Explicit full symplectic packings of the 4-ball

Symplectic maps preserve volume but are far more rigid, so packing balls symplectically is subtle. In dimension four it is known that $k$ equal balls can fill all but an arbitrarily small fraction of a larger ball whenever $k \ge 10$, but the proof gives no picture of such packings. The task is to construct them explicitly, by one algorithm that works for every $k \ge 10$ and every tolerance.

Problem

Work in $\mathbb{R}^4$ with its standard symplectic form. For $i = 1, \dots, k$ let $B_i$ be the closed unit ball centred at $(3i - 3, 0, 0, 0)$, so that the $B_i$ are disjoint unit balls. Let $B(R)$ be the closed ball of radius $R$ centred at the origin.

Task. Give one algorithm that, for every integer $k \ge 10$ and every $\varepsilon \in (0, 1)$, outputs an explicit smooth Hamiltonian $H_\varepsilon \colon \mathbb{R}^4 \times [0, 1] \to \mathbb{R}$ whose time-1 flow $\phi$ satisfies

$$\phi\Bigl(\bigsqcup_{i=1}^{k} B_i\Bigr) \subset B(R) \qquad \text{with} \qquad \frac{k}{R^4} > 1 - \varepsilon,$$

so that the $k$ unit balls fill more than a fraction $1 - \varepsilon$ of the volume of $B(R)$.

Background

McDuff and Polterovich related symplectic packings of the 4-ball to algebraic geometry, including Nagata's conjecture [1]. Biran's work then showed that full packings exist for large numbers of equal balls [2]; the proof uses Seiberg–Witten theory and symplectic inflation and is not constructive. Schlenk's survey stresses this contrast with Euclidean packing: Euclidean packing numbers are usually known only once a maximal packing has been seen, whereas for many known symplectic packing numbers nobody has an idea what a corresponding packing looks like [3].

What is known

• In dimension four, $k$ equal balls fill the volume of a ball up to any $\varepsilon > 0$ for every $k \ge 10$, by non-explicit methods [1, 2] (as summarised by Epoch's problem write-up).
• For $k \le 9$ the optimal packings have explicit constructions; for $k \ge 10$ none are known.
• Epoch's write-up notes that a single approximate embedding for one fixed $\varepsilon$ might be found by numerical optimisation, but that one algorithm valid for all $\varepsilon$ is unlikely to have a brute-force solution and would be a significant advance.

What would count as progress

• An algorithm $(k, \varepsilon) \mapsto H_\varepsilon$, ideally with a proof that the flow embeds the balls as claimed.
• Epoch scores submissions with its own verifier: solution(k, epsilon) must return the source of a function Hamiltonian(Q, P, t) written in PyTorch. The verifier integrates the flow of points sampled on the ball boundaries with a Runge–Kutta method, checks numerically that the flow is symplectic, measures $R$, and tests $k/R^4 > 1 - \varepsilon$ for several $k \ge 10$.
• An explicit family for the single case $k = 10$ and all $\varepsilon$ would already be a significant result.

References

1. D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994), 405–429. doi:10.1007/BF01231766
2. P. Biran, Symplectic packing in dimension 4, Geom. Funct. Anal. 7 (1997), 420–437. doi:10.1007/s000390050014
3. F. Schlenk, Symplectic embedding problems, old and new, Bull. Amer. Math. Soc. 55 (2018), 139–182. doi:10.1090/bull/1587

Source

Imported from FrontierMath: Open Problems, symplectic-ball-packing, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.