January 2026 Archives | The n-Category Café

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January 2026
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January 2026's Entries

Coxeter and Dynkin Diagrams
- Jan 6, 2026
- Dynkin diagrams have always fascinated me. They are magically potent language — you can do so much with them!…

Random Past Entries

TeXnical Issues
- September 2, 2006
- Sage advice on viewing this blog and posting comments thereon.
Lectures on Classical Mechanics
- Dec 21, 2006
- Lectures notes on classical mechanics - now available in printed form!
Lanthanides and the Exceptional Lie Group G₂
- May 27, 2024
- What the connection between the 14 lanthanide elements and the exceptional Lie group G₂?
Shell What?
- Oct 5, 2009
- My ol blog pages
This Week’s Finds in Mathematical Physics (Week 262)
- Mar 30, 2008
- Learn about the Southern Ring Nebula, the frosty dunes of Mars, quantum technology in Singapore, atom chips, graphene transistors, nitrogen-vacancy pairs in diamonds, a new construction of e₈, and a categorification of sl(2).
Summer Research at the Topos Institute
- Nov 2, 2024
- You can now apply for the 2025 Summer Research Associate program at the Topos Institute! The deadline to apply is January 17, 2025.

Recent Comments

Propositions as Some Types and Algebraic Nonalgebraicity (15)
- Mike Shulman
  - Dec 19, 2025 17:54
  - There are many different kinds of type theory that can serve as internal languages for elementary toposes. Some of them ...
- Westlifer
  - Dec 18, 2025 07:58
  - If I understand correctly, the logic enriched type theory is often presented as a 2-layered type theory, with a universe ...
- Peter LeFanu Lumsdaine
  - Jan 16, 2012 19:33
  - On the language issue: I think it’s that “Noun1 verbs noun2” very often carries the additional implicature that noun2 is ...
- Mike Shulman
  - Jan 15, 2012 22:59
  - But I have to say, your calculator has no business telling you that e −999999e^{−999999} is 0 I don’t know ...
- Jesse C. McKeown
  - Jan 15, 2012 20:36
  - OK, exaggerating with the analytic functions with relatively large derivative was I… But I have to say, your calculator has ...

Octonions and the Standard Model (Part 13) (2)
- John Baez
  - Dec 19, 2025 10:12
  - It should be ℂ,ℝ\mathbb{C}, \mathbb{R} and ℍ\mathbb{H}, since typical complex representations are not self-dual. Thanks for noticing that something was ...
- David
  - Dec 19, 2025 10:00
  - Hi, John, Should this “division algebras ℂ,ℂ and ℍ, respectively” be ” division algebras ℝ, ℂ and ℍ, respectively”. David

Octonions and the Standard Model (Part 12) (4)
- John Baez
  - Dec 18, 2025 13:11
  - Thanks! I’ve been discussing these issues with Howard and my impression is that these papers give new more physical reasons ...
- Blake C. Stacey
  - Dec 18, 2025 06:14
  - Barnum and coauthors have a couple more recent papers that look relevant to the question of what properties of quantum ...
- John Baez
  - Dec 17, 2025 14:37
  - Oh no, it’s not a true Hjelmslev plane??? Thanks for warning me. I am indeed building the bioctonionic plane starting ...
- bitdizzy
  - Dec 17, 2025 04:07
  - Let us call it a Hjelmslev-Moufang plane, which we will find out is a parapolar space, and perhaps even a ...

log|x| + C revisited (13)
- Tom Leinster
  - Dec 13, 2025 21:03
  - Right; I probably never knew this complex analysis fact myself, and if I did, I’d long forgotten it.
- Mike Shulman
  - Dec 13, 2025 19:34
  - That’s all technically true, but it’s also not something we necessarily want to get into with calculus students.
- Madeleine Birchfield
  - Dec 13, 2025 19:15
  - My point is rather that as a mathematical matter, for what we think of inside our heads as mathematicians, the ...
- Mike Shulman
  - Dec 13, 2025 18:36
  - The functions in your average calculus class are valued in the real numbers, not the complex numbers, so that those ...
- Mike Shulman
  - Dec 13, 2025 18:31
  - Indeed, it’s common knowledge among many mathematicians (not only to implementers of computer algebra systems) that the statement “∫1xdx=log|x|+C\int \frac{1}{x}dx=log|x|+C” ...

Safeguarded AI Meeting (11)
- Alexander Prähauser
  - Nov 27, 2025 04:08
  - I am writing this (maybe too late but) in the hopes of an open-minded reception, but I do have to ...
- David Roberts
  - Sep 4, 2025 06:26
  - In all cases, the line-by-line proof details presented here were constructed by the author. It seems important to point out ...
- Jeffery Winkler
  - Sep 3, 2025 06:55
  - Here is a physics paper partially written using ChatGPT. https://arxiv.org/pdf/2508.21276 This will be more common in the future. Is this ...
- Jeffery Winkler
  - Sep 3, 2025 06:27
  - Here are some more links about the connections between physics and AI. Physics Today - Nobel Prize https://pubs.aip.org/physicstoday/article-abstract/77/12/12/3320663/Nobel-Prize-highlights-neural-networks-physics 2 Physics ...
- Jeffery Winkler
  - Aug 20, 2025 16:55
  - In 1954, Frederick Brown wrote a science fiction story titled “The Answer”. It’s so short I can reproduce it here. ...

Beyond the Geometry of Music (1)
- Kevin MacDonald
  - Nov 24, 2025 20:38
  - “The Owl of Minerva only flies at dusk”, likely leading to a sleepless night spent with a talking bird.

Dynamics in Jordan Algebras (7)
- John Baez
  - Nov 24, 2025 12:20
  - Great, thanks! That’s good to know.
- Ed
  - Nov 24, 2025 08:15
  - I had a closer look at Braun and Koecher. The result you want is Satz 3.1 on page 281. The ...
- John Baez
  - Nov 10, 2025 14:06
  - Thanks! I’m now quite interested in whether all derivations of finite-dimensional Jordan algebras are inner. Your idea of complexifying the ...
- John Baez
  - Nov 10, 2025 13:56
  - If you tensor the underlying vector spaces of two Jordan algebras, there’s no good way in general to make that ...
- Jesse C. McKeown
  - Nov 8, 2025 20:33
  - It seems so trivial a follow-on, but your philosophical question seems to raise a further question and the [L X,L ...

The Inverse Cube Force Law (4)
- jim cooper
  - Nov 24, 2025 11:46
  - And, of course, the main equation at the beginning should be f+D 2f=cf αf+D^2 f=c f^\alpha.
- jim cooper
  - Nov 23, 2025 19:51
  - DfDf and D 2fD^2f are the derivatives of ff ( dashes didn‘t seem to work). While I am here, „tidely“ ...
- John Baez
  - Nov 23, 2025 14:53
  - Cool! But what’s DD? You didn’t say.
- jim cooper
  - Nov 20, 2025 08:59
  - The following comment might put your remarks in context. If we write the equation for an orbit in the unorthodox ...