In the first talk, I will describe a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models from my joint work with Andrew Harder and Alan Thompson (the so-called "DHT Conjecture"). More precisely, we show that if a Calabi-Yau admits a "Tyurin degeneration" to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev-Nikulin mirror symmetry for K3 surfaces, and explicit families of threefolds that are not realized as complete intersections in toric varieties. This is largely based on our contribution to the String-Math 2015 proceedings, arXiv:1601.08110v3 , with some important updates.

The second talk, based on arXiv:1701.03279v1, provides a classification of threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the appropriate moduli space of K3 surfaces, which we call the "generalized functional invariant". Then we show that if the threefold total space is a smooth Calabi-Yau, there are only finitely many possibilities for the polarizing lattice and the form of the generalized functional invariant. This last makes essential use of our work on "Hodge Numbers from Picard-Fuchs Equations" in arXiv:1612.09439v1 (which builds off of recent work of Eskin, Kontsevich, Moller, and Zorich). Finally, we construct explicit examples of Calabi-Yau threefolds realizing each case and compute their Hodge numbers. A mirror-symmetric interpretation of the classification reveals perfect agreement with the predictions of the DHT Conjecture. Speaker(s): Charles Doran (Alberta)