Forthcoming Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar

Tue, 27/04/2010
15:45 		Jonny Evans (Cambridge) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Isotopy of Lagrangian submanifolds
Lagrangian submanifolds are an important class of objects in symplectic geometry. They arise in diverse settings: as vanishing cycles in complex algebraic geometry, as invariant sets in integrable systems, as Heegaard tori in Heegaard-Floer theory and of course as "branes" in the A-model of mirror symmetry. We ask the difficult question: when are two Lagrangian submanifolds isotopic? Restricting to the simplest case of Lagrangian spheres in rational surfaces we will give examples where this question has a complete answer. We will also give some very pictorial examples (due to Seidel) illustrating how two Lagrangians can fail to be isotopic.
Tue, 11/05/2010
15:45 		Tobias Ekholm (Uppsala) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Symplectic homology of 4-dimensional Weinstein manifolds and Legendrian homology of links
We show how to compute the symplectic homology of a 4-dimensional Weinstein manifold from a diagram of the Legendrian link which is the attaching locus of its 2-handles. The computation uses a combination of a generalization of Chekanov's description of the Legendrian homology of links in standard contact 3-space, where the ambient contact manifold is replaced by a connected sum of $ S^1\times S^2 $'s, and recent results on the behaviour of holomorphic curve invariants under Legendrian surgery.
Tue, 18/05/2010
14:00 		Emanuele Macri (Utah) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
(HoRSe seminar) 'Stability conditions on the local projective plane and $ \Gamma_1(3) $-action I'
We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane. We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland. In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $ \Gamma_1(3) $-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example. In the second hour we will give some details on the proof of the main theorem.
Tue, 18/05/2010
15:45 		Emanuele Macri (Utah) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) ”Stability conditions on the local projective plane and $ \Gamma_1(3) $-action II'
We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane. We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland. In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $ \Gamma_1(3) $-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example. In the second hour we will give some details on the proof of the main theorem.
Tue, 01/06/2010
14:00 		Denis-Charles Cisinski (Paris 13) Algebraic and Symplectic Geometry Seminar Add to calendar L2
(HoRSe seminar) Motivic sheaves over excellent schemes
Starting from Morel and Voevodsky's stable homotopy theory of schemes, one defines, for each noetherian scheme of finite dimension $ X $, the triangulated category $ DM(X) $ of motives over $ X $ (with rational coefficients). These categories satisfy all the the expected functorialities (Grothendieck's six operations), from which one deduces that $ DM $ also satisfies cohomological proper descent. Together with Gabber's weak local uniformisation theorem, this allows to prove other expected properties (e.g. finiteness theorems, duality theorems), at least for motivic sheaves over excellent schemes.
Tue, 01/06/2010
15:45 		Denis-Charles Cisinski (Paris 13) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Realizations of motives
A categorification of cycle class maps consists to define realization functors from constructible motivic sheaves to other categories of coefficients (e.g. constructible $ l $-adic sheaves), which are compatible with the six operations. Given a field $ k $, we will describe a systematic construction, which associates, to any cohomology theory $ E $, represented in $ DM(k) $, a triangulated category of constructible $ E $-modules $ D(X,E) $, for $ X $ of finite type over $ k $, endowed with a realization functor from the triangulated category of constructible motivic sheaves over $ X $. In the case $ E $ is either algebraic de Rham cohomology (with $ char(k)=0 $), or $ E $ is $ l $-adic cohomology, one recovers in this way the triangulated categories of $ D $-modules or of $ l $-adic sheaves. In the case $ E $ is rigid cohomology (with $ char(k)=p>0 $), this construction provides a nice system of $ p $-adic coefficients which is closed under the six operations.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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