The Geometry of Genome Space

Definitions and mathematical statement

This is a research programme rather than a uniquely specified mathematical assertion. A genome space would be a set $G$ of chosen representations of genomes. A metric on $G$ would be a function $d:G\times G\to[0,\infty)$ satisfying, for all $x,y,z\in G$, \[d(x,y)=0\iff x=y,\qquad d(x,y)=d(y,x),\qquad d(x,z)\leq d(x,y)+d(y,z).\] DARPA's challenge is to establish appropriate distance metrics on genome space that incorporate biological utility, rather than merely counting mutations. Neither $G$ nor a mathematical criterion for biological utility or appropriateness is specified in the source. Thus these axioms clarify the word "metric" but do not supply the missing biological requirements or turn the agenda into a definite existence theorem. The supplied literature triage dated 2026-08-17 reports active, non-unique work on genome distances and phylogenetic metrics, labels the situation partially solved, and states that theorem-level solved/open classification and a full solution or refutation are not applicable as written. The underlying registry separately labels the record open. Both are supplied, dated catalogue assessments.

Short English statement

Construct biologically meaningful ways to measure how far apart genomes are. A precise problem first requires a genome representation, a biological utility criterion, and the required properties of the distance; these choices are not fixed by the source.

Sources

• Local UnsolvedMath registry, permanent ID 81, source code catalogue entry, "The Geometry of Genome Space," proposed by DARPA in 2007; supplied statement and background.
• DARPA DSO, BAA 07-68, Mathematical Challenge Fifteen (2007), the primary source identified by the registry: https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html. The supplied review describes the evidence as the original one-sentence metric agenda.
• Supplied literature triage, checked 2026-08-17: classification PARTIAL-PROGRESS; no unique biologically useful genome-space metric specified. It reports no source-text change and cautions that dated triage does not exclude later or unindexed work.