{"topic":"landmark-math-results","sort":"top","limit":25,"offset":0,"count":25,"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","updated":"2005-09-12","pmid":null,"pmcid":null,"openalex_id":"W2164774824","pdf_url":"https://arxiv.org/pdf/math/0509262","abstract":"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 $n0) 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 give an upper bound on the exceptional characteristics and verify its multiplicity one case for all relevant primes.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":"public-domain","is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Algebraic structures and combinatorial models"],"details":null,"provenance":"openalex+arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"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","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W1868517304","pdf_url":null,"abstract":null,"abstract_license":null,"license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex","attribution":"Metadata and citation counts: OpenAlex (CC0)."},{"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","updated":"2003-09-03","pmid":null,"pmcid":null,"openalex_id":"W3104302389","pdf_url":"https://arxiv.org/pdf/math/0110009","abstract":"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.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.MG"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"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","updated":"2011-05-19","pmid":null,"pmcid":null,"openalex_id":"W2008929884","pdf_url":"https://arxiv.org/pdf/1105.3779","abstract":"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.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.MG"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"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","updated":"2014-11-12","pmid":null,"pmcid":null,"openalex_id":"W2963729700","pdf_url":"https://arxiv.org/pdf/1302.2506","abstract":"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.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":true,"retracted":false,"peer_reviewed":false,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex+arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"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. Geom. Topol. 6 (2006) 2051-2116","cited_by":31,"doi":"10.2140/agt.2006.6.2051","arxiv_id":"math/0601265","nct_id":null,"source_url":"https://arxiv.org/abs/math/0601265","archive_url":"https://space-kitty.com/papers/arxiv:math/0601265","updated":"2009-06-29","pmid":null,"pmcid":null,"openalex_id":"W2158064727","pdf_url":"https://arxiv.org/pdf/math/0601265","abstract":"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 determining an unknotting crossing). Among the knots up to 11 crossings in Conway's table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsvath-Szabo, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram. As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.GT"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:math/0401426","kind":"article","title":"Knots with unknotting number one and Heegaard Floer homology","authors":["Peter Ozsvath","Zoltan Szabo"],"author_count":2,"year":2004,"published":"2004-01-30","venue":"Topology","cited_by":53,"doi":"10.1016/j.top.2005.01.002","arxiv_id":"math/0401426","nct_id":null,"source_url":"https://arxiv.org/abs/math/0401426","archive_url":"https://space-kitty.com/papers/arxiv:math/0401426","updated":"2005-03-15","pmid":null,"pmcid":null,"openalex_id":"W2092606789","pdf_url":"https://arxiv.org/pdf/math/0401426","abstract":"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.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.GT","math.SG"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1603.06518","kind":"article","title":"The sphere packing problem in dimension 24","authors":["Henry Cohn","Abhinav Kumar","Stephen D. Miller","Danylo Radchenko","Maryna Viazovska"],"author_count":5,"year":2016,"published":"2016-03-21","venue":"Annals of Mathematics 185 (2017), 1017-1033","cited_by":288,"doi":"10.4007/annals.2017.185.3.8","arxiv_id":"1603.06518","nct_id":null,"source_url":"https://arxiv.org/abs/1603.06518","archive_url":"https://space-kitty.com/papers/arxiv:1603.06518","updated":"2026-09-06","pmid":null,"pmcid":null,"openalex_id":"W3100495456","pdf_url":"https://arxiv.org/pdf/1603.06518","abstract":"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.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.NT","math.MG"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1611.01685","kind":"article","title":"A conceptual breakthrough in sphere packing","authors":["Henry Cohn"],"author_count":1,"year":2016,"published":"2016-11-05","venue":"Notices Amer. Math. Soc. 64 (2017), no. 2, 102-115","cited_by":78,"doi":"10.1090/noti1474","arxiv_id":"1611.01685","nct_id":null,"source_url":"https://arxiv.org/abs/1611.01685","archive_url":"https://space-kitty.com/papers/arxiv:1611.01685","updated":"2016-11-05","pmid":null,"pmcid":null,"openalex_id":"W2552248576","pdf_url":"https://arxiv.org/pdf/1611.01685","abstract":"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.