Ianagol

Mathematician at UC Berkeley.

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

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Last Notes

2026-09-10 15:22:20 UTC

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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 UTC

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

2026-07-27 15:49:35 UTC

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Vasudha has claimed that the Gassner representation is faithful, which is maybe a less well-known problem, but probably a bigger deal (it implies linearity of the braid groups with a much simpler to describe representation than the Lawrence-Krammer representation proved to be faithful by Bigelow & Krammer). https://arxiv.org/abs/2208.12378

2026-07-01 02:04:13 UTC

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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 UTC

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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 UTC

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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 UTC

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 UTC

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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

2026-05-13 16:02:06 UTC

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Quasicrystals were first predicted by Alan Mackay (he knew Penrose and may have been inspired by Penrose tilings)
https://en.wikipedia.org/wiki/Alan_Lindsay_Mackay
I met his son Bob at a climbing gym in Berkeley. http://bobmackay.com/ Bob met Penrose as a teenager through his father and made some of the first computer designs of Penrose tilings.

2026-05-07 20:46:03 UTC

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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 UTC

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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.

2026-04-22 17:04:56 UTC

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This is coming from Humbert’s volume formula for the quotient of hyperbolic space by a Bianchi group.
https://en.wikipedia.org/wiki/Bianchi_group

2026-04-22 15:47:01 UTC

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The figure eight knot group Γ is generated by the Mobius transformations z-> z+ω, z-> 1/(1+z), acting on the upper half-space model of hyperbolic space, with coordinates (z,t), z in C, t>0, and where ω is a third root of unity (-1+sqrt(-3))/2. There is a cusp H (or horoball) centered at infinity, which is the set of points t> 1. Looking at the image of H under the group Γ, one gets balls tangent to the boundary t=0 and of height 1, centered at the lattice Z+Zω, as well as horoballs of various heights tangent to Q(ω). This is what is shown in the image. They are tangent to H, and so are the largest horoball whose interior is disjoint from all of its images. The volume of the image of H in the quotient is sqrt(3), and the volume of the quotient is 2.0298. This ratio is a rational multiple of their value (and I think I had it flipped, 1/ cusp density, so volume / cusp volume).

2026-04-22 01:20:57 UTC

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This is a rational multiple of the cusp density of the hyperbolic structure on the figure eight knot complement.
https://media.mathstodon.xyz/media_attachments/files/116/445/727/091/866/085/original/ea0dcde2a62cd1b8.png

2026-04-09 03:33:13 UTC

How technology in the classroom can distract from learning.
https://www.theatlantic.com/ideas/2026/04/teachers-screens-edtech-students/686681/?gift=G0hU6ohQGLGJfrFtwMSWpVILzN_BAXtiLF-_MUq-2cE

2026-04-05 04:41:41 UTC

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I clicked on it to check that the date wasn’t April 1.

2026-03-15 18:33:19 UTC

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I didn’t look at the petition, but I can imagine a host of reasons not to hold the ICM in the US right now.

2026-01-11 05:14:34 UTC

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Anyone that figures out dark matter will be walking on cloud 9.

2025-12-07 22:24:49 UTC

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

2025-12-07 22:20:22 UTC

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Ken Ono is all in on this.
https://www.wsj.com/tech/ai/math-ken-ono-carina-hong-axiom-startup-649bc417

2025-12-07 22:12:52 UTC

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The classification of finite simple groups begs to differ.

2025-11-06 16:35:00 UTC

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An equivalent way to see it: take a linear transformation sending the parallelopiped to a cube. Then the inscribed ellipse will be sent to a sphere. Now note that the ratio between the volumes is constant.

2025-10-18 21:28:36 UTC

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It’s too bad this hasn’t been worked out completely yet. I’m not sure what the issues are. Someone told me a while ago that there are some tricky differential topology issues that haven’t been resolved.

2025-10-13 15:26:27 UTC

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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

2025-09-28 18:57:39 UTC

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I like that, Ed could be Edinburgh or Education.

2025-09-27 04:57:17 UTC

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TEDinburgh - might run into copyright issues.

2025-09-03 01:39:26 UTC

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Hint: the answer is invariant under area preserving transformation.

2025-08-07 06:44:11 UTC

Math puzzle (inspired by recent posts of James Tanton). Take some points in Euclidean space z_1,…,z_k , what is the locus of points x such that the sum of the squares of the distances of x to z_1 through z_k is constant? I.e. |x-z_1|^2+ … + |x-z_k|^2 = c.

