Joshua Grochow

Assoc. Professor @ CU Boulder Comp. Sci. & Math; Research interests: theoretical computer science, pure mathematics, and complex systems.
Other interests: climate change; accurate (long)COVID info; equity, inclusion, & accessibility; improving academia. #BlackLivesMatter #TransRightsAreHumanRights #LandBack #CovidIsAirborne #CleanTheAir #StillMasking #MaskUp #YallMasking #TCS #TheoryCS #math #ComplexSystems #complexity (both kinds) #academia

Website: https://home.cs.colorado.edu/~jgrochow/
cstheory.stackexchange: https://cstheory.stackexchange.com/users/129/joshua-grochow
Twitter & Bsky: @joshuagrochow@{twitter.com,bsky.social}
Pronouns: He/him

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Published at

2025-03-26T18:23:22Z

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Last Notes

2025-03-27 03:13:59

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I mean, r^4 is central, and acts as -1 in those irreps, so the 8 points of that G-set arise as a generic orbit in the projective space of either of the faithful 2D irreps. (cc @npub1whr…qc2t)

2025-03-27 01:30:14

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I was hoping to see it in a vector space so the "geometry" of the "figure" would be obvious. Notice that r^4 also generates an order 2 subgroup that's not conjugate to s, so that's a different 8-point G-set

2025-03-25 18:43:40

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it's got two (Galois-conjugate) faithful 2D irreps over C:https://people.maths.bris.ac.uk/~matyd/GroupNames/1/SD16.html. A generic orbit there contains 16 points, so will give you a shape, but I have no idea what it is.
A subgroup of order 2 acts as -I on those reps, so each orbit consists of 8 points and their antipodes.
The reps seem to genuinely be complex (bc the characters include sqrt(-2)), so you probably really have to think of it as 16 points in C^2, which might be harder to visualize than R^2.

2025-03-18 19:29:14

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oh, uh, maybe not. Just explains why derivations appeared when computing automorphism of GA(1,Z/8).
But the three types of edges I do have some sort of an explanation for.
Think of B=Z/8=red vertices as the base. The blue vertices and edges between them are something like a global section of a certain 6-fold etale cover of the graph B w edges=odd differences.
Between red verts (in B) there are 3 natural choices for edge set (these are the orbitals of GA(1,Z/8)): nearby, opposite, and distant in @npub12cu…d38p's terminology. Since the edges btwn blue verts cover the distant edges, we should really include nearby edges in B with one color and opposite with another. To make McGee 3-regular, choose to make opposite pairs edges and nearby pairs non-edges in B.
The edges between blue and red vertices are then just the 2-fold "covering map" (it's not quite a cover, because it's 2-to-1 but over each odd edge that would be present in the base there's only one blue edge instead of 2).

2025-03-18 18:53:45

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This totally nerd-sniped me.
For the role of derivations in automorphisms of group extensions, check out the Wells exact sequence (and more recent work on it e.g. Buckley, Malfait, Jin-Liu), nicely exposited (w/ refs) by Jill Dietz in https://doi.org/10.1017/CBO9781316227343.013.
I had some thoughts about building the McGee graph from covering graphs (or etale covers, at least), but I think maybe they are too complicated to be worth writing down.

2025-03-18 01:09:13

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Oh, I don't think you were arguing against what I was saying at all - sorry if I phrased something that way. Just trying to figure out where the cohomology is here!

2025-03-17 23:43:55

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I mean, sure, but I feel like that fits in the following analogy
cohomology class : automorphism coming from a crossed hom
::
nontrivial coho class : non-inner automorphism coming from a crossed hom
And then the thing about class-preserving is a subgroup of automorphisms coming from crossed homs

2025-03-17 06:02:14

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the gif in the first post looks very Mobius-y. Looks like you can view the graph in terms of a 3-cover of the 8-cycle, where you then connect the 3 vertices in each fiber in a path (blue-red-blue) and then add in the red edges after. Definitely feels like something cohomology-y going on.
(But I should be doing other things now - maybe I'll give a shot at making this more precise in a day or two.)

2025-03-17 05:47:22

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those equations are exactly the definition of crossed homomorphism in this context (they all follow from the first equation, using the fact that the group doing the acting is commutative, and the action is faithful).

2025-02-02 20:41:15

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Can you say why you prefer pcloud over Dropbox? I just looked a little and it looks interesting.
For replacing github, have you looked into gitea?

2024-12-11 00:41:04

Just found an English translation of Emmy Noether's 1921 "Idealtheorie in Ringbereichen" ("Ideal Theory in Rings"): https://arxiv.org/abs/1401.2577
(while editing the wikipedia page on subdirect products - my first wiki edit to add an Emmy Noether reference! Turns out there's a direct lineage from Noether to Birkhoff's introduction of subdirect products in universal algebra. Just one more way in which she really revolutionized algebra.)
#math #AlgebraicGeomtry #Algebra #UniversalAlgebra

2024-12-08 21:34:14

We can, and must, do more to protect students and teachers in higher education from the risks of post-COVID condition aka longCOVID https://theconversation.com/we-can-and-must-do-more-to-protect-students-in-higher-education-from-the-risks-of-post-covid-condition-244244
H/t @aeporreca.org on bsky
#academia #LongCovid #covid #covid19

2024-10-11 15:13:45

Thinking about how AI-generated writing isn't only poisoning its own training set, it's also poisoning *ours*.
I'm no expert, but it sure seems like learning good writing often starts with reading good writing. Imagine a generation of kids growing up where half of what they read is AI-generated.
#genAI #writing #reading

2024-10-04 19:28:22

Apparently I missed that Zhuk posted a *simplified* proof of the CSP Dichotomy Conjecture back in January: https://arxiv.org/abs/2404.01080
I'd really love to understand all of this!
#ComputationalComplexity #complexity #math #UniversalAlgebra