Organizers: Ionut Ciocan-Fontanine, Adeel Khan, Y.P. Lee
Wall-crossing formulae for sheaf-counting invariants were recently reformulated by Joyce in terms of natural-looking vertex algebras living on homologies of moduli stacks. One of the downsides is the necesity to work with the full virtual fundamental classes instead of concrete invariants. To remedy this, I introduce additive formal families of vertex algebras that appear, for example, once one includes tautological insertions. In the second half of the talk, I will apply the general framework to conjectures about stable pair invariants while leaving a more detailed discussion for the third talk.
I will start by introducing some notation for Calabi-Yau dg-quivers. This language comes in handy once one starts talking about the obstruction theories of enhanced master spaces for sheaves. These can be constructed explicitly by geometric arguments, and are the key to proving wall-crossing formulae from the previous talk. Because the invariants counting semistable sheaves need to be defined even in the presence of strictly semistable, I will finish with the proof that this definition is independent of choices.
Any unfinished proofs from the second talk will be presented. I will then come back to the wall-crossing for pairs and discuss it in more detail. For example, I will explain why Joyce—Song wall-crossing holds, and how far one can get in proving Fano 3-fold DT/PT correspondences using formal families of vertex algebras.
I'll discuss a certain class of sheaves admitting an action of vector fields by Lie derivatives. These sheaves include D-modules, but there are a great many more natural examples, such as any sheaf of jets of sections of (p,q)-tensors. I'll then introduce a theorem, joint with Henrique Rocha, explaining how these sheaves are in a sense a bounded distance away from D-modules. I will formulate the theorem in a more geometric fashion, with a focus on the Atiyah algebra - the algebra of infinitesimal symmetries of a pair consisting of a space and a bundle on it, and actually prove a strengthened form of it in the complex analytic setting.
Equivariant elliptic cohomology is an equivariant cohomology theory associated to an elliptic curve. After a review of its complexified incarnation, introduced by Grojnowski in 1994, we will give a new perspective on this invariant which involves derived mapping stacks. This naturally gives rise to an algebraic avatar of equivariant elliptic cohomology, that we call (equivariant) elliptic Hochschild homology. Based on joint work with Nicolò Sibilla.
Tautological relation on the moduli space of stable curves were studied by several method. I will reveiw the method used by Pandharipande-Pixton to prove the tautological relation on the moduli space of stable curvse using stable quotient. I will explain how to extend the result to get tautological relations on relative Picard stack over the moduli space of stable curves by constructing the stable quotient over Picard stack. After this I will explain how to get the result on the Picard stack which extends the original form of tautological relation given by Faber and Zagier. This talk is based on the joint work in progress with Younghan Bae.
To the singularities of a hypersurface one may associate increasingly sophisticated invariants: Milnor numbers, sheaves of vanishing cycles and categories of Matrix factorizations. It is an interesting problem to globalize these locally defined invariants. The Milnor numbers associated to the moduli spaces of suitable sheaves on a Calabi-Yau 3-fold leads to the highly influential Donaldson-Thomas invariants. Brav-Bussi-Dupont-Joyce-Szendroi have globalized the sheaf of vanishing cycles to construct a categorification of Donaldson-Thomas invariants (obstructed by some orientation data). I will take about globalizing higher categorical invariants. This depends on the study of (-1)-shifted symplectic structures on derived schemes, but I will not expect the audience to know what those words mean. This is work in progress, joint with B. Hennion and M. Robalo.
I will explain my latest joint work with Tsai, Vilonen and Xue (arXiv:2409.04030) about the nearby-cycle construction of cuspidal character sheaves on graded Lie algebras. These objects are closely related to finite-dimensional representations of double affine Hecke algebras and expected to play a role in the harmonic analysis on symmetric spaces and p-adic groups. I will start by explaining the more classical theory of sheaves on reductive Lie algebra and its Z-graded version. I will then highlight the new phenomena which occur when finite-order gradings on Lie algebras come into play.