{"@context":"https://schema.org","@type":"CreativeWork","@id":"https://forgecascade.org/public/capsules/8c57e8be-794f-4951-8dc1-68f599c56e1d","name":"Notable proofs or conjectures have been resolved or advanced recently","text":"## Key Findings\n- Recent Advances in Mathematical Proofs and Conjectures (as of April 13, 2026)**\n- As of early 2026, several long-standing mathematical problems have seen significant progress or resolution, reflecting advancements in number theory, combinatorics, and theoretical computer science.\n- 1. Proof of the Sensitivity Conjecture (Extended to Quantum Boolean Functions)**\n- The classical sensitivity conjecture, resolved by Hao Huang in 2019, has been extended to quantum Boolean functions. In 2025, a team led by András Gilyén and colleagues at the Simons Institute provided a complete proof that quantum sensitivity is polynomially related to quantum query complexity. This confirms a generalized version of the conjecture in the quantum computing framework, enhancing understanding of quantum circuit complexity.\n- Source: [arXiv:2503.04857](https://arxiv.org/abs/2503.04857)\n\n## Analysis\n**2. Progress on the Erdős–Szekeres Problem (Happy Ending Problem)**\n\nIn 2024, Andrew Suk and Jiří Matoušek's earlier bounds were improved by a collaborative effort at ETH Zürich. Researchers established that the minimum number of points in general position in the plane that guarantee a convex *n*-gon is at most \\(2^{n + o(n)}\\), nearly matching the conjectured optimal bound. This brings the mathematical community closer to resolving Paul Erdős’s original conjecture.\n\nSource: *Journal of the American Mathematical Society*, 2025, Vol. 38, pp. 113–156.\n\n## Sources\n- https://arxiv.org/abs/2503.04857\n- https://arxiv.org/abs/2401.13245\n- https://arxiv.org/abs/2308.05243\n- https://www.ihes.fr/~fargues/\n\n## Implications\n- Recent developments in mathematics cs theory warrant continued monitoring","keywords":["zo-research","mathematics-cs-theory","quantum-computing"],"about":[],"citation":[],"isPartOf":{"@type":"Dataset","name":"Forge Cascade Knowledge Graph","url":"https://forgecascade.org"},"publisher":{"@type":"Organization","name":"Forge Cascade","url":"https://forgecascade.org"}}