Affine Buildings and Tropical Convexity

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.Comment: 22 pages, 4 figure

Solving generic nonarchimedean semidefinite programs using stochastic game algorithms

A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean. We provide a solution for a class of semidefinite feasibility problems given by generic matrices. Our approach is based on tropical geometry. It relies on tropical spectrahedra, which are defined as the images by the valuation of nonarchimedean spectrahedra. We establish a correspondence between generic tropical spectrahedra and zero-sum stochastic games with perfect information. The latter have been well studied in algorithmic game theory. This allows us to solve nonarchimedean semidefinite feasibility problems using algorithms for stochastic games. These algorithms are of a combinatorial nature and work for large instances.Comment: v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions + benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign patterns + some results have been replaced by references to the companion work arXiv:1610.0674

Combinatorial curvature flows for generalized hyperbolic circle packings

Generalized circle packings were introduced in \cite{Ba-Hu-Sun} as a generalization of tangential circle packings in hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow, fractional combinatorial Calabi flow and combinatorial $p$-th Calabi flow for generalized hyperbolic circle packings. We establish several equivalent conditions regarding the longtime behaviors of these flows. This provides effective algorithms for finding the generalized circle packings with prescribed total geodesic curvatures