Abstract: In this talk, we will define the local fundamental group of an isolated singularity as the fundamental group of its link. For algebraic singularities, Kapovich and Kollár showed that any finitely presented group can appear. Although, restrictions appear if one restricts to different families of singularities. For the case of log canonical singularities, we will show that in dimension 3 the free groups cannot appear. Finally, we will present a construction for some local fundamental groups in dimensions 3 and 4. This construction, in particular, will yield the free groups as the local fundamental group of isolated 4-dimensional log canonical singularities. This is based on joint work with Joaquín Moraga

Abstract: Log Calabi-Yau pairs can be thought of as generalizations of Calabi-Yau varieties. Among them, there exists the notion of volume preserving maps. A natural and tough problem is classifying log Calabi-Yau pairs up to volume preserving equivalence. For this task, one can make use of the coregularity, the most important volume preserving invariant of a log Calabi-Yau pair. Recently, other invariants and refinements in the classification have been studied, especially in the coregularity 0 case. In this talk, I will survey some results and comment on works in progress regarding what is known to date about the classification for the case of log Calabi-Yau pairs of the form $(\mathbb{P}^3,D)$.

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