Forthcoming Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar

Tue, 19/01/2010
15:45 		Damiano Testa (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Big rational surfaces
The Cox ring of a variety is an analogue of the homogeneous coordinate ring of projective space. Cox rings are not defined for every variety and even when they are defined, they need not be finitely generated. Varieties for which the Cox ring is finitely generated are called Mori dream spaces and, as the name suggests, they are particularly well-suited for the Minimal Model Program. Such varieties include toric varieties and del Pezzo surfaces. I will report on joint work with T. Várilly and M. Velasco where we introduce a class of smooth projective surfaces having finitely generated Cox ring. This class of surfaces contains toric surfaces and (log) del Pezzo surfaces.
Tue, 26/01/2010
14:00 		Richard Thomas (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
(HoRSe seminar) GW/stable pairs on K3 surfaces
The Katz-Klemm-Vafa formula is a conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. In genus 0 it reduces to the (proved) Yau-Zaslow formula. I will explain how the correspondence between stable pairs and Gromov-Witten theory for toric 3-folds (proved by Maulik-Oblomkov-Okounkov-Pandharipande), some calculations with stable pairs (due to Kawai-Yoshioka) and some deformation theory lead to a proof of the KKV formula. (This is joint work with Davesh Maulik and Rahul Pandharipande. Only they understand the actual formulae. People who like modular forms are not encouraged to come to this talk.)
Tue, 26/01/2010
15:45 		Richard Thomas (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Symmetric and reduced obstruction theories
I will describe some more of the deformation theory necessary for the first talk. This leads to a number of natural questions and counterexamples. This talk requires a strong stomach, or a fanatical devotion to symmetric obstruction theories.
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.    
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, 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, 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, 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, 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, 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.

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

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