Landmark Results in Pure Mathematics
Recent major theorems and their status: geometric Langlands, Kakeya in three dimensions, sphere packing, the unknotting-number additivity counterexample and more.
Landmark Results in Pure Mathematics: State of the Field (October 2026)
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.
Most influential
- On the Multilinear Restriction and Kakeya conjectures
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" probl…
Article308 citationsarXiv:math/0509262doi:10.1007/s11511-006-0006-4 - On two geometric realizations of an affine Hecke algebra
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 shea…
Article72 citationsarXiv:1209.0403doi:10.1007/s10240-015-0077-x - Sphere Packing and Quantum Gravity
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 func…
Article100 citationsarXiv:1905.01319doi:10.1007/jhep12(2019)048 - Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture
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 al…
Article68 citationsarXiv:0711.0871doi:10.1090/s0894-0347-2010-00679-0 - Outline of the proof of the geometric Langlands conjecture for $GL_2$Article42 citationsdoi:10.24033/ast.966
- New upper bounds on sphere packings I
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.
Article311 citationsarXiv:math/0110009doi:10.4007/annals.2003.157.689 - Improved sphere packing lower bounds from Hurwitz lattices
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 pac…
Article53 citationsarXiv:1105.3779doi:10.1016/j.aim.2011.04.016 - Outline of the proof of the geometric Langlands conjecture for GL(2)
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.
Preprint18 citationsarXiv:1302.2506 - Knots with unknotting number 1 and essential Conway spheres
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…
Article31 citationsarXiv:math/0601265doi:10.2140/agt.2006.6.2051 - Knots with unknotting number one and Heegaard Floer homology
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 alternat…
Article53 citationsarXiv:math/0401426doi:10.1016/j.top.2005.01.002 - The sphere packing problem in dimension 24
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 Via…
Article288 citationsarXiv:1603.06518doi:10.4007/annals.2017.185.3.8 - A conceptual breakthrough in sphere packing
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.
Article78 citationsarXiv:1611.01685doi:10.1090/noti1474 - A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture
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…
Preprint14 citationsarXiv:1411.1208 - Preservation of depth in the local geometric Langlands correspondenceArticle13 citationsdoi:10.1090/tran/6794
- The unknotting number and classical invariants I
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 approa…
Article15 citationsarXiv:1203.3225doi:10.2140/agt.2015.15.85
Newest work
- Point Corrections and Wall-Crossing Geometry on the Quintic del Pezzo ThreefoldPreprintarXiv:2610.39336
- Higher order unknotting numbers for knots and linksPreprintarXiv:2610.39474
- Vanishing near the virtual cohomological dimension of $\mathrm{SL}_n(\mathbb Z)$PreprintarXiv:2610.39960
- A lifted entropy splitting lemma and applicationsPreprintarXiv:2610.40145
- Geometric properties of the analytic Emerton-Gee stackPreprintarXiv:2610.38115
- Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime IntersectionsPreprintarXiv:2610.38655
- Primitive flag-transitive generalized quadranglesPreprintarXiv:2610.38819
- Homogeneous polynomial approximation on symmetric convex bodies: A proof of Kroó's conjecturePreprintarXiv:2610.38877
- Counting the Fixed Points of Binary Belief Propagation: Generic Degree, Syzygies, and Real SolutionsPreprintarXiv:2610.36964
- Finite group actions on the non-minimal nilpotent orbit closures in type APreprintarXiv:2610.37396