# Landmark Results in Pure Mathematics Domain: Mathematics & Theoretical Computer Science · 237 papers indexed · newest 2026-10-29 Recent major theorems and their status: geometric Langlands, Kakeya in three dimensions, sphere packing, the unknotting-number additivity counterexample and more. Key concepts: geometric Langlands, Kakeya conjecture, sphere packing, proof of the conjecture, unknotting number, resolution of ## Research dossier (premium) **Landmark Results in Pure Mathematics: State of the Field (October 2026)** (v1, updated 2026-10-30; 3,405 words, 36 sources, 30 evidence-mapped claims) 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. - Preview (free): https://space-kitty.com/api/v1/dossiers/landmark-math-results/preview - Full dossier ($0.20 via x402): https://space-kitty.com/api/v1/dossiers/landmark-math-results - Research pack ($0.20): https://space-kitty.com/api/v1/topics/landmark-math-results/pack ## Most influential - **[On the Multilinear Restriction and Kakeya conjectures](https://space-kitty.com/papers/arxiv:math/0509262)**. Jonathan Bennett, Anthony Carbery, Terence Tao · 2005 · Acta Mathematica (arXiv:math/0509262 · doi:10.1007/s11511-006-0006-4 · 308 citations). Source: > We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n\ > The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with… - **[Sphere Packing and Quantum Gravity](https://space-kitty.com/papers/arxiv:1905.01319)**. Thomas Hartman, Dalimil Mazáč, Leonardo Rastelli · 2019 · Journal of High Energy Physics (arXiv:1905.01319 · doi:10.1007/jhep12(2019)048 · 100 citations). Source: > We establish a precise relation between the modular bootstrap, used to constrain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra $U(1)^c$ maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in $d=2c$ dimensions. We also show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context. For $c=4$ and $c=12$, these functionals exactly reproduce the "magic functions" used recently by Viazovska \[1\] and Cohn et al. \[2\] to solve the… - **[Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture](https://space-kitty.com/papers/arxiv:0711.0871)**. Peter Fiebig · 2010 · Journal of the American Mathematical Society (arXiv:0711.0871 · doi:10.1090/s0894-0347-2010-00679-0 · 68 citations). Source: > We relate a certain category of sheaves of k-vector spaces on a complex affine Schubert variety to modules over the k-Lie algebra (for ch k\>0) or to modules over the small quantum group (for ch k=0) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We gi… - **[Outline of the proof of the geometric Langlands conjecture for $GL\_2$](https://space-kitty.com/papers/doi:10.24033/ast.966)**. Dennis Gaitsgory · 2018 · Astérisque (doi:10.24033/ast.966 · 42 citations). Source: - **[New upper bounds on sphere packings I](https://space-kitty.com/papers/arxiv:math/0110009)**. Henry Cohn, Noam Elkies · 2001 · Annals of Mathematics 157 (2003), 689-714 (arXiv:math/0110009 · doi:10.4007/annals.2003.157.689 · 311 citations). Source: > We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24. - **[Improved sphere packing lower bounds from Hurwitz lattices](https://space-kitty.com/papers/arxiv:1105.3779)**. Stephanie Vance · 2011 · Advances in Mathematics 227 (2011), pp. 2144-2156 (arXiv:1105.3779 · doi:10.1016/j.aim.2011.04.016 · 53 citations). Source: > In this paper we prove an asymptotic lower bound for the sphere packing density in dimensions divisible by four. This asymptotic lower bound improves on previous asymptotic bounds by a constant factor and improves not just lower bounds for the sphere packing density, but also for the lattice sphere packing density and, in fact, the Hurwitz lattice sphere packing density. - **[Outline of the proof of the geometric Langlands conjecture for GL(2)](https://space-kitty.com/papers/arxiv:1302.2506)**. Dennis Gaitsgory · 2013 (arXiv:1302.2506 · 18 citations). Source: > We outline a proof of the categorical geometric Langlands conjecture for GL(2), as formulated in reference \[AG\], modulo a number of more tractable statements that we call Quasi-Theorems. - **[Knots with unknotting number 1 and essential Conway spheres](https://space-kitty.com/papers/arxiv:math/0601265)**. Cameron McA Gordon, John Luecke · 2006 · Algebr. Geom. Topol. 