Around a hundred years ago, Veblen and Young proved what is now known as the Fundamental Theorem of Projective Geometry. One way to interpret this theorem is as a topological characterization of linear geometry. I will report on a new non-linear generalization of this result: a topological characterization of algebraic geometry. This is joint work with Kollár, Olsson, and Sawin. The talk will be aimed at a general mathematical audience and be entirely self-contained.

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