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.MG","math.NT"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1411.1208","kind":"preprint","title":"A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture","authors":["Yasuhiro Wakabayashi"],"author_count":1,"year":2014,"published":"2014-11-05","venue":null,"cited_by":14,"doi":null,"arxiv_id":"1411.1208","nct_id":null,"source_url":"http://arxiv.org/abs/1411.1208","archive_url":"https://space-kitty.com/papers/arxiv:1411.1208","updated":"2021-04-26","pmid":null,"pmcid":null,"openalex_id":"W1876441822","pdf_url":"https://arxiv.org/pdf/1411.1208","abstract":"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 parts of Mochizuki's work were later extended in the case of proper smooth curves by K. Joshi, C. Pauly, and other mathematicians. This manuscript represents an advance in the theory of opers that takes the subject beyond the work of Mochizuki, Joshi, and Pauly. In particular, we provide general unified formulations and the basics of principal bundles and connections defined on families of pointed stable curves. The notion of an oper is accordingly introduced in the context of logarithmic algebraic geometry. Some of the results can be regarded as generalizations of results obtained in the fundamental work on the geometric Langlands program developed by A. Beilinson and V. Drinfeld. We also describe various properties and assertions about (dormant) opers, such as duality, comparison with differential operators, and compactification of the moduli space. Our goal is to give an explicit formula, conjectured by Joshi, for the generic number of dormant $\\mathfrak{sl}_n$-opers. We do so by obtaining a detailed understanding of the moduli space of dormant opers and computing the Gromov-Witten invariants for Quot-schemes in characteristic zero. This formula reveals an interaction between studies in $p$-adic Teichmüller theory and certain areas of mathematics, including Gromov-Witten theory.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":true,"retracted":false,"peer_reviewed":false,"categories":["Algebraic Geometry and Number Theory"],"details":null,"provenance":"openalex+arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"doi:10.1090/tran/6794","kind":"article","title":"Preservation of depth in the local geometric Langlands correspondence","authors":["Tsao-Hsien Chen","Masoud Kamgarpour"],"author_count":2,"year":2015,"published":"2015-08-11","venue":"Transactions of the American Mathematical Society","cited_by":13,"doi":"10.1090/tran/6794","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.1090/tran/6794","archive_url":"https://space-kitty.com/papers/doi:10.1090/tran/6794","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W2962843244","pdf_url":null,"abstract":null,"abstract_license":null,"license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex","attribution":"Metadata and citation counts: OpenAlex (CC0)."},{"id":"arxiv:1203.3225","kind":"article","title":"The unknotting number and classical invariants I","authors":["Maciej Borodzik","Stefan Friedl"],"author_count":2,"year":2012,"published":"2012-03-14","venue":"Algebr. Geom. Topol. 15 (2015) 85-135","cited_by":15,"doi":"10.2140/agt.2015.15.85","arxiv_id":"1203.3225","nct_id":null,"source_url":"https://arxiv.org/abs/1203.3225","archive_url":"https://space-kitty.com/papers/arxiv:1203.3225","updated":"2012-03-14","pmid":null,"pmcid":null,"openalex_id":"W2086910907","pdf_url":"https://arxiv.org/pdf/1203.3225","abstract":"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.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.GT"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1312.1278","kind":"article","title":"Alternating knots with unknotting number one","authors":["Duncan McCoy"],"author_count":1,"year":2013,"published":"2013-12-04","venue":"Adv. Math. 305 (2017), 757-802","cited_by":11,"doi":"10.1016/j.aim.2016.09.033","arxiv_id":"1312.1278","nct_id":null,"source_url":"https://arxiv.org/abs/1312.1278","archive_url":"https://space-kitty.com/papers/arxiv:1312.1278","updated":"2014-04-22","pmid":null,"pmcid":null,"openalex_id":"W1582248218","pdf_url":"https://arxiv.org/pdf/1312.1278","abstract":"We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alternating knots. As a consequence, we also get a converse to the Montesinos trick: an alternating knot has unknotting number one if its branched double cover arises as half-integer surgery on a knot in $S^3$. We also reprove a characterisation of almost-alternating diagrams of the unknot originally due to Tsukamoto.