Algebraic and Symplectic Geometry Seminar
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Tue, 12/10/2010 14:00 |
Daniel Huybrechts (Bonn) |
Algebraic and Symplectic Geometry Seminar  |
SR1 |
| Both parts will deal with spherical objects in the bounded derived category of coherent sheaves on K3 surfaces. In the first talk I will focus on cycle theoretic aspects. For this we think of the Grothendieck group of the derived category as the Chow group of the K3 surface (which over the complex numbers is infinite-dimensional due to a result of Mumford). The Bloch-Beilinson conjecture predicts that over number fields the Chow group is small and I will show that this is equivalent to the derived category being generated by spherical objects (which I do not know how to prove). In the second talk I will turn to stability conditions and show that a stability condition is determined by its behavior with respect to the discrete collections of spherical objects. | |||
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Tue, 12/10/2010 15:45 |
Daniel Huybrechts (Bonn) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Both parts will deal with spherical objects in the bounded derived category of coherent sheaves on K3 surfaces. In the first talk I will focus on cycle theoretic aspects. For this we think of the Grothendieck group of the derived category as the Chow group of the K3 surface (which over the complex numbers is infinite-dimensional due to a result of Mumford). The Bloch-Beilinson conjecture predicts that over number fields the Chow group is small and I will show that this is equivalent to the derived category being generated by spherical objects (which I do not know how to prove). In the second talk I will turn to stability conditions and show that a stability condition is determined by its behavior with respect to the discrete collections of spherical objects. | |||
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Tue, 19/10/2010 15:45 |
Andre Neves (Imperial) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| I will show that given smooth embedded Lagrangian L in a Calabi-Yau, one can find a perturbation of L which lies in the same hamiltonian isotopy class and such that the correspondent solution to mean curvature flow develops a finite time singularity. This shows in particular that a simplified version of the Thomas-Yau conjecture does not hold. | |||
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Thu, 21/10/2010 14:00 |
Tommaso de Fernex (Salt Lake City) |
Algebraic and Symplectic Geometry Seminar |
SR1 |
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Thu, 21/10/2010 15:30 |
Nigel Hitchin (Oxford) |
Algebraic and Symplectic Geometry Seminar |
SR2 |
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Tue, 26/10/2010 15:45 |
Alexander Ritter (Cambridge) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Symplectic cohomology is an invariant of symplectic manifolds with contact type boundary. For example, for disc cotangent bundles it recovers the homology of the free loop space. The aim of this talk is to describe algebraic operations on symplectic cohomology and to deduce applications in symplectic topology. Applications range from describing the topology of exact Lagrangian submanifolds, to proving existence theorems about closed Hamiltonian orbits and Reeb chords. | |||
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Tue, 02/11/2010 15:45 |
Ben Davison (Oxford) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group. | |||
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Tue, 09/11/2010 15:45 |
Amin Gholampour and Yunfeng Jiang (Imperial College London) |
Algebraic and Symplectic Geometry Seminar |
L3 |
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Tue, 16/11/2010 14:00 |
Kai Behrend (Vancouver) |
Algebraic and Symplectic Geometry Seminar |
SR1 |
| On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress. | |||
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Tue, 16/11/2010 15:45 |
Kai Behrend (Vancouver) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exterior differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress. | |||
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Tue, 23/11/2010 15:45 |
Hans-Joachim Hein (Imperial College London) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries. | |||
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Tue, 30/11/2010 15:45 |
Andras Juhasz (Cambridge) |
Algebraic and Symplectic Geometry Seminar |
L3 |
| Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms. | |||
