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*Special Lecture in Differential Geometry and Geometric Analysis*Ìý

This talk centers on the degenerations of Calabi-Yau metrics.WeÌýwill focus on the interactions between algebraic degenerations and metricÌýconvergence with highly singular behaviors in the collapsed contexts. As theÌýcomplex structures degenerate, the collapsing Calabi-Yau metrics may exhibitÌývarious wild geometric properties and highly non-algebraic features.Ìý

First, as motivating examples, we will describe our recent results on theÌýnew collapsing mechanisms of K3 surfaces. Next, we will switch to higherÌýdimensions and we will exhibit some entirely new constructions ofÌýdegenerating Calabi-Yau metrics which are expected to work in more broaderÌýcontexts. Complex structures degeneration will be accurately characterizedÌýby the bubbling and singularity analysis in a geometric manner.