Dan Piponi

Disclaimer:
👽 My opinions are not my own. They're beamed to me by aliens

Current life:
🧠 Thinking at high effort

Previous lives:
🍩 PhD in Riemann theta functions
🧬 Glaxo Group Research
🚀 Argonaut Games
💥 Mass Illusion, Esc, ILM
🎈 Google X, Verily, Google
🎮 Epic Games

Likes:
🚴 I like to bike
🏃 I like to run
🎛️ my musical tastes lie towards the electronic end of the spectrum
🚀 I like Andor *and* The Mandalorian
📈 I have an OEIS entry https://oeis.org/A387361

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dpiponi_at_mathstodon.xyz@momostr.pink

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Publishing to

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nos.lol

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Last Notes

2026-04-16 04:23:28 UTC

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Exactly. Except in this case "folk" means a lot of mathematicians.

2026-04-16 01:52:31 UTC

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That's my reading. This is like a programming language thing: does the symbol mean what the previous definition meant, or is it recursive? I think it's using the old definition inside the new one - but I'm not an expert on this.
You're in Scotland now, right? You're sounding like a native :)

2026-04-16 01:08:28 UTC

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But that paragraph starts "Some have proposed extending the set of elementary functions..."

2026-04-16 00:45:46 UTC

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"Elementary functions typically include arbitrary polynomial root functions"
Yeah, I've never seen that claim before! In fact, people often emphasize that this is *not* an elementary function.

2026-04-14 17:07:46 UTC

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Or a version I experienced yesterday: I had to shut off the water to the house and then switch it on again. You get that nasty hammering phenomenon. You have to bring it back slowly (in this house) to avoid oscillations.

2026-04-14 16:48:06 UTC

When you compute a finite sum f(i) i = 1 .. n, there's a non-obvious subtlety that the sum comes to an abrupt end at i = n - and abrupt ends result in interesting phenomena that aren't always what you're interested in.
For example, it's well know that you get the Gibbs phenomenon when you partially sum the Fourier series for a square wave. But in some sense those wibbles aren't coming from the fact that you haven't summed all of the parts needed. They are, in a sense, because of that abrupt stop.
For example, I link [1] to an alternative way to approach the square wave using exponentials to feather out the sum rather than having a sharp cutoff - inspired by Tao's approach to summation [2]. You get nice smoothed square waves.
Anyway, this technique is called Abel summation and is in all the good Fourier transform textbooks. Engineers (including me when image processing over the years) sort of do this stuff all the time as well, using "windows" to smooth edges.
But in my head I hadn't related it back to the humble sum i=1 .. n in contexts outside of Fourier theory.
The useful thing to learn here is that sometimes it's not good to approach an infinite sum via partial sums of its terms because all you're doing is pushing those edge effects out to infinity. That doesn't remove them, even in the limit. This will change my perspective on a lot of phenomena, especially in physics.
[1] https://www.wolframcloud.com/obj/de1c00c3-3563-4b43-8252-67b4ad33de3b
[2] https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/
https://media.mathstodon.xyz/media_attachments/files/116/404/040/630/018/813/original/c254f31bfe8c4a18.png

2026-04-14 16:11:35 UTC

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Can we get better drugs, I don't like these ones.

2026-04-14 15:18:59 UTC

The news is so beyond plausible I'm beginning to wonder if reality has been replaced by the output of an LLM.

2026-04-08 01:09:40 UTC

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I didn't know that went back to Minkowski. I've thought about this kind of thing a lot but I'm not expert enough to make a useful contribution. I think that there are many problems where if you explore the space of possible solutions it turns out the space is full of good solutions. But, bizarrely, short strings of mathematical language are surprisingly good at finding the bad solutions. The unreasonable ineffectiveness of mathematics!

2026-03-31 16:19:59 UTC

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What I find interesting about this is that during my mathematical training I was taught there is but one way to do mathematics: ZFC. When I got to meet people working in proof verification I learnt that there is a whole world of formalisms out there.

2026-03-23 00:26:49 UTC

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Often the Casimir force is presented as the consequence of summing 1+2+3+... and getting a -1/12 that contributes to the 240 in the standard Casimir force formula. But it's a bit nonsensical to just say we can replace an actual sum with an analytic continuation.
But the sum-to-integral formula is pretty nice and well behaved and fits physical intuition.
And also, you can now understand the Ramanujan number as coming from that 1/(exp(x)-1) which hints at why the analytic continuation isn't so evil after all. I wrote about it long ago and it appeared in This Week's Finds 126 [3].
[3] https://groups.google.com/g/sci.physics/c/bZOBtIhkSpA/m/k30KBIv37rsJ

2026-03-23 00:21:59 UTC

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When you have two plates held *very* close to each other, then the wavefunctions for particles are going to have some kind of boundary condition they need to satisfy at the plates. This means that your wavefunctions now have a discrete spectrum, typically with wavelengths that are an integer divisor of the plate spacing along the axis perpedicular to the plates. You go from a continuous spectrum to a discrete spectrum. S(D) is precisely the operator you need to study this.
There's a catch: in a vacuum in QFT you expect the energy in each harmonic of your wavefunction to be proportional to n, the divisor I mentioned. So you're computing the difference between an infinite sum and an infinite integral. You can apply the operator S(d/dn), but what does it mean?
We can turn to Terry Tao [2]. In the case of divergent sums, S(D) computes the difference between "smoothed sums" and "smoothed integrals". You're basically saying there is some cutoff in n at which point you don't care about the contributions, but then let that cutoff go to infinity. It makes physical sense, high frequency waves "don't care" about comparatively large gaps between plates.
Apply S(d/dn) to the function n->n and you get -1/12 as you see in Mathematica.
(The sign flip is because I'm considering sums n to infinity, not -infinity to n).
So you expect there to be an energy due to the plates and hence a force as you move them. So the Casimir force is basically a consequence of applying S(d/dn).
But I also did the computation a more traditional way as you can see in Mathematica. But there's a reason I did it the operator way...
[2] https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/
https://media.mathstodon.xyz/media_attachments/files/116/275/622/370/376/916/original/e3db58c496d601e5.png

2026-03-23 00:17:56 UTC

Well you know how I like to mess with differential operators. So here's a fun example, arguing somewhat casually:
If D=d/dx
1/D is the inverse, the integration operator (but you need to take some care with the integration bounds to do this properly).
Then exp(D)f computes f(x+1) because it's just a Taylor expansion.
So exp(D)-1 is the finite difference operator.
And 1/(exp(D)-1) is the opposite of the finite difference, ie. sums. (It's the Euler-Maclaurin formula [1])
So S(D)=1/(exp(D)-1)-1/D is the integral-to-sum difference operator. The cool thing about S is that despite the 1/D, if you expand as a series S(D) is a genuine power series with no negative powers.
So where in physics might you need this...?
[1] https://en.wikipedia.org/wiki/Euler–Maclaurin_formula

2026-03-11 05:25:54 UTC

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Both I think, depending on the reactor. I'll know more when I've finished the book. Earlier the book opened by chance momentarily at a sentence that said something like "we got the tour on condition that we didn't describe what we had seen" and then proceeds to describe the tour but I can't find that bit again and surely I misread it!

2026-03-11 05:12:39 UTC

Megawatts and monoliths. A fun pair of guide books that just arrived in the mail. Next trip to Britain I hope I to visit another stone circle and my first nuclear reactor.
https://media.mathstodon.xyz/media_attachments/files/116/208/816/836/188/561/original/e6c310232c38fb17.jpeg

2026-03-07 00:00:38 UTC

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I don't normally suffer from anxiety from looking at pictures of animals but I can't help worrying about these poor little things breaking their fingers. Maybe they're flexible and resist breaking.

2026-02-06 17:28:27 UTC

Where are all the memristors?

2026-01-28 00:25:14 UTC

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I personally take the view that if it's essential for survival we can't really ask why it's there without slipping into religion. So I'm not optimistic about discussion about why there is enough order out there for us to make a living.
But, one could ask how much more order there is than the bare minimum. Our ancestors didn't need general relativity for to climb down from the proverbial trees, nonetheless we do have it now. Should I be surprised that we found this extra regularity that wasn't required for survival? I don't know how to measure how surprised I should be. I certainly wouldn't expect that the only physics we have is precisely the bare minimum for survival - survival just gives the lower bound. But maybe it is surprising we're able to make accurate testable predictions about, say, black hole collisions.
But anyway, none of this changes my belief that pure mathematics was built on empirical principles much like physics.

