Explore Hodge Theory and Lagrangian Fibrations in this comprehensive mathematics colloquium talk by Christian Schnell. Delve into the intricacies of holomorphic symplectic manifolds and their Lagrangian fibrations, examining the conjectures proposed by Junliang Shen, Qizheng Yin, and Davesh Maulik. Learn about Kähler manifolds, holomorphic 2-forms, and the properties of smooth fibers in n-dimensional abelian varieties. Investigate topics such as hard Lefschetz theorems, isomorphisms, symmetries, and decomposition theorems. Gain insights into the non-compact case, symplectic forms, complexes of sheaves, and chromology. Understand the borderline case and its implications for the field of algebraic geometry.