Rethinking Tropical Linear Algebra: Buildings, Bimatroids, and Applications

• Applicant: Professor Martin Ulirsch, Ph.D.
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539867663

Tropical geometry is a dequantized version of classical algebraic geometry, which studies a combinatorial piecewise linear shadow associated to compactifications and degenerations of algebraic varieties. One of its most successful and active parts is tropical linear algebra, a term that stands for the many different incarnations of linearity in tropical geometry. This area ranges from the study of matrices over the min-plus algebra coming to us from optimization to the geometric study of (valuated) matroids. It includes the recent breakthrough results in the Hodge theory of matroids, but also the study of the tropical Grassmannians and Dressians with its numerous applications, e.g. to phylogenetics. The central objectives of this project are: (1) to develop new foundations for tropical linear algebra using techniques from the geometry of affine buildings, the combinatorics of (valuated) bimatroids, and the algebra of hyperstructures; (2) to expand tropical linear algebra beyond the A_n case, integrating valuated Coxeter matroids and affine buildings; and (3) to explore applications of tropical linear algebra, focussing on the combinatorial Hodge theory of bimatroids and Coxeter matroids, the yet-to-be-fully-developed geometry of tropical vector bundles, and the dequantization process in categorical quantum theory.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies