Arithmetic Langlands, Topology, and Geometry

Definitions and mathematical statement

Homotopy theory studies topological spaces and related structures through maps up to homotopy and suitable notions of weak equivalence. The supplied DARPA Mathematical Challenge Eighteen (2007), entitled "Arithmetic Langlands, Topology, and Geometry," asks:

Explore homotopy theory's role in Langlands programs.

This is a research programme, not a quantified conjecture. The source specifies neither a version of the Langlands programme nor a class of arithmetic, geometric, or homotopical objects, and it gives no proposed construction, equivalence, or criterion for completion. Consequently no definite theorem or existence assertion is determined by the supplied wording. A precise instance would require those choices to be stated explicitly; no such choices are made here.

The supplied registry labels the entry open. Its separate literature triage dated 2026-08-17 labels the direction partially solved, records that derived, spectral, and homotopical approaches have become important in modern Langlands research, and says that a full solution or refutation is not applicable as written. These are supplied editorial assessments, not a verified mathematical resolution.

Short English statement

DARPA's 2007 challenge asks for an understanding of homotopy theory in Langlands programmes. The supplied wording is a broad research agenda; a particular Langlands setting and a specific homotopical construction or equivalence are needed before a precise mathematical problem can be stated.

Sources

• Local UnsolvedMath registry, permanent ID 84, source code catalogue entry, "Arithmetic Langlands, Topology, and Geometry"; attributed to DARPA (2007). The supplied background includes literature triage dated 2026-08-17.
• DARPA DSO, BAA 07-68, Mathematical Challenge Eighteen (2007), supplied as the primary source: https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html. The supplied triage identifies this as the original homotopy/Langlands agenda. This draft does not independently verify the linked material.