Ianagol

Mathematician at UC Berkeley.

Homepage: https://math.berkeley.edu/people/faculty/ian-agol

Last Notes

2026-11-05 06:13:08 GMT

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This result now appears in a note of Baldwin-Sivek.
https://arxiv.org/abs/2611.06705

2026-10-28 03:35:29 GMT

The Anthropic paper on a 2^o(n) algorithm for 3-SAT (refuting the “Exponential Time Hypothesis “) has a lemma derived from the 8-square identity coming from octonion multiplication (see Section 2.4). They cite a survey paper of @npub1nf4…nqe4 for this.
https://en.wikipedia.org/wiki/Degen's_eight-square_identity
https://hexagonmath.org/2610.00577v1

2026-10-25 03:30:26 GMT

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This is a result of Kuperberg. https://arxiv.org/abs/1112.0845 A non-trivial knot has a non-cyclic SU(2) representation by a result of Kronheimer-Mrowka. Reducing modulo an ideal, one gets a non-cyclic representation to SL(2,p) for some p. GRH (or now quasi-GRH) allows one to bound the size of the prime polynomially in the crossing number. Baldwin and Sivek are writing a note explaining this new result (assuming the OpenAI result is correct).

2026-10-24 18:40:06 GMT

RE: https://mstdn.social/@QuantaMagazine/117496960947355286
An interesting (to me) consequence of this result (or rather some further advanced building on it) is that the unknot recognition problem is in coNP (ie, there is a certificate that a knot is knotted which is polynomial size in terms of its crossing number). One also has certificates that a knot is hyperbolic or torus. This was previously proved by Kuperberg and Baldwin-Sivek based on the generalized Riemann hypothesis. But the quasi version (for Dedekind zeta functions) is enough. Lackenby proved unknotting is coNP unconditionally using a different approach, but this approach gives different information.
#note1fan…s2ry

2026-09-10 15:22:20 GMT

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Multiple groups wrote up Perelman’s proof of the geometrization conjecture in detail, which held up to scrutiny (modulo the lack of details). So I can imagine a similar thing happening here (although maybe the proof is like the proof of the 4-color theorem, where some parts are too complicated to be human checkable).

2026-08-17 14:50:10 GMT

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That will keep this lady busy
https://youtu.be/OIsN0kiL9fI?is=M1SxFjxTDP27r02w

2026-07-01 02:04:13 GMT

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When the cone angles on the spherical polyhedron are rational, there is a connection with translation surfaces. One can take a branched cover over the vertices so that the cone points all have cone angles a multiple of 2π, and so that there is a global holomorphic 1-form, hence a translation surface. Properties of this translation surface, such as closed geodesics or geodesics connecting vertices, can correspond to those of the polyhedron. For example, it underlies these results: https://arxiv.org/abs/1802.00811
https://arxiv.org/abs/1811.04131
https://mathemalchemy.org/2021/01/19/dodecahedral-trajectory/
https://media.mathstodon.xyz/media_attachments/files/116/842/262/338/660/579/original/cd7a0e21aaf8ea70.png

2026-05-24 09:01:14 GMT

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There’s a textbook devoted to abelian groups: https://link.springer.com/book/10.1007/978-3-319-19422-6

2026-05-22 06:58:27 GMT

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Posted image of a high unit-distance graph from the lattice construction of OpenAI (from Alvaro Lozano-Robledo on Twitter), see comments at the above link.
https://media.mathstodon.xyz/media_attachments/files/116/616/924/632/787/627/original/4ef1f59c0ff1c00e.png

2026-05-22 03:36:13 GMT

I was looking for images of the point configurations maximizing unit distances, and found this post. https://community.wolfram.com/groups/-/m/t/3719376 it would be nice to see some based on the construction in the paper.

2026-05-14 04:11:14 GMT

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Looks like MacKay’s paper appeared in 1981, and Levine-Steinhardt 1984. The three jointly received the Buckley prize for this in 2010.
https://media.mathstodon.xyz/media_attachments/files/116/570/970/515/304/201/original/cf9c8009ff7bb275.png

2026-05-07 20:46:03 GMT

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It looks like the continued fraction expansion of the Friedman constant has period 66754.
https://mathoverflow.net/questions/457086/simple-continued-fraction-of-freimans-constant
It would be nice if there is a geometric interpretation of this constant.

2026-05-06 16:10:59 GMT

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I hadn’t seen this before. Presumably these are the orbits of Z_10 x Z_10 subgroups of the 14400 - order symmetry group. The orientation preserving subgroup is generated by the product of binary icosahedral groups acting on the right and left on the unit quaternions (with center acting trivially). This is then extended by an orientation-reversing element. In the icosahedral group (which is A_5), there is a cyclic element of order 5. The preimage of this element in the binary icosahedral group is order 10. So one gets (Z_10xZ_10)/(-1,-1). Maybe one can adjoin an orientation-reversing element to get the full Z_10 x Z_10? Or maybe it’s some sort of Z/2-extension? In any case, this suggests that on the Clifford torus, the tetrahedra come in two mirror image groups of 50. This seems to be the case looking at the visualization.

2025-12-07 22:24:49 GMT

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This seems impressive.
https://www.linkedin.com/posts/axiommath_yesterday-thousands-of-bright-undergrads-activity-7403482884361093120---sC/

2025-12-07 22:12:52 GMT

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The classification of finite simple groups begs to differ.

2025-10-13 15:26:27 GMT

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Another option is to consider “long knots”. I’m not sure who invented this, but I learned about it from Ryan Budney. This is just where you imagine the two ends going off to infinity, asymptotic to lines. https://arxiv.org/abs/math/0309427

2025-09-27 04:57:17 GMT

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TEDinburgh - might run into copyright issues.

2025-03-24 04:00:58 GMT

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Deleting the red edges (and erasing the red vertices), one gets the Cayley graph for the semidihedral group of order 16 with respect to the generator of the cylic group Z8 and the involution automorphism of Z8 (in the Cayley graph, one may draw one green edge for each involution instead of two directed edges). This group is a subgroup of the McGee group. https://groupprops.subwiki.org/wiki/Semidihedral_group:SD16 Pairs of opposite green edges, fixed by the other involutions of the McGee group (corresponding to the D16 subgroup, or extension of Z8 by the -1 involution) have midpoints of edges connected by a red edge to recover the McGee graph from the Cayley graph of SD16.
https://media.mathstodon.xyz/media_attachments/files/114/215/401/119/908/336/original/9dc95725475a0ff4.png