Algebraic and Symplectic Geometry Seminar (past)
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Tue, 15/05/2012 15:45 |
Balazs Szendroi (Oxford) |
Algebraic and Symplectic Geometry Seminar  |
L3 |
I revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on  , and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. The main example will be the case of the resolved conifold, corresponding to the gauge group  . I will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to a structural result about vector spaces. |
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Tue, 08/05/2012 15:45 |
Algebraic and Symplectic Geometry Seminar |
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Tue, 01/05/2012 15:45 |
Jonathan Pridham (Cambridge) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Derived moduli stacks extend moduli stacks to give families over simplicial or dg rings. Lurie's representability theorem gives criteria for a functor to be representable by a derived geometric stack, and I will introduce a variant of it. This establishes representability for problems such as moduli of sheaves and moduli of polarised schemes. | |||
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Tue, 24/04/2012 15:45 |
Algebraic and Symplectic Geometry Seminar |
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Wed, 14/03/2012 15:45 |
Nikita Nekrasov (Paris) |
Algebraic and Symplectic Geometry Seminar |
L2 |
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String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities). In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear. In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topologica vertex formalism, for all toric Calabi-Yau's admitting the latter. |
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Wed, 14/03/2012 14:00 |
Nikita Nekrasov (Paris) |
Algebraic and Symplectic Geometry Seminar |
L2 |
| String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities). In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear. In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topological vertex formalism, for all toric Calabi-Yau's admitting the latter. | |||
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Tue, 13/03/2012 15:45 |
Yohsuke Imagi (Kyoto) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Special Lagrangian submanifolds are area minimizing Lagrangian submanifolds discovered by Harvey and Lawson. There is no obstruction to deforming compact special Lagrangian submanifolds by a theorem of Mclean. It is however difficult to understand singularities of special Lagrangian submanifolds (varifolds). Joyce has studied isolated singularities with multiplicity one smooth tangent cones. Suppose that there exists a compact special Lagrangian submanifold M of dimension three with one point singularity modelled on the Clliford torus cone. We may apply the gluing technique to M by a theorem of Joyce. We obtain then a compact non-singular special Lagrangian submanifold sufficiently close to M as varifolds in Geometric Measure Theory. The main result of this talk is as follows: all special Lagrangian varifolds sufficiently close to M are obtained by the gluing technique. The proof is similar to that of a theorem of Donaldson in the Yang-Mills theory. One first proves an analogue of Uhlenbeck's removable singularities theorem in the Yang-Mills theory. One uses here an idea of a theorem of Simon, who proved the uniqueness of multiplicity one tangent cones of minimal surfaces. One proves next the uniqueness of local models for desingularizing M (see above) using symmetry of the Clifford torus cone. These are the main part of the proof. | |||
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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. | |||
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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, 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, 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, 14/02/2012 15:45 |
Karen Vogtmann (Cornell) |
Algebraic and Symplectic Geometry Seminar |
L3 |
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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, 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, 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, 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, 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, 10/01/2012 15:45 |
Roman Bezrukavnikov (MIT) |
Algebraic and Symplectic Geometry Seminar |
SR1 |
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I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others |
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Tue, 10/01/2012 14:00 |
Roman Bezrukavnikov (MIT) |
Algebraic and Symplectic Geometry Seminar |
SR1 |
| I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others | |||
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Thu, 01/12/2011 16:30 |
Al Kasprzyk (Imperial) |
Algebraic and Symplectic Geometry Seminar |
L1 |