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.GT"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1509.00112","kind":"article","title":"Quantum K-theoretic geometric Satake: the case","authors":["Sabin Cautis","Joel Kamnitzer"],"author_count":2,"year":2017,"published":"2017-10-30","venue":"Compositio Mathematica","cited_by":12,"doi":"10.1112/s0010437x17007564","arxiv_id":"1509.00112","nct_id":null,"source_url":"https://doi.org/10.1112/s0010437x17007564","archive_url":"https://space-kitty.com/papers/arxiv:1509.00112","updated":"2017-06-04","pmid":null,"pmcid":null,"openalex_id":"W1937528024","pdf_url":"https://arxiv.org/pdf/1509.00112","abstract":"The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $ G $ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group. Bezrukavnikov-Finkelberg developed a derived version of this equivalence which relates the derived category of $ G^\\vee$-equivariant constructible sheaves on $ Gr $ with the category of $G$-equivariant ${\\mathcal O}(\\mathfrak g)$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group $ U_q \\mathfrak g $. We define a convolution category $ KConv(Gr) $ whose morphism spaces are given by the $ G^\\vee \\times \\mathbb C^\\times $-equivariant algebraic K-theory of certain fibre products. We conjecture that $KConv(Gr)$ is equivalent to a full subcategory of the category of $ U_q \\mathfrak g $-equivariant $ \\mathcal O_q(G) $-modules. We prove this conjecture when $G = SL_n$. A key tool in our proof is the $SL_n$ spider, which is a combinatorial description of the category of $U_q \\mathfrak{sl}_n$ representations. By applying horizontal trace, we show that the annular $SL_n$ spider describes the category of $ U_q \\mathfrak{sl}_n $-equivariant $ \\mathcal O_q(SL_n) $-modules. Then we use quantum loop algebras to relate the annular $SL_n $ spider to $ KConv(Gr) $. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Algebraic structures and combinatorial models"],"details":null,"provenance":"openalex+arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1211.0180","kind":"preprint","title":"Geometric structure and the local Langlands conjecture","authors":["Anne‐Marie Aubert","Paul Baum","Roger Plymen","Maarten Sander Solleveld"],"author_count":4,"year":2012,"published":"2012-11-01","venue":"MIMS EPrints (University of Southampton)","cited_by":11,"doi":null,"arxiv_id":"1211.0180","nct_id":null,"source_url":"http://eprints.maths.manchester.ac.uk/1918/1/Introv34.pdf","archive_url":"https://space-kitty.com/papers/arxiv:1211.0180","updated":"2013-05-20","pmid":null,"pmcid":null,"openalex_id":"W1947573928","pdf_url":"https://arxiv.org/pdf/1211.0180","abstract":"We prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive $p$-adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":true,"retracted":false,"peer_reviewed":false,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex+arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1712.02108","kind":"article","title":"On the arithmetic Kakeya conjecture of Katz and Tao","authors":["Ben Green","Imre Ruzsa"],"author_count":2,"year":2017,"published":"2017-12-06","venue":"Periodica Mathematica Hungarica","cited_by":6,"doi":"10.1007/s10998-018-0270-z","arxiv_id":"1712.02108","nct_id":null,"source_url":"https://arxiv.org/abs/1712.02108","archive_url":"https://space-kitty.com/papers/arxiv:1712.02108","updated":"2017-12-06","pmid":null,"pmcid":null,"openalex_id":"W2963302741","pdf_url":"https://arxiv.org/pdf/1712.02108","abstract":"The arithmetic Kakeya conjecture, formulated by Katz and Tao in 2002, is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in $\\mathbb{R}^n$ is $n$. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.NT","math.CA","math.CO"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1206.6249","kind":"article","title":"The Surgery Unknotting Number of Legendrian Links","authors":["A. Bianca Boranda","Lisa Traynor","Shuning Yan"],"author_count":3,"year":2012,"published":"2012-06-27","venue":"Involve 6 (2013) 273-299","cited_by":7,"doi":"10.2140/involve.2013.6.273","arxiv_id":"1206.6249","nct_id":null,"source_url":"https://arxiv.org/abs/1206.6249","archive_url":"https://space-kitty.com/papers/arxiv:1206.6249","updated":"2012-06-27","pmid":null,"pmcid":null,"openalex_id":"W2026007552","pdf_url":"https://arxiv.org/pdf/1206.6249","abstract":"The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting number is calculated for every Legendrian link that is topologically a twist knot or a torus link and for every positive, Legendrian rational link. In addition, the surgery unknotting number is calculated for every Legendrian knot in the Legendrian knot atlas of Chongchitmate and Ng whose underlying smooth knot has crossing number 7 or less. In all these calculations, as long as the Legendrian link of $j$ components is not topologically a slice knot, its surgery unknotting number is equal to the sum of $(j-1)$ and twice the smooth 4-ball genus of the underlying smooth link.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.SG","math.GT"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1404.0598","kind":"preprint","title":"Preservation of depth in local geometric Langlands correspondence","authors":["Tsao-Hsien Chen","Masoud Kamgarpour"],"author_count":2,"year":2014,"published":"2014-04-02","venue":null,"cited_by":8,"doi":null,"arxiv_id":"1404.0598","nct_id":null,"source_url":"http://arxiv.org/abs/1404.0598","archive_url":"https://space-kitty.com/papers/arxiv:1404.0598","updated":"2014-04-02","pmid":null,"pmcid":null,"openalex_id":"W2275878567","pdf_url":"https://arxiv.org/pdf/1404.0598","abstract":"It is expected that, under mild conditions, local Langlands correspondence preserves depths of representations. In this article, we formulate a conjectural geometrisation of this expectation. We prove half of this conjecture by showing that the depth of a categorical representation of the loop group is less than or equal to the depth of its underlying geometric Langlands parameter. A key ingredient of our proof is a new definition of the slope of a meromorphic connection, a definition which uses opers. In the appendix, we consider a relationship between our conjecture and Zhu's conjecture on non-vanishing of the Hecke eigensheaves produced by Beilinson and Drinfeld's quantisation of Hitchin's fibration for non-constant groups.