Past Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar (past)

Tue, 22/02/2011
14:00 		Tony Pantev (Univesity of Pennsylvania) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Mirror symmetry and mixed Hodge structures I

I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative mixed Hodge structure on the cohomology of the Fukaya category. I will discuss computable Hodge theoretic invariants arising from twist functors, and from geometric extensions. I will also explain how the instanton-corrected Chern-Simons theory fits in the framework of normal functions in non-commutative Hodge theory and will give applications to explicit descriptions of quantum Lagrangian branes. This is a joint work with L. Katzarkov and M. Kontsevich.
Tue, 08/02/2011
15:45 		Nicolas Addington (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Derived Categories of Cubic 4-Folds
If $ X $ is a Fano variety with canonical bundle $ O(-k) $, its derived category has a semi-orthogonal decomposition (I will say what that means)
\[ D(X) = \langle O(-k+1), ..., O(-1), O, A \rangle, \]
where the subcategory $ A $ is the "interesting piece" of $ D(X) $. In the previous talk we saw that $ A $ can have very rich geometry. In this talk we will see a less well-understood example of this: when $ X $ is a smooth cubic in $ P^5 $, $ A $ looks like the derived category of a K3 surface. We will discuss Kuznetsov's conjecture that $ X $ is rational if and only if $ A $ is geometric, relate it to Hassett's earlier work on the Hodge theory of $ X $, and mention an autoequivalence of $ D(Hilb^2(K3)) $ that I came across while studying the problem.
Tue, 08/02/2011
14:00 		Nicolas Addington (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Complete Intersections of Quadrics
There is a long-studied correspondence between intersections of two quadrics and hyperelliptic curves, first noticed by Weil and since used as a testbed for many fashionable theories: Hodge theory, motives, and moduli of vector bundles in the '70s and '80s, derived categories in the '90s, non-commutative geometry and mirror symmetry today. The story generalizes to three, four, and more quadrics, exhibiting new geometric behaviour at each step. The case of four quadrics nicely illustrates the modern theory of flops and derivced categories and, as a special case, gives a pair of derived-equivalent Calabi-Yau 3-folds.
Tue, 25/01/2011
15:45 		Jun Li (Stanford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Localized virtual cycles, and applications to GW and DT invariants II
We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).
Tue, 25/01/2011
14:00 		Jun Li (Stanford) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
(HoRSe seminar) Localized virtual cycles, and applications to GW and DT invariants I
We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).
Tue, 18/01/2011
15:45 		Artan Sheshmani (University of Illinois at Urbana Champaign) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Wall-crossing and invariants of higher rank stable pairs
We introduce a higher rank analog of Pandharipande-Thomas theory of stable pairs. Given a Calabi-Yau threefold $ X $, we define the higherrank stable pairs (which we call frozen triples) given by the data $ (F,\phi) $ where $ F $ is a pure coherent sheaf with one dimensional support over $ X $ and $ \phi:{\mathcal O}^r\rightarrow F $ is a map. We compute the Donaldson-Thomas type invariants associated to the frozen triples using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman. This work is a sequel to arXiv:1011.6342, where we gave a deformation theoretic construction of a higher rank enumerative theory of stable pairs over a Calabi-Yau threefold, and we computed similar invariants using Graber-Pandharipande virtual localization technique.
Tue, 30/11/2010
15:45 		Andras Juhasz (Cambridge) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Cobordisms of sutured manifolds
Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms.
Tue, 23/11/2010
15:45 		Hans-Joachim Hein (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Gravitational instantons from rational elliptic surfaces
Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.
Tue, 16/11/2010
15:45 		Kai Behrend (Vancouver) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) On the calculus underlying Donaldson-Thomas theory II
On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exterior differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.
Tue, 16/11/2010
14:00 		Kai Behrend (Vancouver) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
(HoRSe seminar) On the calculus underlying Donaldson-Thomas theory I
On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.
Tue, 09/11/2010
15:45 		Amin Gholampour and Yunfeng Jiang (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Counting invariants for the ADE McKay quivers
Tue, 02/11/2010
15:45 		Ben Davison (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Motivic Donaldson-Thomas invariants and 3-manifolds
I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.
Tue, 26/10/2010
15:45 		Alexander Ritter (Cambridge) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Topological quantum field theory structure on symplectic cohomology
Symplectic cohomology is an invariant of symplectic manifolds with contact type boundary. For example, for disc cotangent bundles it recovers the homology of the free loop space. The aim of this talk is to describe algebraic operations on symplectic cohomology and to deduce applications in symplectic topology. Applications range from describing the topology of exact Lagrangian submanifolds, to proving existence theorems about closed Hamiltonian orbits and Reeb chords.
Tue, 19/10/2010
15:45 		Andre Neves (Imperial) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Finite time singularities for Lagrangian mean curvature flow
I will show that given smooth embedded Lagrangian L in a Calabi-Yau, one can find a perturbation of L which lies in the same hamiltonian isotopy class and such that the correspondent solution to mean curvature flow develops a finite time singularity. This shows in particular that a simplified version of the Thomas-Yau conjecture does not hold.
Tue, 12/10/2010
15:45 		Daniel Huybrechts (Bonn) Algebraic and Symplectic Geometry Seminar Add to calendar L3
(HoRSe seminar) Spherical objects on K3 surfaces II
Both parts will deal with spherical objects in the bounded derived category of coherent sheaves on K3 surfaces. In the first talk I will focus on cycle theoretic aspects. For this we think of the Grothendieck group of the derived category as the Chow group of the K3 surface (which over the complex numbers is infinite-dimensional due to a result of Mumford). The Bloch-Beilinson conjecture predicts that over number fields the Chow group is small and I will show that this is equivalent to the derived category being generated by spherical objects (which I do not know how to prove). In the second talk I will turn to stability conditions and show that a stability condition is determined by its behavior with respect to the discrete collections of spherical objects.
Tue, 12/10/2010
14:00 		Daniel Huybrechts (Bonn) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
(HoRSe seminar) Spherical objects on K3 surfaces I
Both parts will deal with spherical objects in the bounded derived category of coherent sheaves on K3 surfaces. In the first talk I will focus on cycle theoretic aspects. For this we think of the Grothendieck group of the derived category as the Chow group of the K3 surface (which over the complex numbers is infinite-dimensional due to a result of Mumford). The Bloch-Beilinson conjecture predicts that over number fields the Chow group is small and I will show that this is equivalent to the derived category being generated by spherical objects (which I do not know how to prove). In the second talk I will turn to stability conditions and show that a stability condition is determined by its behavior with respect to the discrete collections of spherical objects.

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

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