Georgia Institute of Technology College of Sciences

 In their seminal paper on combinatorial Hodge theory, Adiprasito, Huh, and Katz showed, among other things, that a very specific set of toric varieties has Chow rings that satisfy Poincaré duality, even though the varieties are not compact. In joint work with Farbod Shokrieh, we generalize this statement to all toric varieties whose fans are supported on a tropically smooth set. This has several consequences in tropical intersection theory; most notably it allows us to prove the long-suspected duality between tropical cycles and cocycles.

In my talk I will assume no prior knowledge of tropical intersection theory. I will define tropical cycles and cocycles explicitly and explain how they are connected to the intersection theory of toric varieties and the Chow rings of fans appearing in combinatorial Hodge theory. Finally, we will see how to use the duality statement mentioned above to define the tropical intersection product.