Past Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar (past)

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, 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, 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, 01/12/2009
15:00 		Simon Donaldson (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar Imperial College
(HoRSe seminar at Imperial College) Moduli of Calabi-Yau 3-folds and instantons on $ G_2 $ manifolds
This talk will be largely speculative. First we consider the formal properties that could be expected of a "topological field theory" in 6+1 dimensions defined by $ G_2 $ instantons. We explain that this could lead to holomorphic bundles over moduli spaces of Calabi-Yau 3-folds whose ranks are the DT-invariants. We also discuss in more detail the compactness problem for $ G_2 $ instantons and associative submanifolds. The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.
Tue, 01/12/2009
13:30 		Simon Donaldson (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar Imperial College
(HoRSe seminar at Imperial college) Gauge theory and exceptional holonomy
This talk will review material, well-known to specialists, on calibrated geometry and Yang-Mills theory over manifolds with holonomy $ SU(3) $, $ G_2 $ or $ Spin(7) $. We will also describe extensions of the standard set-up, modelled on Gromov's "taming forms" for almost-complex structures. The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.
Tue, 24/11/2009
15:45 		Julius Ross (Cambridge) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Constant scalar curvature orbifold metrics and stability of orbifolds through embeddings in weighted projective spaces
There is a conjectural relationship due to Yau-Tian-Donaldson between stability of projective manifolds and the existence of canonical Kahler metrics (e.g. Kahler-Einstein metrics). Embedding the projective manifold in a large projective space gives, on one hand, a Geometric Invariant Theory stability problem (by changing coordinates on the projective space) and, on the other, a notion of balanced metric which can be used to approximate the canonical Kahler metric in question. I shall discuss joint work with Richard Thomas that extends this framework to orbifolds with cyclic quotient singularities using embeddings in weighted projective space, and examples that show how several obstructions to constant scalar curvature orbifold metrics can be interpreted in terms of stability.
Tue, 10/11/2009
15:45 		Christian Pauly (Montpellier) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Opers, Quot-schemes and Frobenius-destabilised vector bundles over curves
In this talk I will introduce and study opers over a smooth projective curve X defined over a field of positive characteristic. I will describe a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F^*(E) under the Frobenius map F of X has maximal Harder-Narasimhan polygon and the set of opers having zero p-curvature. These sets turn out to be finite, which allows us to derive dimensions of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X.
Tue, 20/10/2009
15:45 		Balazs Szendroi (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Hilbert schemes of threefolds: virtual motives, Hodge modules and deformed MacMahon formulae II
I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.
Tue, 20/10/2009
14:00 		Balazs Szendroi (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar SR2
(HoRSe seminar) Hilbert schemes of threefolds: virtual motives, Hodge modules and deformed MacMahon formulae I
I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.
Thu, 15/10/2009
15:45 		Gregory Sankaran (Bath) Algebraic and Symplectic Geometry Seminar Add to calendar SR2
(COW seminar) Moduli of irreducible symplectic manifolds
I shall describe joint work with Gritsenko and Hulek in which we study the moduli spaces of polarised holomorphic symplectic manifolds via their periods. There are strong similarities with moduli spaces of K3 surfaces, but also some important differences, notably that global Torelli fails. I shall explain (conjecturally) why and show how the techniques used to obtain general type results for K3 moduli can be modified to give similar, and quite strong, results in this case. Mainly I shall concentrate on the case of deformations of Hilbert schemes of K3 surfaces.
Thu, 15/10/2009
14:00 		Timothy Logvinenko (Liverpool) Algebraic and Symplectic Geometry Seminar Add to calendar DH 3rd floor SR
(COW seminar) Reid's recipe and derived categories
We give a three dimensional generalization of the classical McKay correspondence construction by Gonzales-Sprinberg and Verdier. This boils down to computing for the Bridgeland-King-Reid derived category equivalence the images of twists of the point sheaf at the origin of C^3 by irreducible representations of G. For abelian G the answer turns out to be closely linked to a piece of toric combinatorics known as Reid's recipe.
Tue, 13/10/2009
15:45 		Matt Kerr (Durham) Algebraic and Symplectic Geometry Seminar Add to calendar L3
A Tour of Normal Functions and Algebraic Cycles
Associated to a pencil of algebraic curves with singular fibres is a bundle of Jacobians (which are abelian varieties off the discriminant locus of the family and semiabelian varieties over it). Normal functions, which are holomorphic sections of such a Jacobian bundle, were introduced by Poincare and used by Lefschetz to prove the Hodge Conjecture (HC) on algebraic surfaces. By a recent result of Griffiths and Green, an appropriate generalization of these normal functions remains at the center of efforts to establish the HC more generally and understand its implications. (Furthermore, the nature of the zero-loci of these normal functions is related to the Bloch-Beilinson conjectures on filtrations on Chow groups.) Abel-Jacobi maps give the connection between algebraic cycles and normal functions. In this talk, we shall discuss the limits and singularities of Abel-Jacobi maps for cycles on degenerating families of algebraic varieties. These two features are strongly connected with the issue of graphing admissible normal functions in a Neron model, properly generalizing Poincare's notion of normal functions. Some of these issues will be passed over rather lightly; our main intention is to give some simple examples of limits of AJ maps and stress their connection with higher algebraic K-theory. A very new theme in homological mirror symmetry concerns what the mirror of a normal function should be; in work of Morrison and Walcher, the mirror is related to counting holomorphic disks in a CY 3-fold bounding on a Lagrangian. Along slightly different lines, we shall briefly describe a surprising application of "higher" normal functions to growth of enumerative (Gromov-Witten) invariants in the context of local mirror symmetry.
Tue, 22/09/2009
14:00 		Kentaro Nagao (Oxford and Kyoto) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Cluster algebra via DT theory
Tue, 22/09/2009
12:00 		Tom Bridgeland (University of Sheffield) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Tilting and cluster transfromations
I'll explain (following Kontsevich and Soibelman) how cluster transformations intertwine non-commutative DT invariants for CY3 algebras related by a tilt.

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

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