Tame geometry and transcendence in Hodge theory (TameHodge)

Hodge theory is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It occupies a central position in mathematics through its relations to differential geometry, algebraic geometry, differential equations and number theory. At heart Hodge theory is NOT algebraic. On the other hand, some of the deepest conjectures in mathematics suggest that this transcendence is severely constrained. Recent work of myself and others suggests that tame geometry is the natural setting for understanding these constraints. Tame geometry, developed by model-theorist as ominimal geometry, has for prototype real semi-algebraic geometry, but is much richer. As a spectacular application of tame geometry, Bakker, Tsimerman and I recently reproved a famous result of Cattani-Deligne-Kaplan, often considered as the most serious evidence for the Hodge conjecture: the algebraicity of Hodge loci. We will explore this striking new connection between tame geometry and Hodge theory.