{"topic":{"slug":"landmark-math-results","name":"Landmark Results in Pure Mathematics","domain":"math-cs","description":"Recent major theorems and their status: geometric Langlands, Kakeya in three dimensions, sphere packing, the unknotting-number additivity counterexample and more.","keywords":["geometric Langlands","Kakeya conjecture","sphere packing","proof of the conjecture","unknotting number","resolution of"],"paper_count":237,"latest_published":"2026-10-29","url":"https://space-kitty.com/topics/landmark-math-results","api_url":"https://space-kitty.com/api/v1/topics/landmark-math-results"},"dossier":{"slug":"landmark-math-results","topic":"landmark-math-results","title":"Landmark Results in Pure Mathematics: State of the Field (October 2026)","summary":"This dossier surveys recent landmark results in pure mathematics: the geometric Langlands proof series, the three-dimensional Kakeya theorem, optimal sphere packings in dimensions 8 and 24, and the 2025 counterexample to additivity of the unknotting number. The main takeaway is that the newest headline results are preprints or unrefereed announcements, while the dimension 8 and 24 sphere-packing theorems are the most firmly settled.","version":1,"published_at":"2026-10-30","updated_at":"2026-10-30","stats":{"words":3405,"citations":36,"claims":30,"timeline":19,"benchmarks":7,"open_problems":6,"key_papers":15,"glossary":15},"verification":{"status":"verified","method":"automated figure and citation-relevance triage against cited abstracts, plus targeted AI review with web checks of flagged statements; automated citation resolution","claims_checked":25,"issues_fixed":7,"verified_at":"2026-10-30"},"ai_assisted":true,"price":{"usd":"0.20","protocol":"x402","network":"eip155:84532","asset":"USDC"},"url":"https://space-kitty.com/dossiers/landmark-math-results","preview_url":"https://space-kitty.com/api/v1/dossiers/landmark-math-results/preview","purchase_url":"https://space-kitty.com/api/v1/dossiers/landmark-math-results"},"top_papers":[{"id":"arxiv:math/0509262","kind":"article","title":"On the Multilinear Restriction and Kakeya conjectures","authors":["Jonathan Bennett","Anthony Carbery","Terence Tao"],"author_count":3,"year":2005,"published":"2005-09-12","venue":"Acta Mathematica","cited_by":308,"doi":"10.1007/s11511-006-0006-4","arxiv_id":"math/0509262","nct_id":null,"source_url":"https://arxiv.org/abs/math/0509262","archive_url":"https://space-kitty.com/papers/arxiv:math/0509262"},{"id":"arxiv:1209.0403","kind":"article","title":"On two geometric realizations of an affine Hecke algebra","authors":["Roman Bezrukavnikov"],"author_count":1,"year":2015,"published":"2015-09-22","venue":"Publications mathématiques de l IHÉS","cited_by":72,"doi":"10.1007/s10240-015-0077-x","arxiv_id":"1209.0403","nct_id":null,"source_url":"https://doi.org/10.1007/s10240-015-0077-x","archive_url":"https://space-kitty.com/papers/arxiv:1209.0403"},{"id":"arxiv:1905.01319","kind":"article","title":"Sphere Packing and Quantum Gravity","authors":["Thomas Hartman","Dalimil Mazáč","Leonardo Rastelli"],"author_count":3,"year":2019,"published":"2019-05-03","venue":"Journal of High Energy Physics","cited_by":100,"doi":"10.1007/jhep12(2019)048","arxiv_id":"1905.01319","nct_id":null,"source_url":"https://arxiv.org/abs/1905.01319","archive_url":"https://space-kitty.com/papers/arxiv:1905.01319"},{"id":"arxiv:0711.0871","kind":"article","title":"Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture","authors":["Peter Fiebig"],"author_count":1,"year":2010,"published":"2010-09-23","venue":"Journal of the American Mathematical Society","cited_by":68,"doi":"10.1090/s0894-0347-2010-00679-0","arxiv_id":"0711.0871","nct_id":null,"source_url":"https://doi.org/10.1090/s0894-0347-2010-00679-0","archive_url":"https://space-kitty.com/papers/arxiv:0711.0871"},{"id":"doi:10.24033/ast.966","kind":"article","title":"Outline of the proof of the geometric Langlands conjecture for $GL_2$","authors":["Dennis Gaitsgory"],"author_count":1,"year":2018,"published":"2018-11-06","venue":"Astérisque","cited_by":42,"doi":"10.24033/ast.966","arxiv_id":null,"nct_id":null,"source_url":"http://dx.doi.org/10.24033/ast.966","archive_url":"https://space-kitty.com/papers/doi:10.24033/ast.966"},{"id":"arxiv:math/0110009","kind":"article","title":"New upper bounds on sphere packings I","authors":["Henry Cohn","Noam Elkies"],"author_count":2,"year":2001,"published":"2001-10-01","venue":"Annals of Mathematics 157 (2003), 689-714","cited_by":311,"doi":"10.4007/annals.2003.157.689","arxiv_id":"math/0110009","nct_id":null,"source_url":"https://arxiv.org/abs/math/0110009","archive_url":"https://space-kitty.com/papers/arxiv:math/0110009"},{"id":"arxiv:1105.3779","kind":"article","title":"Improved sphere packing lower bounds from Hurwitz lattices","authors":["Stephanie Vance"],"author_count":1,"year":2011,"published":"2011-05-19","venue":"Advances in Mathematics 227 (2011), pp. 2144-2156","cited_by":53,"doi":"10.1016/j.aim.2011.04.016","arxiv_id":"1105.3779","nct_id":null,"source_url":"https://arxiv.org/abs/1105.3779","archive_url":"https://space-kitty.com/papers/arxiv:1105.3779"},{"id":"arxiv:1302.2506","kind":"preprint","title":"Outline of the proof of the geometric Langlands conjecture for GL(2)","authors":["Dennis Gaitsgory"],"author_count":1,"year":2013,"published":"2013-02-11","venue":null,"cited_by":18,"doi":null,"arxiv_id":"1302.2506","nct_id":null,"source_url":"http://arxiv.org/abs/1302.2506","archive_url":"https://space-kitty.com/papers/arxiv:1302.2506"},{"id":"arxiv:math/0601265","kind":"article","title":"Knots with unknotting number 1 and essential Conway spheres","authors":["Cameron McA Gordon","John Luecke"],"author_count":2,"year":2006,"published":"2006-01-11","venue":"Algebr. 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