Algebraic and Symplectic Geometry Seminar
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Tue, 17/01/2012 14:00 |
Helge Ruddat (Universität Mainz) |
Algebraic and Symplectic Geometry Seminar  |
SR1 |
| Assuming the natural compactification X of a hypersurface in (C^*)^n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Mark Gross and Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, I will explain relations to homological mirror symmetry and the Gross-Siebert construction. | |||
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Tue, 17/01/2012 15:45 |
Helge Ruddat (Universität Mainz) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Assuming the natural compactification X of a hypersurface in (C^*)^n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Mark Gross and Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, I will explain relations to homological mirror symmetry and the Gross-Siebert construction. | |||
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Tue, 24/01/2012 14:00 |
Bertrand Toen (Montpelier) |
Algebraic and Symplectic Geometry Seminar |
SR1 |
| This is a report on a joint work (in progress) with Pantev, Vaquie and Vezzosi. After some reminders on derived algebraic geometry, I will present the notion of shifted symplectic structures, as well as several basic examples. I will state existence results: mapping spaces towards a symplectic targets, classifying spaces of reductive groups, Lagrangian intersections, and use them to construct many examples of (derived) moduli spaces endowed with shifted symplectic forms. In a second part, I will explain what "Quantization" means in the shifted context. The general theory will be illustrated by the particular examples of moduli of sheaves on oriented manifolds, in dimension 2, 3 and higher. | |||
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Tue, 24/01/2012 15:45 |
Bertrand Toen (Montpelier) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| This is a report on a joint work (in progress) with Pantev, Vaquie and Vezzosi. After some reminders on derived algebraic geometry, I will present the notion of shifted symplectic structures, as well as several basic examples. I will state existence results: mapping spaces towards a symplectic targets, classifying spaces of reductive groups, Lagrangian intersections, and use them to construct many examples of (derived) moduli spaces endowed with shifted symplectic forms. In a second part, I will explain what "Quantization" means in the shifted context. The general theory will be illustrated by the particular examples of moduli of sheaves on oriented manifolds, in dimension 2, 3 and higher. | |||
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Tue, 31/01/2012 15:45 |
Andre Henriques (Utrecht) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| There is a beautiful classification of full (rational) CFT due to Fuchs, Runkel and Schweigert. The classification says roughly the following. Fix a chiral algebra A (= vertex algebra). Then the set of full CFT whose left and right chiral algebras agree with A is classified by Frobenius algebras internal to Rep(A). A famous example to which one can successfully apply this is the case when the chiral algebra A is affine su(2): in that case, the Frobenius algebras in Rep(A) are classified by A_n, D_n, E_6, E_7, E_8, and so are the corresponding CFTs. Recently, Kapustin and Saulina gave a conceptual interpretation of the FRS classification in terms of 3-dimentional Chern-Simons theory with defects. Those defects are also given by Frobenius algebras in Rep(A). Inspired by the proposal of Kapustin and Saulina, we will (partially) construct the three-tier CFT associated to a given Frobenius algebra. | |||
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Tue, 14/02/2012 15:45 |
Karen Vogtmann (Cornell) |
Algebraic and Symplectic Geometry Seminar |
L3 |
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Tue, 21/02/2012 15:45 |
Tom Bridgeland (Oxford) |
Algebraic and Symplectic Geometry Seminar |
L3 |
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I will explain how moduli spaces of quadratic differentials on Riemann surfaces can be interpreted as spaces of stability conditions for certain 3-Calabi-Yau triangulated categories. These categories are defined via quivers with potentials, but can also be interpreted as Fukaya categories. This work (joint with Ivan Smith) was inspired by the papers of Gaiotto, Moore and Neitzke, but connections with hyperkahler metrics, Fock-Goncharov coordinates etc. will not be covered in this talk. |
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Tue, 28/02/2012 15:45 |
Oliver Fabert (Freiburg) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Symplectic field theory (SFT) can be viewed as TQFT approach to Gromov-Witten theory. As in Gromov-Witten theory, transversality for the Cauchy-Riemann operator is not satisfied in general, due to the presence of multiply-covered curves. When the underlying simple curve is sufficiently nice, I will outline that the transversality problem for their multiple covers can be elegantly solved using finite-dimensional obstruction bundles of constant rank. By fixing the underlying holomorphic curve, we furthermore define a local version of SFT by counting only multiple covers of this chosen curve. After introducing gravitational descendants, we use this new version of SFT to prove that a stable hypersurface intersecting an exceptional sphere (in a homologically nontrivial way) in a closed four-dimensional symplectic manifold must carry an elliptic orbit. Here we use that the local Gromov-Witten potential of the exceptional sphere factors through the local SFT invariants of the breaking orbits appearing after neck-stretching along the hypersurface. | |||
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Tue, 06/03/2012 14:00 |
Jacopo Stoppa (Cambridge) |
Algebraic and Symplectic Geometry Seminar |
SR1 |
| Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the first talk. | |||
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Tue, 06/03/2012 15:45 |
Jacopo Stoppa (Cambridge) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the first talk. | |||
