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Algebraic and Symplectic Geometry Seminar (past)

Tue, 22/06/2010
15:45 		Duiliu Diaconescu (Rutgers) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) ADHM Sheaves, Wallcrossing, and Cohomology of the Hitchin Moduli Space II
The second talk will present conjectural motivic generalizations of ADHM sheaf invariants as well as their wallcrossing formulas. It will be shown that these conjectures yield recursive formulas for Poincare and Hodge polynomials of moduli spaces of Hitchin pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.
Tue, 22/06/2010
14:00 		Duiliu Diaconescu (Rutgers) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) ADHM Sheaves, Wallcrossing, and Cohomology of the Hitchin Moduli Space I
The first talk will present a construction of equivariant virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.
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.
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, 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, 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, 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, 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, 02/03/2010
15:45 		Gergely Berczi (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Thom polynomials and the Green-Griffiths conjecture
The Green-Griffiths conjecture from 1979 says that every projective algebraic variety $ X $ of general type contains a certain proper algebraic subvariety $ Y $ such that all nonconstant entire holomorphic curves in $ X $ must lie inside $ Y $. In this talk we explain that for projective hypersurfaces of degree $ d>dim(X)^6 $ this is the consequence of a positivity conjecture in global singularity theory.
Tue, 23/02/2010
15:45 		Kentaro Nagao (Oxford and Kyoto) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Cluster category and applications
I will introduce the theory of cluster categories after Amiot and Plamondon. For a quiver with a potential, the cluster category is defined as the quotient of the category of perfect dg-modules by the category of dg-modules with finite dimensional cohomologies. We can show the existence of the equivalence in the first talk as an application of the cluster category. I will also propose a definition of a counting invariant for each element in the cluster category.
Tue, 23/02/2010
14:00 		Kentaro Nagao (Oxford and Kyoto) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
(HoRSe seminar) Quiver mutations and stability conditions
Let $ (Q',w') $ be a quiver with a potential given by successive mutations from a quiver with a potential $ (Q,w) $. Then we have an equivalence of the derived categories of dg-modules over the Ginzburg dg-algebras satisfying the following condition: a simple module over the dg-algebra for $ (Q',w') $ is either concentrated on degree 0 or concentrated on degree 1 as a dg-module over the dg-algebra for $ (Q,w) $. As an application of this equivalence, I will give a description of the space of stability conditions.
Tue, 16/02/2010
15:45 		Martijn Kool (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Moduli Spaces of Sheaves on Toric Varieties
Extending work of Klyachko, we give a combinatorial description of pure equivariant sheaves on a nonsingular projective toric variety X and use this description to construct moduli spaces of such sheaves. These moduli spaces are explicit and combinatorial in nature. Subsequently, we consider the moduli space M of all Gieseker stable sheaves on X and describe its fixed point locus in terms of the moduli spaces of pure equivariant sheaves on X. As an application, we compute generating functions of Euler characteristics of M in case X is a toric surface. In the torsion free case, one finds examples of new as well as known generating functions. In the pure dimension 1 case using a conjecture of Sheldon Katz, one obtains examples of genus zero Gopakumar-Vafa invariants of the canonical bundle of X.
Tue, 09/02/2010
15:45 		Tom Coates (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Gromov-Witten Invariants and Modular Forms II
I will show that generating functions for certain non-compact Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi Iritani.
Tue, 09/02/2010
14:00 		Tom Coates (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
(HoRSe seminar) Gromov-Witten Invariants and Modular Forms I
I will show that generating functions for certain non-compact Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi Iritani.
Tue, 02/02/2010
15:45 		Michael Wemyss (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Mutations of Quivers in the Minimal Model Programme
Following work of Bridgeland in the smooth case and Chen in the terminal singularities case, I will explain a proposal that extends the existence of flops for threefolds (and the required derived equivalences) to also cover canonical singularities.  Moreover this technique conjecturally says much more than just the existence of the flop, as it shows how the dual graph changes under the flop and also which curves in the flopped variety contract to points without contracting divisors.  This allows us to continue the Minimal Model Programme on the flopped variety in an easy way, thus producing many varieties birational to the given input.    

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

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