6 (2006) 2051-2116 (arXiv:math/0601265 · doi:10.2140/agt.2006.6.2051 · 31 citations). Source: > For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM-knot (of Eudave-Munoz) or (S^3,K) contains an EM-tangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (ie algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determin… - **[Knots with unknotting number one and Heegaard Floer homology](https://space-kitty.com/papers/arxiv:math/0401426)**. Peter Ozsvath, Zoltan Szabo · 2004 · Topology (arXiv:math/0401426 · doi:10.1016/j.top.2005.01.002 · 53 citations). Source: > We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with ten crossings and unknotting number equal to one. - **[The sphere packing problem in dimension 24](https://space-kitty.com/papers/arxiv:1603.06518)**. Henry Cohn, Abhinav Kumar, Stephen D. Miller et al. · 2016 · Annals of Mathematics 185 (2017), 1017-1033 (arXiv:1603.06518 · doi:10.4007/annals.2017.185.3.8 · 288 citations). Source: > Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case. - **[A conceptual breakthrough in sphere packing](https://space-kitty.com/papers/arxiv:1611.01685)**. Henry Cohn · 2016 · Notices Amer. Math. Soc. 64 (2017), no. 2, 102-115 (arXiv:1611.01685 · doi:10.1090/noti1474 · 78 citations). Source: > This expository paper describes Viazovska's breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kumar, Miller, Radchenko, and Viazovska. - **[A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture](https://space-kitty.com/papers/arxiv:1411.1208)**. Yasuhiro Wakabayashi · 2014 (arXiv:1411.1208 · 14 citations). Source: > This manuscript presents a detailed and original account of the theory of opers defined on pointed stable curves in arbitrary characteristic and their moduli. In particular, it includes the development of the study of dormant opers, which are opers of a certain sort in positive characteristic. The theory of dormant opers (or more generally, opers in positive characteristic) on pointed stable curves, which has proved to be rather rich and deep, was born in the work of S. Mochizuki, who developed the theory for $\\mathfrak{sl}\_2$-opers and used it to establish $p$-adic Teichmüller theory. Some pa… - **[Preservation of depth in the local geometric Langlands correspondence](https://space-kitty.com/papers/doi:10.1090/tran/6794)**. Tsao-Hsien Chen, Masoud Kamgarpour · 2015 · Transactions of the American Mathematical Society (doi:10.1090/tran/6794 · 13 citations). Source: - **[The unknotting number and classical invariants I](https://space-kitty.com/papers/arxiv:1203.3225)**. Maciej Borodzik, Stefan Friedl · 2012 · Algebr. Geom. Topol. 15 (2015) 85-135 (arXiv:1203.3225 · doi:10.2140/agt.2015.15.85 · 15 citations). Source: > Given a knot K we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of K. This lower bound subsumes the lower bounds given by the Levine-Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotting number being equal to one. Our approach in particular allows us to show for 25 knots with up to 12 crossings that their unknotting number is at least three, most of which are very difficult to treat otherwise. ## Newest - **[Point Corrections and Wall-Crossing Geometry on the Quintic del Pezzo Threefold](https://space-kitty.com/papers/arxiv:2610.39336)**. Zheyuan Fu · 2026 (arXiv:2610.39336). Source: - **[Higher order unknotting numbers for knots and links](https://space-kitty.com/papers/arxiv:2610.39474)**. Jean-Baptiste Meilhan, Harumi Ogita, Akira Yasuhara · 2026 (arXiv:2610.39474). Source: - **[Vanishing near the virtual cohomological dimension of $\\mathrm{SL}\_n(\\mathbb Z)$](https://space-kitty.com/papers/arxiv:2610.39960)**. Taiwang Deng · 2026 (arXiv:2610.39960). Source: - **[A lifted entropy splitting lemma and applications](https://space-kitty.com/papers/arxiv:2610.40145)**. Gabriel Coutinho, Luis Higino · 2026 (arXiv:2610.40145). Source: - **[Geometric properties of the analytic Emerton-Gee stack](https://space-kitty.com/papers/arxiv:2610.38115)**. Yutaro Mikami · 2026 (arXiv:2610.38115). Source: - **[Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections](https://space-kitty.com/papers/arxiv:2610.38655)**. Chwas Ahmed, Mohammed Rafiq Namiq · 2026 (arXiv:2610.38655). Source: - **[Primitive flag-transitive generalized quadrangles](https://space-kitty.com/papers/arxiv:2610.38819)**. Timothy C. Burness, Tao Feng, Jianbing Lu · 2026 (arXiv:2610.38819). Source: - **[Homogeneous polynomial approximation on symmetric convex bodies: A proof of Kroó's conjecture](https://space-kitty.com/papers/arxiv:2610.38877)**. Matthew J. Colbrook · 2026 (arXiv:2610.38877). Source: - **[Counting the Fixed Points of Binary Belief Propagation: Generic Degree, Syzygies, and Real Solutions](https://space-kitty.com/papers/arxiv:2610.36964)**. Yusuke Watanabe · 2026 (arXiv:2610.36964). Source: - **[Finite group actions on the non-minimal nilpotent orbit closures in type A](https://space-kitty.com/papers/arxiv:2610.37396)**. Gwyn Bellamy, Misha Feigin, Pavel Shlykov · 2026 (arXiv:2610.37396). Source: --- Space Kitty Archive · https://space-kitty.com · API: https://space-kitty.com/api/v1 · MCP: https://space-kitty.com/mcp · Premium: $0.20/request via x402 (Base Sepolia (testnet)). Metadata: arXiv (CC0), OpenAlex (CC0), Europe PMC (CC BY/CC0 works), ClinicalTrials.gov.