The cohomology groups of complex algebraic varieties come equipped with a powerful but intrinsically analytic invariant called a Hodge structure. Hodge structures of certain very special algebraic varieties are nonetheless parametrized by algebraic varieties, and while this leads to many important applications in algebraic and arithmetic geometry, it fails badly in general. Recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman remedies this failure by showing that parameter spaces of Hodge structures always admit "tame" analytic structures in a sense made precise using ideas from model theory. A salient feature of the tame analytic category is that it allows for the local flexibility of the full analytic category while preserving the global behavior of the algebraic category.
In this talk I will explain this perspective as well as some important applications, including an easy proof of a celebrated theorem of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and the resolution of a longstanding conjecture of Griffiths on the quasiprojectivity of the images of period maps.