# P3072: Birkhoff conjecture for integrable convex billiards

- ID: `P3072`
- Reference: `birkhoff-billiard-conjecture`
- Page: https://theoremdb.org/statements/P3072
- Record maturity: Reviewed problem with recorded work

## Problem

If a strictly convex \(C^\infty\) planar billiard table has an invariant essential caustic for every rotation number \(\rho\in(0,\tfrac12)\), must its boundary be an ellipse?

### Context

Known frontier: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.

Open boundary: The global classification of all smooth strictly convex integrable tables remains open.

### Problem setup

- **Definition (billiard map).** The boundary phase-space map taking one reflection to the next.
- **Definition (integrable).** The phase annulus is foliated by invariant essential curves, equivalently caustics in the stated formulation.
- **Remark.** A caustic is a curve tangent to every segment of any orbit that starts tangent to it. Ellipses have a full foliation by confocal caustics; the conjecture says they are the only smooth strictly convex tables with this complete integrability.

### What counts as a solution

- Prove every table satisfying the invariant-caustic hypothesis is an ellipse.
- Or construct a nonelliptic smooth strictly convex table with the full caustic foliation.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.

The exact unresolved remainder is: The global classification of all smooth strictly convex integrable tables remains open.

A complete resolution must meet the following acceptance conditions:
- Prove every table satisfying the invariant-caustic hypothesis is an ellipse.
- Or construct a nonelliptic smooth strictly convex table with the full caustic foliation.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: Birkhoff billiard conjecture integrable ellipse open 2026; local strong Birkhoff conjecture almost every ellipse 2025
- Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.
- Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.

### Other known results

- **Claim 2** (supported): Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): The global classification of all smooth strictly convex integrable tables remains open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `birkhoff-billiard-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse”. Inventiones mathematicae 244(1) (2026), 221-298. DOI 10.1007/s00222-025-01397-y. main local strong Birkhoff theorem https://doi.org/10.1007/s00222-025-01397-y
   - Also cited at Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse,” Inventiones Mathematicae 244(1) (2026), 221–298. main local strong Birkhoff theorem
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves the local strong Birkhoff conjecture near almost every ellipse under the paper's perturbative hypotheses.
   - Source used to assess the problem's recorded status.
   - For Birkhoff conjecture for integrable convex billiards: This is the dated publication status for the canonical target Birkhoff conjecture for integrable convex billiards.
   - Source named by the research packet.
2. <a id="reference-2"></a>M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov, and S. Tabachnikov, “Open problems on billiards and geometric optics”. arXiv:2110.10750 (2021). integrable billiards section https://arxiv.org/abs/2110.10750
   - preprint; reference source; arXiv:2110.10750, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Places Birkhoff's conjecture among current billiard problems and records known cases.
   - Source used to assess the problem's recorded status.
   - For Birkhoff conjecture for integrable convex billiards: Places Birkhoff's conjecture among current billiard problems and records known cases.