2026-01-27 22:37:38 UTC

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Mathematicians also add axioms as needed. ZFC didn't appear by magic ex nihilo. It's the result of a centuries long process of trying out different approaches until something was good enough to do most of what was needed - and the thing that was needed was very often motivated by physics even though mathematicians sometimes deny it.

2026-01-24 21:35:27 UTC

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It certainly affects non-white people disproportionately. Depressing though it is, I'm glad the brutality of American law enforcement is finally being recognised by more Americans.

2026-01-24 20:38:03 UTC

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Cops have always shot plenty of white people. In this case it's clear why it's being reported but mostly it only gets reported in local news.

2026-01-24 00:46:40 UTC

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Apparently they're cognate.

2026-01-23 19:24:20 UTC

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Their fish and chips is fecking good.

2026-01-23 19:09:48 UTC

Yelp review of local "Irish" pub:
"Love my potato-leek soup. Their homemade shepherd pie is delicious. My 11 year old son heard the people in booth said the F word ... it's hard for me to teach him about bad word but he heard it here at the restaurant"
https://media.mathstodon.xyz/media_attachments/files/115/945/984/116/498/958/original/864771ae04032e5c.png

2026-01-18 04:29:29 UTC

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I don't know. I never agreed with any decision Janeway made. Drove me crazy.

2026-01-04 01:12:34 UTC

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Ah, Humphreys, I used to own a physical copy of that book. I managed to read most of it which probably means I thought it was interesting. Can't remember much about it now though I recognised the paragraph right away. Maybe I got some intuition about Dynkin diagrams from it.

2026-01-03 17:42:40 UTC

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My wife caught me doing the capturing
https://media.mathstodon.xyz/media_attachments/files/115/832/398/868/556/456/original/8c9f5a8b99857a73.jpeg

2026-01-03 17:41:17 UTC

Took a break in an early morning run a little while ago to capture this.
https://media.mathstodon.xyz/media_attachments/files/115/832/389/333/825/238/original/c72fb749a6f60af9.jpeg

2025-12-30 20:06:08 UTC

Three beautiful things we saw in New York:
* Christina's World at the Met
* Grand Central Station
* New York Public Library, Jefferson Market Branch
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2025-12-11 06:11:13 UTC

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One good thing about my printer is that it's a laser printer and the toner cartridges seem to last an eternity.

2025-12-11 04:33:05 UTC

I printed something! That might not sound like much but judging by the documents the printer spat out, the last time I lost something in the print buffer was back in the 2010s. I did try to print something else, but like the dragons of Melniboné it's done its stuff and it's now back in "Deep Sleep", probably until some time in the 2030s.

2025-12-09 20:50:03 UTC

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This has been my conclusion about most detectives!

2025-12-09 16:10:19 UTC

Apparently the hardest part of writing detective fiction is contriving a way for a detective to be present at a murder scene. Which is why so much detective fiction has implausible coincidences to frame the story: detectives who just happen to be invited to the country manor when a murder happens, detectives that happen to have a relative in trouble, detectives who just happen to be visiting the village fair as someone is shot, and so on.

2025-12-01 23:20:11 UTC

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It sounds more like something you might get at a bar, I like that kind of shot.

2025-12-01 23:10:18 UTC

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Yes, it's very off-putting. But I think it's now the default term used by British media.
A lot of people seem to be phobic about very minor procedures so I think naming may actually sway people one way or the other.
I just discovered someone I know has never had an eye test because he was worried that his eyeball would be touched. You might get a puff of air for a glaucoma test, but that's about it.

2025-12-01 23:01:43 UTC

I think the British really need to stop using the word "jab" as a synonym for "(hypodermic) injection".

2025-11-27 22:38:20 UTC

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Entropy of mixing is confusing. If entropy is based on probabilities and probabilities are subjective, what does it mean when you think something has high entropy because it was mixed but then you learn that you can undo it?
(I'm sure there's a concrete answer here although I remember reading about physicists confused about MRI that can mix spin states and then undo them, confounding naive definitions of entropy.)

2025-11-27 21:31:22 UTC

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Also, while on the confectionery theme, when I was an undergrad our algebraic topology lecturer was Lickorish (who was one of those rare lecturers students look forward to) who gave his name to the Lickorish twist, a theorem about Dehn twists, and I wonder if what the the taffy machine is doing can be viewed as (some kind of slice through?) a Dehn twist.

