Talk

We will consider arbitrary holomorphic families of Kähler structures on a fixed prequantizable symplectic manifold and provided the corresponding family of Kähler quantisations for level k form a vector bundle over the chosen family of Kähler structures, we will for large enough k construct a large family of Hitchin connections in this bundle. We will also consider a natural family of Hermitian structures on the bundle of quantum spaces and understand which (unique) Hitchin connection is compatible with a given one of these Hermitian structures.

Cubic fourfold plays a central role in algebraic geometry because of its close relation to hyper-Kahler geometry. In this talk I will start with this relation and discuss the Hodge theory and moduli theory for cubic fourfolds. Then I will introduce an application of the above theories, namely, a complete classification of the symplectic automorphism groups of smooth cubic fourfolds. This work can be regarded as a higher dimensional analogue of Mukai's celebrated result on classification of finite symplectic automorphism groups of K3 surfaces. This is a joint work with Radu Laza.

In the first hour, I will describe the standard trisection of CP^2 and how to put complex curves in CP^2 into bridge position with respect to this trisection.  Time permitting, I will also present trisections of projective surfaces obtained as a branched covers over these curves.  In the second hour, I will describe how to compute the homology and intersection form from a trisection diagram.

In the first hour, I will describe the standard trisection of CP^2 and how to put complex curves in CP^2 into bridge position with respect to this trisection.  Time permitting, I will also present trisections of projective surfaces obtained as a branched covers over these curves.  In the second hour, I will describe how to compute the homology and intersection form from a trisection diagram.

The classical mapping class group Γ(g) of a surface of genus g shares many features with its higher dimensional analogue Γ(g,n)—the group of isotopy classes of diffeomorphisms of #ᵍ(Sⁿ x Sⁿ)—but some aspects become easier in high dimensions. This enabled Kreck in the 70’s to describe Γ(g,n) for n>2 in terms of an arithmetic group and the group of exotic spheres. His answer, however, left open two extension problems which were later understood in some dimensions, but remained unsettled in most cases.