Abstracts of the courses

Toric deformations  Klaus Altmann:
In the first lecture we study the infinitesimal theory and present formulae for the vector spaces T1 and T2 in different situations. While (affine) toric varieties are given by cones over polyhedra, we will see that T1 is related to the set of Minkowski summands of these gadgets.
In the second part we focus on deformations over ``true'' base spaces. This includes mparameter families or even the entire versal deformation being compatible with a given character. To be able to describe the nearby fibers (e.g. smoothings) we need the language of Tvarieties and polyhedral divisors.
The last lecture is thought to get a closer look to the easiest special case, namely to the twodimensional cyclic quotient singularities. For them we describe the socalled Presolutions which are in a 1to1 correspondence to the irreducible components of the versal base space (Kollár/ShephardBarron) and discuss the subfunctor of qGdeformations, i.e. of those satisfying Kollár's condition.

Obstruction to formality  Manfred Lehn:

Homotopy theory of DGLie algebras and formality criteria  Marco Manetti:
In the first part of the lectures we give an elementary description of the homotopy categories of differential graded Lie algebras and Linfinity algebras over a field of characteristic 0, together with some applications to deformation theory.
In the second part we study formality problems. In particular we give some necessary and/or sufficient conditions on a DGLie algebra in order to be homotopy equivalent to its cohomology Lie algebra.
Abstracts of the talks

Deformation functor associated with a DGLie algebra and applications  Donatella Iacono:
In this seminar, we focus our attention on deformation theory via Differential Graded Lie Algebras (DGLA). In particular, we describe how to associate a deformation functor with a DGLA and we describe some properties. Finally, we analyze some examples.

Computational aspects of toric deformations  Lars Kastner:
Besides giving classical deformation theory a new combinatorial interpretation, toric deformation theory also has many algorithmic advantages. Many examples that are infeasible to treat with computer algebra can be dealt with with combinatorial software frameworks. This talk will compare some classical algorithms for computing the spaces $T^1$ and $T^2$ with their toric counterpart. We will give an overview of the different steps needed to assemble these spaces according to the
toric construction. Additionally we will look at toric translations of algebraic properties of certain toric singularities that come for free
with this construction.

Formality and singularities of some moduli spaces of sheaves on K3 surfaces  Ziyu Zhang:
We consider the moduli space of Hsemistable sheaves of a not necessarily primitive class on a projective K3 surface (X,H). To understand its singularities, Kaledin and Lehn conjectured that a certain differential graded algebra that controls the deformations of these sheaves is formal, and developed a technique that reduces the problem to the computation of some vector bundles over the base of the twistor family associated to (X,H). I will show how to carry out such computation in some explicit examples.