Past Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar (past)

Tue, 03/02/2009
15:45 		Greg Berczi (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Multidegrees of singularities
Tue, 27/01/2009
15:45 		Dominic Joyce (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Hamiltonian stationary submanifolds of compact symplectic manifolds
Let $ (M,\omega) $ be a symplectic manifold, and $ g $ a Riemannian metric on $ M $ compatible with $ \omega $. If $ L $ is a compact Lagrangian submanifold of $ (M,\omega) $, we can compute the volume Vol$ (L) $ of $ L $ using $ g $. A Lagrangian $ L $ is called Hamiltonian stationary if it is a stationary point of the volume functional amongst Lagrangians Hamiltonian isotopic to $ L $. Suppose $ L' $ is a compact Lagrangian in $ {\mathbb C}^n $ which is Hamiltonian stationary and rigid, that is, all infinitesimal Hamiltonian deformations of $ L $ as a Hamiltonian stationary Lagrangian come from rigid motions of $ {\mathbb C}^n $. An example of such $ L' $ is the $ n $-torus $  \bigl\{(z_1,\ldots,z_n)\in{\mathbb C}^n:\vert z_1\vert=a_1, \ldots,\vert z_n\vert=a_n\bigr\} $, for small $ a_1,\ldots,a_n>0 $. I will explain a construction of Hamiltonian stationary Lagrangians in any compact symplectic manifold $ (M,\omega) $, which works by `gluing in' $ tL' $ near a point $ p $ in $ M $ for small $ t>0 $.
Tue, 20/01/2009
15:45 		Frances Kirwan (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Non-reductive GIT
Fri, 12/12/2008
15:30 		Masahiro Futaki (University of Tokyo) Algebraic and Symplectic Geometry Seminar Add to calendar L3
The directed Fukaya category of Landau-Ginzburg models
Tue, 02/12/2008
15:45 		Jon Woolf (Liverpool) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Tilting and the space of stability conditions
Bridgeland's notion of stability condition allows us to associate a complex manifold, the space of stability conditions, to a triangulated category $ D $. Each stability condition has a heart - an abelian subcategory of $ D $ - and we can decompose the space of stability conditions into subsets where the heart is fixed. I will explain how (under some quite strong assumpions on $ D $) the tilting theory of $ D $ governs the geometry and combinatorics of the way in which these subsets fit together. The results will be illustrated by two simple examples: coherent sheaves on the projective line and constructible sheaves on the projective line stratified by a point and its complement.
Tue, 18/11/2008
15:45 		Jeff Giansiracusa (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Introduction to String Topology
Tue, 04/11/2008
15:45 		Tom Coates (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Higher-Genus Gromov-Witten Invariants and Crepant Resolutions
Let X be a Gorenstein orbifold and Y a crepant resolution of X. Suppose that the quantum cohomology algebra of Y is semisimple. We describe joint work with Iritani which shows that in this situation the genus-zero crepant resolution conjecture implies a higher-genus version of the crepant resolution conjecture. We expect that the higher-genus version in fact holds without the semisimplicity hypothesis.
Tue, 28/10/2008
15:45 		Andras Szenes (Université de Genève) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Residues and the intersection ring of toric varieties
Thu, 23/10/2008
16:30 		Soenke Rollenske (Imperial) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Smoothing nodal Calabi-Yau n-folds
Tue, 14/10/2008
15:45 		Jason Lotay (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Ruled Lagrangian submanifolds of the almost symplectic 6-sphere
There is a non-degenerate 2-form on S^6, which is compatible with the almost complex structure that S^6 inherits from its inclusion in the imaginary octonions. Even though this 2-form is not closed, we may still define Lagrangian submanifolds. Surprisingly, they are automatically minimal and are related to calibrated geometry. The focus of this talk will be on the Lagrangian submanifolds of S^6 which are fibered by geodesic circles over a surface. I will describe an explicit classification of these submanifolds using a family of Weierstrass formulae.
Wed, 17/09/2008
16:00 		Fernando Rodriguez-Villegas (UT Austin) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Quiver representations and the enumeration of graphs
We show that the leading terms of the number of absolutely indecomposable representations of a quiver over a finite field are related to counting graphs. This is joint work with Geir Helleloid.
Mon, 07/07/2008
14:15 		Yng-Ing Lee (National Taiwan University) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Lagrangian Mean Curvature Flow
Mean curvature vector is the negative gradient of the area functional. Thus if we deform a submanifold along its mean curvature vector, which is called mean curvature flow (MCF), the area will decrease most rapidly. When the ambient manifold is Kahler-Einstein, being Lagrangian is preserved under MCF. It is thus very natural trying to construct special Lagrangian/ Lagrangian minimal through MCF. In this talk, I will make a brief introduction and report some of my recent works with my coauthors in this direction, which mainly concern the singularities of the flow.
Mon, 30/06/2008
14:15 		Jim Bryan (UBC, Vancouver) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Donaldson-Thomas and Gromov-Witten theory of Calabi-Yau orbifolds
There are two basic theories of curve counting on Calabi-Yau threefolds. Donaldson-Thomas theory arises by considering curves as subschemes; Gromov-Witten theory arises by considering curves as the image of maps. Both theories can also be formulated for orbifolds. Let X be a dimension three Calabi-Yau orbifold and let Y –> X be a Calabi-Yau resolution. The Gromov-Witten theories of X and Y are related by the Crepant Resolution Conjecture. The Gromov-Witten and Donaldson-Thomas theories of Y are related by the famous MNOP conjecture. In this talk I will (with some provisos) formulate the remaining equivalences: the crepant resolution conjecture in Donaldson-Thomas theory and the MNOP conjecture for orbifolds. I will discuss examples to illustrate and provide evidence for the conjectures.
Tue, 03/06/2008
15:45 		Dominic Joyce (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Generalized Donaldson-Thomas invariants. II. Invariants and transformation laws.
This is the second of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. (Still work in progress.) Behrend showed that conventional Donaldson-Thomas invariants can be written as the Euler characteristic of the moduli space of semistable sheaves weighted by a "microlocal obstruction function" \mu. In previous work, the speaker defined Donaldson-Thomas type invariants "counting" coherent sheaves on a Calabi-Yau 3-fold using Euler characteristics of sheaf moduli spaces, and more generally, of moduli spaces of "configurations" of sheaves. However, these invariants are not deformation-invariant. We now combine these ideas, and insert Behrend's microlocal obstruction \mu into the speaker's previous definition to get new generalized Donaldson-Thomas invariants. Microlocal functions \mu have a multiplicative property implying that the new invariants transform according to the same multiplicative transformation law as the previous invariants under change of stability condition. Then we show that the invariants counting pairs in the previous seminar are sums of products of the new generalized Donaldson-Thomas invariants. Since the pair invariants are deformation invariant, we can deduce by induction on rank that the new generalized Donaldson-Thomas invariants are unchanged under deformations of the underlying Calabi-Yau 3-fold.

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

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