Affine Buildings and Tropical Convexity

TL;DR

This paper explores the connection between tropical convexity and affine buildings, introducing algorithms for convex hulls and computations in Bruhat--Tits buildings with applications across mathematics.

Contribution

It presents new algorithms for convex hulls and computations in affine buildings, bridging tropical geometry with combinatorial and computational methods.

Findings

Developed a convex hull algorithm for Bruhat--Tits buildings

Created techniques for computing with apartments and membranes

Algorithms applicable to various mathematical fields

Abstract

The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat--Tits building of SL$_d(K)$ and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics.