Dominic Joyce (Oxford)
A Darboux theorem for shifted symplectic derived schemes; d-critical loci

Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209 introduced the notion of "k-shifted symplectic structure" on a derived scheme or derived stack, for integers k. Derived moduli schemes or stacks of coherent sheaves, or complexes of coherent sheaves, on a Calabi-Yau m-fold are (2-m)-shifted symplectic. Thus, this is a new geometric structure on Calabi-Yau moduli spaces.
We prove a "Darboux Theorem" for k-shifted symplectic derived schemes for k<0, writing them Zariski locally in a standard form (joint work with Vittoria Bussi and Chris Brav). For example, when k = - 1 (the Calabi-Yau 3-fold case), a - 1-shifted symplectic derived scheme is Zariski locally a derived critical locus Crit ( f : U --> A^1) for a regular function f on a smooth scheme U.
Next, I explain "d-critical loci". An (algebraic) d-critical locus (X,s) is a classical scheme X which may be written Zariski locally as a classical critical locus Crit ( f : U --> A^1), together with a global section s of a certain sheaf on X, which remembers some information about the function f (basically, it remembers f up to second order in the ideal of Crit(f) in U).
There is a truncation functor from - 1-shifted symplectic derived schemes to d-critical structures on the underlying classical schemes. Hence, any Calabi-Yau 3-fold classical moduli scheme is a d-critical locus.