2025-11-27 21:21:50 UTC

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😵‍💫
The next video youtube suggested was pretty good too. A different mechanism
https://www.youtube.com/watch?v=olnLaOA27L0

2025-11-27 18:03:38 UTC

Not having grown up in the US I'm not 100% sure what does and doesn't count as "taffy" but it's a great subject for a paper. Connects to the topic of rational tangles that I touched on at my ICFP talk years ago.
https://arxiv.org/abs/2511.19461
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2025-11-26 15:59:04 UTC

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My thesis was "borrowing" results from conformal field theory papers and dressing them up as pure mathematics about Riemann surfaces.
I always thought of Riemann surfaces as the meeting point of 3 disciplines but not the same 3 as you :) A Riemann surface is a certain type of field (in the algebraic sense) giving a purely algebraic definition. It's the same as the algebraic geometry definition really but shifts the emphasis.

2025-11-25 23:15:51 UTC

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"I was under the impression that one needed to specialize in algebraic geometry to know it well."
This was my problem as a student. I studied Riemann surfaces (not Riemannian manifolds). You can view Riemann surfaces as complex analysis things or as algebraic geometry things. Technically the same object. But they look very different. Sometimes constructions are better from one viewpoint or the other and I was limited mostly to the former.
Eg. the Riemann-Roch theorem is a classic algebraic geometry result that I learnt via an analytic argument.
Doesn't matter now - of course :)

2025-11-24 15:21:28 UTC

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I ought to know that material much better than I do because M_g is the home of stuff I studied as a grad student but sometimes you can get away with knowing the local structure without understanding how it all fits together.

2025-11-24 01:06:06 UTC

When I called the Yoneda lemma the hardest trivial thing in mathematics it must have resonated with a lot of people. My claim popped up in Notices of the AMS.
https://www.ams.org/journals/notices/202511/rnoti-p1248.pdf

2025-11-22 04:28:11 UTC

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I've mentioned this before but I really enjoyed it. All kinds of interesting optical systems including guanine in various optical systems.
https://academic.oup.com/book/10639

2025-11-20 22:18:03 UTC

The US is the laughing stock of the world
https://www.cdc.gov/vaccine-safety/about/autism.html

2025-11-08 05:17:36 UTC

I can't believe I didn't know this fact before today - it seems so fundamental.
If you have a real symmetric matrix A, and pick some unit vector x, then it's standard textbook material that x.Ax (the Rayleigh quotient) is bounded by the smallest and largest eigenvalues. So what happens if I pick a random unit vector, how is x.Ax distributed?
Let's jump to the case of a Hermitian matrix A and a random unit vector on the complex sphere. How is T=conj(x).A.x distributed? Again it's standard that it's bounded between the smallest and largest eigenvalue.
We can start with a 2x2 matrix A. First surprise to me: T is distributed *uniformly* between the two eigenvalues.
The first plot below is the histogram of values of T for 1,000,000 vectors x and a randomly chosen Hermitian matrix A. The yellow lines mark the two eigenvalues.
Now try 3x3. Remarkably the distribution is piecewise linear with the break points at the eigenvalues.
And let's jump to 5x5. We get the third plot, whose curve is given by a cubic B-spline with control points given by the eigenvalues.
Bizarrely you can prove you always get B-splines making this one of the few times I've seen B-splines in "nature", so to speak.
This is a purely mathematical result but I found it in this paper on wave transport through random media: https://arxiv.org/abs/2511.04602
My calculations (pretty much the same as the paper) are here: https://colab.research.google.com/drive/1g7ImBblYAsninp5-sr6uWQTVfTYurerE?usp=sharing
https://media.mathstodon.xyz/media_attachments/files/115/512/340/864/407/810/original/908f0decce0300a1.png
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2025-10-23 21:35:04 UTC

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https://media.mathstodon.xyz/media_attachments/files/115/425/627/261/958/506/original/3da19324bca26913.png

2025-10-23 20:31:43 UTC

Prepare to have your mind blown
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2025-10-06 20:11:03 UTC

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This is the issue throughout all of QM that bothers me - the use of classical looking fields for quantum particles. How can you get even a rough idea of the energy levels of an atom with more than one electron when the electron whose energy you're computing is embedded in clouds of all the other electrons?
There are arguments like https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation
More generally though I don't doubt the AB effect is real simply because AB is just probing one tiny part of a body of theory that works. But I can sympathise with someone worrying about this specific experiment.

