Johannes Rau

The general aim of the project is to extend and deepen the fruitful interactionbetween the geometry of smooth varieties and the combinatorial properties ofmatroids that lies at the heart of tropical geometry. More explicitly, in my projectI want to tackle three problems which are central to the further development ofthis theory:1. Using the matroidal intersection theory developed in [FR13; Sha13] and thetropical Hodge groups developed in [Ite+16], I want to prove a tropical traceformula in analogy to its classical counterparts by Lefschetz, Weil, Grothendieck,Deligne. The goal is apply this formula to the study of matroid invariants(characteristic polynomial, g-polynomial [AHK15; ADH20; Spe09]) as well as tothe study of classical non-compact trace formulas [GM03].2. I want to generalize the existing tropical-Lagrangian correspondences tohigher dimensions/codimensions using the language of Lagrangian matroids[Mat18; Mik19; Hic19; ADH20].3. The last problems aims to solve further cases of the tropical Hodge conjec-ture, particularly in the case of tropical abelian varieties [JRS18; MZ14; Zha20].

The general aim of the project is to extend and deepen the fruitful interactionbetween the geometry of smooth varieties and the combinatorial properties ofmatroids that lies at the heart of tropical geometry. More explicitly, in my projectI want to tackle three problems which are central to the further development ofthis theory:1. Using the matroidal intersection theory developed in [FR13; Sha13] and thetropical Hodge groups developed in [Ite+16], I want to prove a tropical traceformula in analogy to its classical counterparts by Lefschetz, Weil, Grothendieck,Deligne. The goal is apply this formula to the study of matroid invariants(characteristic polynomial, g-polynomial [AHK15; ADH20; Spe09]) as well as tothe study of classical non-compact trace formulas [GM03].2. I want to generalize the existing tropical-Lagrangian correspondences tohigher dimensions/codimensions using the language of Lagrangian matroids[Mat18; Mik19; Hic19; ADH20].3. The last problems aims to solve further cases of the tropical Hodge conjec-ture, particularly in the case of tropical abelian varieties [JRS18; MZ14; Zha20].