We will be holding a small one-day geometry event in Glasgow, on August 24th 2022.
Location: ARC Building, Room 237a (on the main entrance floor, right besides the reception).
Speakers and schedule:
10:00 - 11:00: Martin Kalck (Freiburg)
11:30 - 12:30: Søren Gammelgaard (Oxford)
14:30 - 15:30: Johannes Schmitt (Kaiserslautern)
16:00 - 17:00: Ben Wormleighton (Washington U. St. Louis)
For any question, do not hesitate to contact Franco Rota (franco dot rota at glasgow dot ac dot uk) or Wahei Hara (wahei dot hara at glasgow dot ac dot uk).
We acknowledge support from the University of Glasgow, the ERC and the EPSRC, the latter through the Programme Grant: Enhancing Representation Theory, Noncommutative Algebra and Geometry
Ben Wormleighton: Reconciling Fano and cluster mutations
This is a report on joint work with Jonathan Lai and Tim Magee. I will describe a precise, explicit comparison between mirror symmetry constructions for orbifold del Pezzo surfaces coming from the Fanosearch program and for log Calabi--Yau pairs coming from the Gross--Hacking--Keel theory of cluster varieties. In particular, we identify polytopes found via scattering diagrams with Fano polytopes of toric degenerations. In many cases, our methods also recover Landau-Ginzburg models associated to toric degenerations in cluster-theoretic terms.
Søren Gammelgaard: Quiver varieties and moduli spaces attached to Kleinian singularities
We discuss how Nakajima's quiver varieties can be used to understand the Hilbert schemes of points on Kleinian singularities, that is, singular surfaces isomorphic to C^2/G, for G a finite subgroup of SL_2(C). This approach lets us show that each such Hilbert scheme is irreducible, has symplectic singularities, and has a unique symplectic resolution. Time permitting, we may touch upon a type of "equivariant Quot scheme" associated to the same singularity, or possibly a generalisation to sheaves on a stacky compactification of the same singular surface. This is joint work with A. Craw, Á. Gyenge, and B. Szendrői.
Martin Kalck: Derived categories of singular varieties and finite dimensional algebras
We give new examples of semiorthogonal decompositions and tilting for derived categories of
certain projective varieties with isolated quotient singularities. The tilting objects and their endomorphism rings admit explicit descriptions. This generalises (and partly refines) earlier work of Kuznetsov, Kawamata and Karmazyn, Kuznetsov & Shinder.
It also provides new examples for Kuznetsov & Shinder’s recent concept of "categorical absorption of singularities“.
This is an ongoing joint work with Yujiro Kawamata and Nebojsa Pavic.
Johannes Schmitt: Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution
Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 6 exceptional groups, for which it is still open. In this talk, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. This is joint work with Gwyn Bellamy and Ulrich Thiel.