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SUMMARY:D-manifolds\, a new theory of derived differential geometry - Joyc
 e\, D (Oxford)
DTSTART:20110414T153000Z
DTEND:20110414T163000Z
UID:TALK30785@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:I describe a new class of geometric objects I call "d-manifold
 s". D-manifolds are a kind of "derived" smooth manifold\, where "derived" 
 is in the sense of the derived algebraic geometry of Jacob Lurie\, Bertran
 d Toen\, etc. The definition draws on ideas of Jacob Lurie\, David Spivak.
  The original aim of the project\, which I believe I have achieved\, is to
  find the "right" definition of the Kuranishi spaces of Fukaya\, Oh\, Ohta
  and Ono\, which is the geometric structure on moduli spaces of J-holomorp
 hic curves in a symplectic manifold. \n\nThe definition of d-manifolds inv
 olves doing algebraic geometry over smooth functions (C-infinity rings)\; 
 roughly speaking\, a d-manifold is a differential-geometric analogue of a 
 scheme with a perfect obstruction theory. D-manifolds form a strict 2-cate
 gory dMan. It is a 2-subcategory of the larger 2-category of "d-spaces" dS
 pa. The definition does not involve localization of categories\, so we hav
 e very good control of what 1-morphisms and 2-morphisms are. \n\nThe 2-cat
 egories dMan and dSpa have some very nice properties. All fibre products e
 xist in dSpa\, and a fibre product of d-manifolds is a d-manifold under we
 ak transversality condition. For example\, any fibre product of two d-mani
 folds over a manifold is a d-manifold. You can glue d-manifolds by equival
 ences of open d-submanifolds (a kind of pushout in dMan) provided the glue
 d topological space is Hausdorff. There is a notion of "virtual cotangent 
 bundle" of a d-manifold\, which lives in a 2-category of virtual vector bu
 ndles\, and a 1-morphism of d-manifolds is etale (a local equivalence) iff
  it induces an equivalence of virtual cotangent bundles. And so on. \n\nTh
 ere are also good notions of d-manifolds with boundary and d-manifolds wit
 h corners\, and orbifold versions of all this\, d-orbifolds. \n\nD-manifol
 ds and d-orbifolds have applications to moduli spaces and enumerative inva
 riants in both differential and algebraic geometry. Almost any moduli spac
 e which is used to define some kind of counting invariant should have a d-
 manifold or d-orbifold structure. Any moduli space of solutions of a smoot
 h nonlinear elliptic p.d.e. on a compact manifold has a d-manifold structu
 re. Any C-scheme with a perfect obstruction theory has a d-manifold struct
 ure. In symplectic geometry\, Kuranishi spaces and polyfold structures on 
 moduli spaces of J-holomorphic curves induce d-orbifold structures. So muc
 h of Gromov-Witten theory\, Donaldson-Thomas theory\, Lagrangian Floer coh
 omology\, Symplectic Field Theory\,... can be rewritten in this language.\
 n
LOCATION:Seminar Room 1\, Newton Institute
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