2025-10-06 16:55:04 UTC

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But surely this isn't really an analogy, it's HoTT doing the work it was intended to do. The magnetic field in the AB effect comes from a cocycle in a cohomology which means it acts as a mapping from paths to some base ring or field and HoTT was built to work with those paths. You'll get good mileage out of this :)

2025-09-25 19:13:35 UTC

Woah!
https://www.quantamagazine.org/a-simple-way-to-measure-knots-has-come-unraveled-20250922/
https://media.mathstodon.xyz/media_attachments/files/115/266/522/229/846/247/original/bb233e5b876cada6.png

2025-09-25 03:31:10 UTC

A curious formula from a curious paper
https://arxiv.org/abs/2509.19324
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2025-09-22 16:34:33 UTC

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"Sounds very 60's-ish"
Is that code for "many drugs were consumed during the construction of this book"?

2025-09-12 03:27:35 UTC

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As I said years ago, for the last couple of centuries we've lived in a bubble of rationality, but it's eventually going to collapse in a flash of sonoluminescence.

2025-09-07 21:44:09 UTC

Debugging the irrigation system. Best explanation came from a book on capacitors. Complicated by the fact that there's a backflow valve that keeps injecting Heaviside steps into the system.
https://media.mathstodon.xyz/media_attachments/files/115/165/179/496/189/024/original/02386adaf963890b.png

2025-08-30 04:27:32 UTC

Long press everything because the UI isn’t going to give you any clue that long press does anything and if you don’t long press everything you’re never going to know.

2025-07-29 20:43:10 UTC

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Exactly. If a non-expert really wants to make these claims I'll maybe mention something about how mathematicians mostly use real numbers and in those 0.99999... = 1 but there are alternative number systems and maybe they can tell me why they're using the one they're using and how it works (or maybe not as I'll probably get bored of hearing how complicated it is to patch over all the flaws that keep popping up).

2025-07-29 20:36:23 UTC

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I'm a little more tolerant than I used to be about these things. I don't mind if someone wants to say 0.999... ≠ 1 but the onus is on them to answer the myriad of questions I have about their system that have already been settled in the system I use. And while I won't simply say they are wrong I'll happily say they are non-standard.

2025-07-29 17:41:16 UTC

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I find mathematical beliefs held by non-mathematicians to be fascinating. They're often held very strongly even when actual mathematicians argue otherwise. I'm not sure but I think in many cases these beliefs can be traced back to something as simple as a passing comment by a teacher at a young age.

2025-07-29 16:22:45 UTC

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What I mean is that on Earth I expect parabolae and the reason this explosion looks unusual to my eyes is that it's all straight lines.

2025-07-29 15:01:18 UTC

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Am I interpreting this correctly? We're seeing straight lines of debris rather than parabolae.

2025-07-26 15:56:46 UTC

My experience so far with LLMs when coding is that they don't write great code, and trying to get them to write working code is a massive time sink which I quickly learnt not to get caught in.
But their suggestions often use libraries, APIs and techniques that I was previously unaware of. They give good examples of use and they're also very good at showing idiomatic ways of doing things. So I think I myself will benefit a lot if I have the sense not to directly use the code they generate.
Generate and test and learn and understand, don't simply generate.

2025-07-26 15:39:13 UTC

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I couldn't get this to work. I wonder if it requires a genuine Mac keyboard. I'll try some more variations.

2025-07-26 05:27:06 UTC

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I found a command line tool to pair a Bluetooth device. Oh...it pops up a "terminal would like to use Bluetooth" window that requires a mouse click. Obviously nobody at Apple has ever considered the possibility of using a wireless mouse with a Mac. I hate computers.

2025-07-26 05:19:57 UTC

Am I the first person to ever use a Bluetooth mouse with a Mac? I searched the web and asked ChatGPT and nothing can tell me how to do it. So I'm asking here. (If your instructions require me to use a mouse I'll get grumpy.)

2025-07-16 17:44:44 UTC

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I have a hard time with American documentaries that use famous actors as hosts. It always feels like someone reading a script rather than telling us about this cool thing they want to share.

