Title: 

1. Lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces. Applications.

2. Homogeneous and uniform vector bundles on projective spaces.

3. Ulrich bundles on general surfaces in a projective space.

Abstract: 

1. In the first lecture we will present a new and useful method to construct isotròpic and lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces.

2. In this lecture we will address the long standing open conjecture: Any uniform vector bundle of rank r<2n on a projective space of dimension n is homogeneous.

3: In the 3rd lecture we will discuss the problem of determining the pairs (r,d) such that a general Surface of degree d in the 3-dimensional projective space supports an Ulrich bundle of rank r.

Title: Hyperelliptic curves and Ulrich sheaves on the complete intersection of two quadrics

Subtitle1: Vector bundles on hyperelliptic curves. Koszul algebras and BGG.

Subtitle2: Clifford algebra, Morita equivalence and Reid's theorem

Subtitle3: Tate resolutions of O_X-modules and Ulrich bundles

Abstract: Using the connection between hyperelliptic curves, Clifford algebras, and smooth complete intersections X of two quadrics, we describe Ulrich bundles on X and construct some of minimal possible rank. 

We will start out describing vector bundles on hyperelliptic curves via matrix factoriztions. The Jacobian of the hyperelliptic curve associated to X can be identified with the space of maximal isotropic subspaces of the pencil by Reid's theorem. We give a proof of Reid's theorem and apply these techniques to describe Ulrich bundles on X. Finally we will construct Ulrich bundles of smallest possible rank on X.