2025-08-05 16:48:32 UTC

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A reminder that the reason we don’t have unique representations for real numbers is that the Cantor set is totally disconnected but the real numbers are connected (I think I didn’t learn about this explanation until I was an assistant professor).

2025-08-04 18:38:21 UTC

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Giving a new meaning to Hamiltonian mechanics.

2025-07-21 18:18:55 UTC

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It feels like this should be related to Faulhaber’s formula, but I’m not sure how.

2025-07-13 11:50:10 UTC

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I shared a gifted version of this article here: (for people without a subscription). https://mathstodon.xyz/@Ianagol/114835860905798385

2025-07-11 17:53:29 UTC

Record lattice sphere packings discovered. https://www.quantamagazine.org/new-sphere-packing-record-stems-from-an-unexpected-source-20250707/?mc_cid=a20fe07f36&mc_eid=284cf6a352 This also seems to optimize the “lattice kissing” bound.

2025-06-14 20:00:22 UTC

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More signs
https://media.mathstodon.xyz/media_attachments/files/114/683/483/426/704/602/original/1b1d76c6f5020bd0.png
https://media.mathstodon.xyz/media_attachments/files/114/683/483/628/763/708/original/57c89bce25621aa5.png
https://media.mathstodon.xyz/media_attachments/files/114/683/483/833/523/040/original/473284744f6c62bb.png
https://media.mathstodon.xyz/media_attachments/files/114/683/484/061/024/015/original/81a2607e02891a49.png

2025-06-14 19:56:46 UTC

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No Kings protest at Larkspur Landing
https://media.mathstodon.xyz/media_attachments/files/114/683/470/524/300/530/original/f8b5eebca8776d2f.png
https://media.mathstodon.xyz/media_attachments/files/114/683/470/751/067/343/original/294000509bc9ef2c.png
https://media.mathstodon.xyz/media_attachments/files/114/683/471/055/396/484/original/6eb14015e049623e.png
https://media.mathstodon.xyz/media_attachments/files/114/683/471/390/206/074/original/27ee94864a67e82f.png

2025-06-12 23:06:56 UTC

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This is the whole point, these deportations are a drop in the bucket compared to the number of immigrants, but news of these deportations are intended to scare current and potential immigrants into leaving or not coming. That was also the point of separating children from their families - who would want to risk that?

2025-04-05 18:04:37 UTC

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Thanks, the link was at the end of the article so I didn’t include it in the posting.
A conceptually simpler formula is to compute the arc of the circle between tangent circles in the ring. Then the sum of these angles is 2π. Of course, this is not an algebraic equation, but could be made algebraic via the application of trig identities.
https://media.mathstodon.xyz/media_attachments/files/114/286/674/233/640/598/original/03abd54b517b7f9d.png

2025-04-05 01:02:33 UTC

Formula relating the radii of a ring of circles.
https://scitechdaily.com/descartes-unfinished-mystery-mathematicians-solve-380-year-old-geometry-problem/
I recall learning about the ring lemma as a graduate student, but I didn’t realize that there wasn’t an explicit formula known relating the radii.
https://en.m.wikipedia.org/wiki/Ring_lemma

2025-03-24 15:38:52 UTC

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I hadn’t heard of it before, I just looked at the gif of the McGee graph and identified the Cayley graph minor via groupprops.

2025-03-24 04:32:27 UTC

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Cool, are you visiting the topos institute? We are on spring break this week.

2025-03-24 04:00:58 UTC

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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

2025-03-02 21:49:48 UTC

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Rent an electric car? Driving up the 101 from SB is nice. Highway 1 is still closed in Big Sur I think.

2025-02-28 03:43:39 UTC

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Not sure if this counts, but first and second fundamental theorems of calculus are pretty famous.

2025-02-27 05:54:29 UTC

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My wife canceled our subscription too (I would only read it occasionally).

2025-01-21 03:05:55 UTC

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reading the question, it wasn’t immediately clear to me that a solution should exist, since one is given one parameter of freedom (α) to try to hit the identity, but the Lie group SO(3) has dimension 3. So potentially this could trace out a curve in SO(3) that doesn’t pass through the identity (except for α=0). But somehow the symmetry allows a non-trivial solution.