Hodge theory, as developed by Deligne and Griffiths, is one of the main tools for analysing the geometry and arithmetic of complex algebraic varieties, that is, solution sets of algebraic equations over the complex numbers. It associates to any complex algebraic variety an apparently simple linear algebra gadget: a finite dimensional vector space over the rationals, whose complexification is naturally endowed with two filtrations. Hodge theory occupies a central position in mathematics through its relations to differential geometry, algebraic geometry, differential equations and number theory.

It is an essential fact that at heart, Hodge theory is not algebraic but rather the transcendental comparison of two algebraic structures. On the other hand, some of the deepest conjectures in mathematics (the Hodge conjecture and the Grothendieck period conjecture) suggest that this transcendence is severely constrained. In these lectures, we survey the recent advances bounding this transcendence, mainly due to the introduction of tame geometry as a natural framework for Hodge theory.