Forthcoming Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar

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, 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, 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, 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, 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, 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, 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
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.

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

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