We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible fundamental groups of complex algebraic varieties. This was first described by Goldman and Millson in the case of compact Kähler manifold, using formal deformation theory and differential graded Lie algebras. We review this using methods of Hodge theory and of derived deformation theory and we are able to describe locally the representation variety for non-compact smooth varieties and representations underlying a variation of Hodge structure.[-]

Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L∞-algebras, giving in general ∞-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a correspondence between infinitesimal deformation problems and partition Lie algebras over a positive characteristic field. In this talk, I will explain how to construct an analogue of the integration functor for certain point-set models of (spectral) partition Lie algebras, and how this integration functor can recover the associated deformation problem under some assumptions. Furthermore, I will discuss some applications of these constructions to unstable p-adic homotopy theory.[-]

I am going to define toroidal crossing singularities and toroidal crossing varieties and explain how to produce them in large quantities by subdividing lattice polytopes. I will then explain the statement of a global smoothing theorem proved jointly with Felten and Filip. The theorem follows the tradition of well-known theorems by Friedman, Kawamata-Namikawa and Gross-Siebert. In order to apply a variant of the theorem to construct (conjecturally all) projective Fano manifolds with non-empty anticanonical divisor, Corti and Petracci discovered the necessity to allow for particular singular log structures that are known by the inspiring name 'admissible'. I will explain the beautiful classical geometric curve-in-surface geometry that underlies this notion and hint at why we believe that we can feed these singular log structures into the smoothing theorem in order to produce all 98 Fano threefolds with very ample anticanonical class by a single method.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]