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Explore a comprehensive lecture on comparing geometric and Hodge-theoretic compactifications, delivered by Philip Engel from the University of Georgia. Delve into the intricacies of classical period domains, Griffiths' transversality condition, and the identification of moduli spaces with arithmetic quotients. Examine the algebro-geometric compactifications of D/Gamma, including Baily-Borel and toroidal compactifications, and their generalization to semitoroidal compactifications in weight 2 cases. Investigate the relationship between distinguished (semi)toroidal compactifications and moduli spaces of generalized geometric objects, such as stable abelian varieties and stable K3 surfaces. Review the works of Mumford, Namikawa, and Alexeev on abelian varieties, focusing on the second Voronoi fan and its connection to Delaunay-Voronoi decompositions. Discover recent joint work on extending these concepts to stable K3 surfaces, exploring log general type pairs, recognizable divisors, and their associated semifans. Gain insights into the K3 analogue of Delaunay-Voronoi decompositions and their relationship to polyhedral decompositions of singular integral-affine spheres.
