Geometry and algorithms for upper triangular tropical matrix identities.

We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed $m$-letter alphabet) that form identities in the semigroup $\ut{n}$ of $n\times n$ upper triangular tropical matrices. In the case $n=2$ these identities are precisely those satisfied by the bicyclic monoid, whilst in the case $n=3$ they form a subset of the identities which hold in the plactic monoid of rank $3$. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for $\ut{n}$ if and only if their signatures are equal. Our algorithms are thus based on polyhedral computations and achieve optimal complexity in some cases. For $n=m=2$ we prove a Structural Theorem, which allows us to quickly enumerate the pairs of words of fixed length which form identities for $\ut{2}$. This allows us to recover a short proof of Adjan's theorem on minimal length identities for the bicyclic monoid, and to construct minimal length identities for $\ut{3}$, providing counterexamples to a conjecture of Izhakian in this case. We conclude with six conjectures at the intersection of semigroup theory, probability and combinatorics, obtained through analysing the outputs of our algorithms.