Dominic Joyce (Oxford)
Extensions of Donaldson-Thomas theory of Calabi-Yau 3-folds

I discuss several projects beginning from the previous talk, joint work with Lino Amorim, Oren Ben-Bassat, Dennis Borisov, Chris Brav, Vittoria Bussi, Delphine Dupont, Sven Meinhardt, and Balazs Szendroi.
Firstly, we show that given a d-critical locus (X,s) with an "orientation" we can construct a canonical perverse sheaf P_{X,s} on X (also a D-module, and a mixed Hodge module if X is over C), such that if (X,s) is locally modelled on Crit ( f : U --> A^1) then P_{X,s} is locally modelled on the perverse sheaf of vanishing cycles of f.
The pointwise Euler characteristic of P_{X,s} is the "Behrend function" of X from Donaldson-Thomas theory.
Thus, if (X,s) is a Calabi-Yau 3-fold moduli scheme, then the hypercohomology H^*(P_{X,s}) has graded dimension the Donaldson-Thomas invariant of X. So this perverse sheaf provides a "categorification" of Donaldson-Thomas theory.
We hope in future to define an associative multiplication on such hypercohomologies H^*(P_{X,s}) to give a Kontsevich-Soibelman style "Cohomological Hall algebra".
We can also apply this to give a perverse sheaf P_{L,M} on intersections L\cap M of (oriented) complex/algebraic Lagrangians L,M in a complex/algebraic symplectic manifold (S,\omega). The hypercohomology H^*(P_{L,M}) is analogous to Lagrangian Floer cohomology HF^*(L,M).
We hope in future to define a "Fukaya category" of (S,\omega) with objects (complexes of) (oriented) Lagrangians L, and morphisms the hypercohomology of these perverse sheaves.
Secondly, given an oriented d-critical locus (X,s), we can define a motive MF_{X,s} on X, such that if (X,s) is locally modelled on Crit ( f : U --> A^1) then P_{X,s} is locally modelled on the (normalized) motivic Milnor fibre of f. For a Calabi-Yau 3-fold moduli space, this MF_{X,s} is basically the Kontsevich-Soibelman motivic DT invariant.
Thirdly, we can extend all the above from (derived) schemes to (derived) Artin stacks.
If I have time I will discuss some other projects we are working on, including defining new D-T style invariants "counting" stable coherent sheaves on a Calabi-Yau 4-fold X, unchanged under deformations of X.