Atiyah’s Minkowski Space Conjecture

For every integer $n\ge 2$ and every admissible marked configuration of $n$ complete pairwise disjoint timelike affine worldlines in Minkowski space, let $F_1,\ldots,F_n$ be the associated binary forms of degree $n-1$ , whose roots are the ordered retarded celestial directions. The conjecture asserts that $F_1,\ldots,F_n$ are linearly independent over the coefficient field; equivalently, the coefficient matrix of these forms has nonzero determinant.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Atiyah's Minkowski Space Conjecture Fails for Every n≥3 — arXiv — Ziran Liu

3 days ago

Progress summary

Refreshed 3 days ago

Claimed solved

A new unrefereed manuscript claims the conjecture is false in three or more dimensions, while the two-dimensional case holds.

Atiyah’s Minkowski Space Conjecture predicts universal independence in dimension $n=2$ but failure in every dimension $n\ge 3$ . Ziran Liu’s manuscript claims to establish this complete threshold.

August 2026 counterexample claim

Liu gives a one-parameter planar family whose determinant has rank exactly $2$ at $n=3$ , then extends the construction to counterexamples for every $n\ge 3$ . The manuscript therefore claims the conjecture is settled, with universal independence at $n=2$ and failure thereafter, but the calculation and counterexamples have not been independently verified.

Current status (as of August 2026): The conjecture is claimed to hold at $n=2$ and fail for every $n\ge 3$ , but this remains unverified pending independent checking of the unrefereed preprint.

Sources

arxiv.org

No solutions have been posted yet.