Past Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar (past)

Thu, 01/12/2011
13:30 		Sara Pasquetti (Imperial) Algebraic and Symplectic Geometry Seminar Add to calendar Gibson 1st Floor SR
Topological strings and dualities from M5 branes (COW seminar)
Tue, 29/11/2011
15:45 		Ben Davison (Université Paris Diderot - Paris 7) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Motivic DT invariants of the one loop quiver with potential
Tue, 22/11/2011
15:45 		Alexei Oblomkov (Massachusetts) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Representation theory of DAHAs
In the talk I plan to overview several constructions for finite dimensional represenations of DAHA: construction via quantization of Hilbert scheme of points in the plane (after Gordon, Stafford), construction via quantum Hamiltonian reduction (after Gan, Ginzburg), monodromic construction (after Calaque, Enriquez, Etingof). I will discuss the relations of the constructions to the conjectures from the first lecture.
Tue, 15/11/2011
15:45 		Raf Bocklandt (Newcastle) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Noncommutative mirror symmetry for punctured surfaces

A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any dimer model Q we can associate two categories: A wrapped Fukaya category F(Q), and a category of matrix factorizations M(Q). In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in F(Q) and will give us certain matrix factorizations of a potential on the Jacobi algebra of the dimer in M(Q).

We show that there is a duality D on the set of all dimers such that for consistent dimers the category of matrix factorizations M(Q) is isomorphic to the Fukaya category of its dual,  F((DQ)). We also discuss the connection with classical mirror symmetry.
Tue, 08/11/2011
15:45 		Vittoria Bussi (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Donaldson-Thomas theory: generalizations and related conjectures
Generalized Donaldson-Thomas invariants $ \bar{DT}^\alpha(\tau) $ defined by Joyce and Song are rational numbers which 'count' both $ \tau $-stable and $ \tau $-semistable coherent sheaves with Chern character $ \alpha $ on a Calabi-Yau 3-fold X, where $ \tau $ denotes Gieseker stability for some ample line bundle on X. The theory of Joyce and Song is valid only over the field $ \mathbb C $. We will extend it to algebraically closed fields $ \mathbb K $ of characteristic zero. We will describe the local structure of the moduli stack $ \mathfrak M $ of coherent sheaves on X, showing that an atlas for $ \mathfrak M $ may be written locally as the zero locus of an almost closed 1-form on an étale open subset of the tangent space of $ \mathfrak M $ at a point, and use this to deduce identities on the Behrend function $ \nu_{\mathfrak M} $ of $ \mathfrak M $. This also yields an extension of generalized Donaldson-Thomas theory to noncompact Calabi-Yau 3-folds. Finally, we will investigate how our argument might yield generalizations of the theory to a even wider context, for example the derived framework using Toen's theory and to motivic Donaldson-Thomas theory in the style of Kontsevich and Soibelman.
Tue, 25/10/2011
15:45 		Agnes Gadbled (Cambridge) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Exotic monotone Lagrangian tori
There exist two constructions of families of exotic monotone Lagrangian tori in complex projective spaces and products of spheres, namely the one by Chekanov and Schlenk, and the one via the Lagrangian circle bundle construction of Biran. It was conjectured that these constructions give Hamiltonian isotopic tori. I will explain why this conjecture is true in the complex projective plane and the product of two two-dimensional spheres.
Wed, 14/09/2011
14:00 		Alexander Beilinson (Univesity of Chicago) Algebraic and Symplectic Geometry Seminar Add to calendar L3
On the crystalline period map
Tue, 21/06/2011
15:45 		Yanki Lekili (Cambridge) Algebraic and Symplectic Geometry Seminar Add to calendar L3
The Fukaya category of the once-punctured torus
In joint work with Tim Perutz, we give a complete characterization of the Fukaya category of the punctured torus, denoted by $ Fuk(T_0) $. This, in particular, means that one can write down an explicit minimal model for $ Fuk(T_0) $ in the form of an A-infinity algebra, denoted by A, and classify A-infinity structures on the relevant algebra. A result that we will discuss is that no associative algebra is quasi-equivalent to the model A of the Fukaya category of the punctured torus, i.e., A is non-formal. $ Fuk(T_0) $ will be connected to many topics of interest: 1) It is the boundary category that we associate to a 3-manifold with torus boundary in our extension of Heegaard Floer theory to manifolds with boundary, 2) It is quasi-equivalent to the category of perfect complexes on an irreducible rational curve with a double point, an instance of homological mirror symmetry.
Tue, 07/06/2011
15:45 		Aaron Bertram (Utah) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Birational models of the Hilbert Scheme of Points in $ P^2 $ as Moduli of Bridgeland-stable Objects
The effective cone of the Hilbert scheme of points in $ P^2 $ has finitely many chambers corresponding to finitely many birational models. In this talk, I will identify these models with moduli of Bridgeland-stable two-term complexes in the derived category of coherent sheaves on $ P^2 $ and describe a map from (a slice of) the stability manifold of $ P^2 $ to the effective cone of the Hilbert scheme that would explain the correspondence. This is joint work with Daniele Arcara and Izzet Coskun.
Tue, 31/05/2011
15:45 		Bernhard Keller (Paris 7) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Quiver mutation and quantum dilogarithms
Tue, 17/05/2011
15:45 		Arend Bayer (University of Connecticut) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Towards Bridgeland stability conditions on threefolds
I will discuss a conjectural Bogomolov-Gieseker type inequality for "tilt-stable" objects in the derived category of coherent sheaves on smooth projective threefolds. The conjecture implies the existence of Bridgeland stability conditions on threefolds, and also has implications to birational geometry: it implies a slightly weaker version of Fujita's conjecture on very ampleness of adjoint line bundles.
Tue, 03/05/2011
15:45 		Martijn Kool (Imperial) Algebraic and Symplectic Geometry Seminar Add to calendar L3
A short proof of the Göttsche conjecture

Counting the number of curves of degree $d$ with $n$ nodes (and no further singularities) going through $(d^2+3d)/2 - n$ points in general position in the projective plane is a problem which was already considered more than 150 years ago. More recently, people conjectured that for sufficiently large $d$ this number should be given by a polynomial of degree $2n$ in $d$. More generally, the Göttsche conjecture states that the number of $n$-nodal curves in a general $n$-dimensional linear subsystem of a sufficiently ample line bundle $L$ on a nonsingular projective surface $S$ is given by a universal polynomial of degree $n$ in the 4 topological numbers $L^2, L.K_S, (K_S)^2$ and $c_2(S)$. In a joint work with Vivek Shende and Richard Thomas, we give a short (compared to existing) proof of this conjecture.
Tue, 01/03/2011
15:45 		Vivek Shende (Princeton) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Cohomology of Hilbert schemes of plane curve singularities and the triply graded Khovanov-Rozansky homology of their links
I describe a conjecture equating the two items appearing in the title.
Tue, 22/02/2011
15:45 		Tony Pantev (University of Pennsylvania) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Mirror symmetry and mixed Hodge structures II

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

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