[1st week]

Cristina Manolache - Moduli Spaces of Sheaves (introductory course)

This course will be an introduction to Moduli Spaces of vector bundles. A moduli space of stable vector bundles on a smooth, algebraic variety X is a scheme whose points are in ”natural bijection” to isomorphic classes of stable vector bundles on X. Using Geometric Invariant Theory the moduli space can be constructed as a quotient of certain Quot-scheme by a natural group action. We introduce the crucial concept of stability of vector bundles over smooth projective varieties and we give a cohomological characterization of the (semi)stability. The notion of (semi)stability is needed to ensure that the set of vector bundles one wants to parameterize is small enough to be parameterized by a scheme of finite type. We introduce the formal definition of moduli functor, fine moduli space and coarse moduli space and we recall some generalities on moduli spaces of vector bundles. Then we focus on vector bundles on algebraic surfaces. Quite a lot is known in this case and we will review the main results, some of which will illustrate how the geometry of the surface is reflected in the geometry of the moduli space. Going beyond surfaces, we introduce the notion of monad, which allow the classification of vector bundles on, for instance, P^3. Monads appeared in a wide variety of contexts within algebraic geometry, and they are very useful when we want to construct vector bundles with prescribed invariants like rank, determinant, Chern classes, etc. Finally, we will study moduli spaces of vector bundles on higher dimensional varieties. As we will stress, the situation drastically differs and results like the smoothness and irreducibility of moduli spaces of stable vector bundles on algebraic surfaces turn to be false for moduli spaces of stable vector bundles on higher dimensional algebraic varieties.

Laura Pertusi - Derived Categories of Sheaves (introductory course)

Derived category of coherent sheaves is a convenient environment of investigating algebraic geometry of a variety. It provides useful techniques and gives a perspective point of view. On the level of derived categories one can see unexpected connections which are not visible on the classical level. I will try to give an introduction into the techniques of derived categories and semiorthogonal decomposition and (hopefully) will present many examples of applications of derived categories in algebraic geometry.

Michel Brion - Geometric Invariant Theory (introductory course)

Geometric Invariant Theory (GIT) studies the actions of reductive algebraic groups on algebraic varieties and the constructions of quotients in algebraic geometry. One of the most relevant aspect of GIT is its application to moduli theory: we can indeed approach moduli problems trying to construct moduli spaces as GIT quotients. In this course we will present the basic notions of geometric invariant theory, we will illustrate its connections to moduli and provide concrete examples of moduli spaces obtained as GIT quotients.

Abdelmoubine Amar Henni - Quiver Moduli Spaces (exercise session)

These exercise sessions will be divided in three main parts; the first part is about representations of a given quiver and morphisms between different representations. This allows one to introduce and check irreducibility of a given quiver representation. In the second part we introduce the notion of stability of a quiver representation and as an application we give some examples of the construction of the GIT quotient as done by A. King. The last part will be related to the construction and properties of moduli spaces of representations of quivers such as the Hilbert scheme of points on the affine plane C^2.

[2nd week]

Cristian Martinez - Bridgeland stability: the General Theory (introductory course)

Bridgeland stability is a powerful tool for extracting geometry from homological algebra. In particular, it gives a framework for studying moduli spaces of objects in a triangulated category, such as the derived category of an algebraic variety. The subject was born as a mathematical interpretation of work in string theory, but has since impacted many areas, including classical algebraic geometry, derived categories of coherent sheaves, enumerative geometry, homological mirror symmetry, and symplectic geometry. The goal of this course is to develop the foundations of Bridgeland stability, covering the following topics: 1) The theory of t-structures on triangulated categories, including examples via tilting. 2) The definition of stability conditions and the stability manifold, as well as Bridgeland's deformation theorem. 3) Constructions of stability conditions. 4) Moduli spaces of stable objects.

Marcos Jardim - Instanton Objects and Bridgeland Stability Conditions (introductory course)

Instantons, which emerged in particle physics, have been intensely studied since the 1970's and had an enormous impact on mathematics since then. In this paper, we focus on one particular way in which ideas originating in mathematical physics have guided the development of algebraic geometry in the past 40+ years. To be precise, we examine how the notion of mathematical instanton bundles in algebraic geometry has evolved from a class of vector bundles over the 3-dimensional projective space both to a class of torsion-free sheaves on projective varieties of arbitrary dimension, and to a class of objects in the derived category of Fano threefolds. The final goal of the course is to determine suitable chambers in the space of Bridgeland stability conditions that contain these instanton objects, which include the classical rank 2 instanton bundles over the projective space, as Bridgeland stable objects, and describe the corresponding Bridgeland moduli space.

Xiaolei Zhao -  Noncommutative abelian surfaces and Kummer type kyperkähler varieties  (advanced course)

Examples of noncommutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkähler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these noncommutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkähler manifolds deformation equivalent to a generalized Kummer variety is not available from classical geometry.

In this lecture series, we will construct families of noncommutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkähler manifolds of Kummer type. Based on joint work with Arend Bayer, Alex Perry and Laura Pertusi.

Hannah Dell - Counting Invariants through Bridgeland Stability (advanced course)

I will start by providing a brief overview of (generalized) DT invariants and their wall-crossing formulae. Following that, I will delve into how (weak) Bridgeland stability conditions are applied to explore these invariants. Specifically, I will explain joint work with R. Thomas demonstrating that all DT theory is governed by rank 1 DT theory.

James Hotchkiss - Bridgeland Stability (exercise session)