Program - December 19, 2018 (Aula C, second floor)
You can download the poster and the program.
14:00-14:50
Emma Brakkee
Two polarized K3 surfaces associated to the same cubic fourfold
Abstract:
For infinitely many d, Hassett showed that special cubic fourfolds of
discriminant d are related to polarized K3 surfaces of degree d via
their Hodge structures. For half of the d, a generic special cubic has
not one but two different associated K3 surfaces. This induces an
involution on the moduli space of polarized K3 surfaces of degree d. We
give a geometric description of this involution. As an application, we
obtain examples of Hilbert schemes of two points on K3 surfaces that are
derived equivalent but not birational.
15:00-15:50
Andrea Ricolfi
A higher rank local DT/PT correspondence
Abstract:
Donaldson-Thomas invariants are virtual counts of stable objects in the derived category of a Calabi-Yau 3-fold X.
Toda proved the DT/PT correspondence in arbitrary rank, linking DT invariants to PT invariants via wall-crossing in a suitable heart in D(X).
We will show a local version of Toda's correspondence, centered at a fixed slope-stable sheaf of a given rank.
This generalises the (rank one) local DT/PT correspondence centered at a Cohen-Macaulay curve embedded in X.
Our wall-crossing formula is the virtual analogue of a recent Euler characteristic calculation for Quot schemes on 3-folds proved by Gholampour-Kool.
Joint work with Sjoerd Beentjes.
16:30-17:20
Jeff Hicks
Tropical Lagrangian Submanifolds from Lagrangian Cobordisms
Abstract:
Mirror symmetry conjectures that there is a relation between the symplectic geometry of a Calabi-Yau manifold X, and the complex geometry of a mirror Calabi-Yau manifold Y. These spaces are expected to have dual torus fibrations over a common affine base Q. One proposed mechanism for this duality comes from comparing symplectic geometry on X and complex geometry on Y to the affine geometry on Q.
The recent work of Matessi and Mikhalkin provides a method to lift tropical varieties on Q to Lagrangian submanifolds on X, filling in one of the steps of this comparison.
We will construct a version of these tropical Lagrangian submanifolds using a Lagrangian surgery cobordism. If time permits, we will discuss how these Lagrangian submanifolds are homologically mirror to hypersurfaces in toric varieties.