolution; G-Hilb for abelian subgroups of SL3: Craw-Reid triangulation of the junior simplex; G-Hilb for the trihedral subgroups of SL3: the “boats”, G-Hilb for 1/r(1,1,a,b) subgroups of SL4: the “traps”.
The McKay correspondence:
The representation graphs of finite subgroups of SL2 and their relations to the McKay quiver; Minimal resolution of dim=2 quotient singularities and the intersection graphs of their exceptional divisors; McKay's observation and GSp-V construction; BKR equivalence in dim=3; Extracting geometry out of BKR: derived Reid's recipe; Abelian subgroups of SL3: sink-source graphs, their classification and relation to the local toric picture; Reid's recipe for abelian subgroups of SL3 and how to compute it.
Quiver Reps and Stability:
Quivers; Representations of quivers; Finite-dimensional algebra reps as quiver reps; King's stability conditions for quiver reps; the McKay quiver; Reps of the McKay quiver as G-sheaves on C^n, in particular G-clusters and G-constellations; Theta-stability for G-constellations; BKR result: fine module spaces of G-cons in dim=2,3 are smooth and crepant; Chamber structure on the space of theta-stability conditions for subgroups of SL2: the Weil group action; Chamber structure on the space of theta-stability conditions for abelian subgroups of SL3: get all projective crepant resolutions as per Craw-Ishii.
McKay correspondence à la Auslander:
Auslander's algebraic picture of the McKay correspondence; maximal Cohen-Maucalay modules: what are they and what are they good for; Auslander-Reiten quiver; Seki-Yamaura interpretation of SL2 McKay via quiver mutations; Threefold MMP picture: MMAs, NCCRs, et cetera.
Chris Brav (Oxford): Classical and derived monodromy of a quintic threefold
We show that the monodromy group of the quintic mirror family splits as a free product $\ZZ \ast \ZZ/5\ZZ$. Under homological mirror symmetry, monodromy operators correspond to autoequivalences of the derived category of coherent sheaves of a smooth quintic threefold. We show that the corresponding derived monodromy group is a two-generator Artin group surjecting onto the classical monodromy group $\ZZ \ast \ZZ/5\ZZ$.
Constantin Shramov(Steklov Institute RAS): Jordan property for Cremona groups of low rank
A group G is called a Jordan group if there exists a constant J such that for any finite subgroup F in G there is an abelian subgroup H in F of index [F:H]< J. I will survey the known results concerning the Jordan property of the groups of the form Bir(S), where S is a surface (in this case the only birational type of S
that violates this property is a product of a projective line and an elliptic curve), and prove it for the Cremona group of rank 3.
Evgeny Smirnov(HSE, Moscow): Schubert calculus and Gelfand-Zetlin polytopes
We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope. We also prove a formula for all Demazure characters of a given representation of GL(n) via exponential sums over integral points in faces of the Gelfand-Zetlin polytope associated with the
representation. This is a joint work with V.Kiritchenko and V.Timorin.