Motivic Donaldson-Thomas theory and singularity theory

This workshop, sponsored by AIM, the Renyi Institute, and the NSF, will bring together experts from Donaldson-Thomas theory and singularity theory to explore the exciting connections which have recently emerged between these two fields. While both disciplines have been the subject of intense research in the last few years, they have been pursued by essentially disjoint communities of researchers. However, recent breakthroughs have made it clear that DT theory and singularity theory are fundamentally connected.

Donaldson-Thomas theory has blossomed in recent years as an algebro-geometric enumerative theory on Calabi-Yau and other threefolds, with connections to topological string theory, Gromov-Witten theory, combinatorics, representation theory, symmetric functions and integrable systems. While these invariants are defined in great generality and are expected to have a nice conjectural structure, their computation has posed a major challenge in all but the simplest cases.

A major advance was the work of Behrend, who observed that, in the Calabi-Yau case, DT invariants arise from a symmetric obstruction theory, whose obstruction cone is a Lagrangian subcone of a cotangent bundle. This fits nicely with the microlocal geometry of characteristic cycles due to Kashiwara et al. and the theory of MacPherson-Chern classes for constructible functions, so that the corresponding DT invariant is given by the Euler characteristic of an intrinsic constructible function attached to the moduli space. If the DT-type moduli space arises locally as the critical locus of a holomorphic function on a complex manifold, then the value of Behrend's distinguished constructible function at a point is the reduced Euler characteristic of the Milnor fibre of the function at the point. This observation brings the methods of singularity theory to bear on the problem, since the Milnor fibre is one of the basic invariants of a singularity. Its (reduced) Euler characteristic or cohomology can be codified by the powerful language of nearby/vanishing cycles of constructible functions or sheaves.

This point of view also allows for local "motifications" and "categorifications", respectively, where one replaces a numerical invariant (an Euler characteristic) by a motivic class or a Hodge theoretical cohomology group coming from the motivic class or mixed Hodge module of vanishing cycles. This point of view has inspired a great deal of recent activity in the field. A key technical result is a corresponding Thom-Sebastiani theorem for vanishing cycles.

In a parallel development, the singularity theory community has been tackling closely related problems about characteristic Chern- resp. Hirzebruch classes of singular complex hypersurfaces in terms of nearby/vanishing cycles of constructible functions resp. motivic Grothendieck groups or mixed Hodge modules.

The aim of the workshop is to bring all these developments together in a creative way. We aim to discuss the following topics during the workshop.

1. Nearby and vanishing cycles on the level of constructible functions and sheaves, mixed Hodge modules and motivic Grothendieck groups
2. DT-style moduli spaces, their geometry and cohomology with coefficients in vanishing cycles
3. Recent work of Kontsevich-Soibelman: exponential Hodge structures, critical and fast-decay cohomology, cohomological Hall algebra.
4. Microlocal geometry and invariants of singularities
5. Characteristic classes of singular hypersurfaces, and their relation to DT theory

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org