I will report on aspects of work with Sheridan and Ganatra in which we show how homo- logical mirror symmetry for Calabi-Yau manifolds implies equality of Yukawa couplings on the A- and B-sides. On the A-side, these couplings are generating functions for genus-zero GW invariants. On the B-side, one has a degenerating family of CY manifolds, and the couplings are fiberwise integrals involving a holomorphic volume form. We show that the Fukaya category implicitly "knows" the correct normalization of this volume form, as well as the mirror map.[-]

There are five superstring theories, all formulated in 9+1 spacetime dimensions; lower-dimensional theories are studied by taking some of the spatial dimensions to be compact (and small). One of the remarkable features of this setup is that the same lower-dimensional theory can often be realized by pairing different superstring theories with different geometries. The focus of these lectures will be on the mathematical implications of some of these physical “dualities.”
Our main focus from the string theory side will be the superstring theories known as type IIA and type IIB. The duality phenomenon occurs for compact spaces of various dimensions and types. We will begin by discussing “T-duality” which uses tori as the compact spaces. We will then digress to introduce M-theory as a strong-coupling limit of the type IIA string theory, and F-theory as a variant of the type IIB string theory whose existence is motivated by T-duality. The next topic is compactifying the type IIA and IIB string theories on K3 surfaces (where the duality involves a change of geometric parameters but not a change of string theory).
By the third lecture, we will have turned our attention to Calabi-Yau manifolds of higher dimension, and the “mirror symmetry” which relates pairs of them. Various aspects of mirror symmetry have various mathematical implications, and we will explain how these are conjecturally related to each other.[-]

There are five superstring theories, all formulated in 9+1 spacetime dimensions; lower-dimensional theories are studied by taking some of the spatial dimensions to be compact (and small). One of the remarkable features of this setup is that the same lower-dimensional theory can often be realized by pairing different superstring theories with different geometries. The focus of these lectures will be on the mathematical implications of some of these physical “dualities.”
Our main focus from the string theory side will be the superstring theories known as type IIA and type IIB. The duality phenomenon occurs for compact spaces of various dimensions and types. We will begin by discussing “T-duality” which uses tori as the compact spaces. We will then digress to introduce M-theory as a strong-coupling limit of the type IIA string theory, and F-theory as a variant of the type IIB string theory whose existence is motivated by T-duality. The next topic is compactifying the type IIA and IIB string theories on K3 surfaces (where the duality involves a change of geometric parameters but not a change of string theory).
By the third lecture, we will have turned our attention to Calabi-Yau manifolds of higher dimension, and the “mirror symmetry” which relates pairs of them. Various aspects of mirror symmetry have various mathematical implications, and we will explain how these are conjecturally related to each other.[-]

Homological mirror symmetry asserts that the connection, discovered by physicists, between a count of rational curves in a Calabi-Yau manifold and period integrals of its mirror should follow from an equivalence between the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. Physicists' observation can be reformulated as, or rather upgraded to, a statement about an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the above assertion would be to try to attach similar Hodge-like data to abstract derived categories. The aim of the talk is to report on some recent progress in this direction and illustrate the approach in the context of what physicists call Landau-Ginzburg B-models.[-]

This partly expository talk focuses on the notion of ”symplectic Landau-Ginzburg models”, i.e. symplectic manifolds equipped with maps to the complex plane, ”stops”, or both, as they naturally arise in the context of mirror symmetry. We describe several viewpoints on these spaces and their Fukaya categories, their monodromy, and the functors relating them to other flavors of Fukaya categories. (This touches on work of Abouzaid, Seidel, Ganatra, Hanlon, Sylvan, Jeffs, and others).[-]

The intimate relation between the arithmetic properties of varieties varying in families and the properties of the associated Picard-Fuchs differential is subject with a long and rich history that can be traced back to Deuring, Igusa, Dwork, Honda, Katz from which the notion of crystals emerged. A particular nice situation arises from families of Calabi-Yau motives, which can arise via various constructions, most notably via Mirror-Symmetry. In the two talks I will try to give a rough overview of this field, and illustrate it with specific examples. In particular, I will indicate how Calabi-Yau operators can be used to realise certain rank 4 motives attached Siegel paramodular forms by specific Calabi-Yau threefolds.[-]