Tropical Hodge Theory

The Hodge theory on complex manifolds is a classical example of application of analytical methods in algebraic geometry. One of the main ideas of the tropical geometry is that there is should be a tropical analog of an object from the complex geometry. Following this idea Lagerberg introduced tropical currents and differential forms, Itenberg, Katzarkov, Mikhalkin, Zharkov introduced tropical cohomology theory. Based on these works, Jell, Shaw, Smacka constructed tropical de Rham cohomology theory. I my talk I will discuss how to construct a Hodge theory on Tropical varieties and how it resembles classical Hodge theory.