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":true,"retracted":false,"peer_reviewed":false,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex+arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:2210.09581","kind":"article","title":"Sticky Kakeya sets and the sticky Kakeya conjecture","authors":["Hong Wang","Joshua Zahl"],"author_count":2,"year":2022,"published":"2022-10-18","venue":"J. Amer. Math. Soc. 39 (2026), pp. 515-585","cited_by":6,"doi":"10.1090/jams/1067","arxiv_id":"2210.09581","nct_id":null,"source_url":"https://arxiv.org/abs/2210.09581","archive_url":"https://space-kitty.com/papers/arxiv:2210.09581","updated":"2025-10-07","pmid":null,"pmcid":null,"openalex_id":"W4415933284","pdf_url":"https://arxiv.org/pdf/2210.09581","abstract":"A Kakeya set is a compact subset of $\\mathbb{R}^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff and Minkowski dimension $n$. There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate multi-scale self-similarity, and sets of this type played an important role in Katz, Łaba, and Tao's groundbreaking 1999 work on the Kakeya problem. We propose a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have Hausdorff and Minkowski dimension $n$. We prove this conjecture in three dimensions.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.CA"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1810.05125","kind":"article","title":"Knot Floer homology and the unknotting number","authors":["Akram Alishahi","Eaman Eftekhary"],"author_count":2,"year":2018,"published":"2018-10-11","venue":"Geom. Topol. 24 (2020) 2435-2469","cited_by":17,"doi":"10.2140/gt.2020.24.2435","arxiv_id":"1810.05125","nct_id":null,"source_url":"https://arxiv.org/abs/1810.05125","archive_url":"https://space-kitty.com/papers/arxiv:1810.05125","updated":"2018-10-18","pmid":null,"pmcid":null,"openalex_id":"W2898039440","pdf_url":"https://arxiv.org/pdf/1810.05125","abstract":"Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the ν^-(K). Moreover, the difference l(K)-ν^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.GT"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:1311.6702","kind":"article","title":"Immersed disks, slicing numbers and concordance unknotting numbers","authors":["Brendan Owens","Saso Strle"],"author_count":2,"year":2013,"published":"2013-11-26","venue":"Communications in Analysis and Geometry","cited_by":29,"doi":"10.4310/cag.2016.v24.n5.a8","arxiv_id":"1311.6702","nct_id":null,"source_url":"https://arxiv.org/abs/1311.6702","archive_url":"https://space-kitty.com/papers/arxiv:1311.6702","updated":"2015-11-04","pmid":null,"pmcid":null,"openalex_id":"W2963096590","pdf_url":"https://arxiv.org/pdf/1311.6702","abstract":"We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordance unknotting number, which is the minimal unknotting number in a smooth concordance class. Using Heegaard Floer homology we obtain bounds that can be used to determine two of these invariants for all prime knots with crossing number ten or less, and to determine the concordance unknotting number for all but thirteen of these knots. As a further application we obtain some new bounds on Gordian distance between torus knots. We also give a strengthened version of Ozsvath and Szabo's obstruction to unknotting number one.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":null,"categories":["math.GT"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."},{"id":"arxiv:2505.07695","kind":"preprint","title":"Introduction to the proof of the Kakeya conjecture","authors":["Larry Guth"],"author_count":1,"year":2025,"published":"2025-05-12","venue":null,"cited_by":1,"doi":null,"arxiv_id":"2505.07695","nct_id":null,"source_url":"https://arxiv.org/abs/2505.07695","archive_url":"https://space-kitty.com/papers/arxiv:2505.07695","updated":"2025-05-12","pmid":null,"pmcid":null,"openalex_id":"W4417517294","pdf_url":"https://arxiv.org/pdf/2505.07695","abstract":"Recently, Hong Wang and Joshua Zahl announced a proof of the 3-dimensional Kakeya conjecture. This is a survey article on the proof of Kakeya. We introduce the problem, discuss previous work and some of the difficulties of the problem, and describe the new ideas in the recent work.","abstract_license":"CC0-1.0 (arXiv descriptive metadata)","license":null,"is_open_access":false,"retracted":false,"peer_reviewed":false,"categories":["math.CA"],"details":null,"provenance":"arxiv","attribution":"Abstract: arXiv descriptive metadata (CC0). Thank you to arXiv for use of its open access interoperability. This service was not reviewed or approved by, nor does it necessarily express or reflect the policies or opinions of, arXiv."}],"next":"https://space-kitty.com/api/v1/topics/landmark-math-results/papers?sort=top&limit=25&offset=25"}