Mathematical Colloquium

Condensed Mathematics is a new approach to the study of topological Abelian groups, rings, and modules recently introduced by Clausen and Scholze. One of the main problems encountered in commutative algebra and algebraic geometry is that certain fundamental tools such as biduality are only available under very strong finiteness conditions. These are usually not preserved even by the most common operations, such as pushforward along a non-proper morphism. One can try to rectify this by considering topological (instead of abstract) Abelian groups, but this category is no longer Abelian, thus causing further problems.
I shall explain the notions of condensed Abelian group and solid condensed Abelian group (solidity should be thought of as an analogue of completeness), using elementary examples throughout. Finally, I shall explain how this approach solves some of the issues outlined above.

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For complex varieties, both algebraic and analytic, the Kodaira dimension is one of the fundamental invariants governing their rough geometry. It encodes a coarse measure of the positivity of the canonical bundle. Conjecturally, all varieties are built from positively curved, negatively curved, or flat varieties, the latter corresponding to varieties with (numerically) trivial canonical bundle.
We give an overview of the structure theory of complex varieties, with a focus on the $K$-trivial case. In particular, we explain the advances of the last fifteen years concerning generalizations of the Beauville-Bogomolov decomposition theorem to the singular setting.

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