WorkshopsGeometric structures on moduli spaces and derived differential geometry
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Many important areas in both differential and algebraic geometry which involve forming moduli spaces of geometric objects and then "counting" the points in the moduli space to get a number (or homology class, etc.). To do this "counting" one must first define some kind of geometric structure on the moduli space, and then use this structure to define a virtual class or virtual chain in homology. In algebraic geometry one generally gives the moduli space the structure of a scheme (or Deligne-Mumford stack) with a perfect obstruction theory. For moduli spaces of J-holomorphic curves in symplectic geometry, the situation is more messy. For moduli spaces without strong simplifying assumptions, there are two main geometric structures in use: the "Kuranishi spaces" of Fukaya-Oh-Ohta-Ono, and the "polyfolds" of Hofer-Wysocki-Zehnder. This talk will describe new classes of geometric spaces we call "d-manifolds". They are "derived" versions of smooth manifolds, where "derived" is in the sense of the derived algebraic geometry programme of Jacob Lurie. D-manifolds form a 2-category, and are a simplified version of the "derived manifolds" of David Spivak. Manifolds are examples of d-manifolds, but d-manifolds also include many spaces one would regard classically as singular or obstructed. Almost all the main ideas of differential geometry have analogues for d-manifolds -- submersions, immersions, embeddings, submanifolds, orientations, transverse fibre products, and so on. There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds. Virtual classes and cycles may be constructed for compact, oriented d-manifolds and d-orbifolds. Almost any moduli space which is used to define some kind of counting invariant will have a d-manifold or d-orbifold structure, including moduli spaces of J-holomorphic curves in symplectic geometry. This project began as an attempt to find the "correct" definition of Kuranishi spaces in the work of Fukaya-Oh-Ohta-Ono. I claim that Kuranishi spaces should really be d-orbifolds with corners. The d-orbifold framework can be used to simplify FOOO-style Lagrangian Floer cohomology, and other parts of symplectic geometry.
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Dominic Joyce
2011-07-05
09:30:00 - 10:30:00
R101 , Astronomy and Mathematics Building
Many important areas in both differential and algebraic geometry which involve forming moduli spaces of geometric objects and then "counting" the points in the moduli space to get a number (or homology class, etc.). To do this "counting" one must first define some kind of geometric structure on the moduli space, and then use this structure to define a virtual class or virtual chain in homology. In algebraic geometry one generally gives the moduli space the structure of a scheme (or Deligne-Mumford stack) with a perfect obstruction theory. For moduli spaces of J-holomorphic curves in symplectic geometry, the situation is more messy. For moduli spaces without strong simplifying assumptions, there are two main geometric structures in use: the "Kuranishi spaces" of Fukaya-Oh-Ohta-Ono, and the "polyfolds" of Hofer-Wysocki-Zehnder. This talk will describe new classes of geometric spaces we call "d-manifolds". They are "derived" versions of smooth manifolds, where "derived" is in the sense of the derived algebraic geometry programme of Jacob Lurie. D-manifolds form a 2-category, and are a simplified version of the "derived manifolds" of David Spivak. Manifolds are examples of d-manifolds, but d-manifolds also include many spaces one would regard classically as singular or obstructed. Almost all the main ideas of differential geometry have analogues for d-manifolds -- submersions, immersions, embeddings, submanifolds, orientations, transverse fibre products, and so on. There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds. Virtual classes and cycles may be constructed for compact, oriented d-manifolds and d-orbifolds. Almost any moduli space which is used to define some kind of counting invariant will have a d-manifold or d-orbifold structure, including moduli spaces of J-holomorphic curves in symplectic geometry. This project began as an attempt to find the "correct" definition of Kuranishi spaces in the work of Fukaya-Oh-Ohta-Ono. I claim that Kuranishi spaces should really be d-orbifolds with corners. The d-orbifold framework can be used to simplify FOOO-style Lagrangian Floer cohomology, and other parts of symplectic geometry.
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