2025-06-04 21:03:37 UTC

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The "you have to insert a USB device twice to get it noticed" bug has persisted in some form for at least 20 years. Only happens with some devices though. Nearly 100% guaranteed to happen with my iPod nano, which I still use. I should just be glad it's still supported (just barely).

2025-06-04 20:53:55 UTC

If I remove my AirPods and put them back in the case then there is, I estimate, about a 1-5% chance the audio will continue (possibly loudly, sometimes embarrassingly so) from the iPhone itself a few seconds later. This bug has sat unfixed for about 10 years now. Sometimes I wonder what do they actually do at 1 Infinite Loop.

2025-06-03 21:25:38 UTC

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I'd like to add a data point.
Google X (where I worked for several years) was explicitly created to emulate Bell Labs. To some extent that may have been real, but it may also have played some more nefarious marketing purpose.
It doesn't sound as freewheeling as AT&T. Nobody would have said "I’ve always pursued my interests without much regard for final value or value to the world" - well not aloud anyway. But they did hire tinkerers and mad scientists and it was pretty open to employees suggesting new lines of inquiry without burdensome cost-benefit analyses. Sergey Brin was pretty adamant that it was his job to worry about expenses, not ours.
I don't think Google X is widely perceived as a success, and in more recent times my ex-colleagues there have been less happy, but I can't walk more than a minute in San Francisco without seeing a Waymo taxi, and more significantly "Attention is all You Need" is probably one of the most influential papers ever published and IMO will likely eventually have a major impact on the future of humanity. (Not necessarily for the better, of course.)

2025-05-27 04:41:58 UTC

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TL;DR the thing that makes LLMs work is a sort of cubic function that’s mixed up with a nonlinear function.

2025-05-27 04:35:04 UTC

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I must be losing my mind but the parentheses look balanced to me.

2025-05-25 03:59:45 UTC

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Getting it to play the episode after the one you just listened to can be a challenge. If I crash, it's because I was doing it in the car. If I'm late, it's because I spent the first 10 minutes of the journey trying to get to that episode.

2025-05-25 03:39:33 UTC

Was beginning to wonder if I'd ingested too much lead in my childhood and that's why I'm too stupid to work Apple's Podcast app, but it seems like even the LLMs will tell you it's one of the most hated Apple apps. Definitely created by some designer with a hatred of users.

2025-05-19 22:24:56 UTC

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Fussiness about LaTeX. That's another curse I was inflicted with!

2025-05-19 22:18:33 UTC

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Some people are cursed with hating all Google visual design and a decade working there never fixed me. I greatly respect what Google have under the hood though.

2025-05-19 21:59:27 UTC

Given a choice between garbage Android software and garbage iPhone software I'll pick iPhone every day. At least it doesn't hurt my eyes to look at it.

2025-05-12 03:31:10 UTC

I'm an actual imposter and I get a bit fed up of the people faking it.

2025-05-10 23:15:19 UTC

I've never hate-watched a TV show before but I do intend to get to the end of season 1 of Shrinking.

2025-04-30 21:35:57 UTC

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I think of Huygens' principle being about convolution where each point provides another translated/scaled copy of the function you're convolving with. You get the same classically but with inf-convolution instead. And of course you can use Legendre transforms to study these things just as you use Fourier transforms to study ordinary convolutions.

2025-04-30 20:03:16 UTC

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There's a long exercise I did years ago that goes something like this:
You usually think of classical mechanics as the classical limit of quantum mechanics but this isn't quite true in the form classical mechanics is written.
In QM the field is exp(i∫L(x,∂x)Dx) where the integral is a path integral over all paths that satisfy some constraints at the start end end point of the dynamics you're considering. If you hold the start point constant this field is a function of the end point and is the thing we usually solve for in the Schrodinger equation.
You can also write classical (point particle) mechanics in terms of fields, but this time the field is given by the inf of L(x,∂x) over all paths. You can extract actual motions from these fields or you can study these "fields" in their own right. Usually we discard the actual value of L. You can mimic much of the theory of QM in this classical setting.
If you're like me you have better intuition about Fourier transforms than Legendre transforms but this alternative classical frameworks allows you to use knowledge about Fourier transforms to reason using Legendre transforms. Partly it comes down to looking closely at what the boundary conditions are, ie. the constraints you impose at start and end of the trajectory - eg. you have to work with a different field if your boundary specifies momentum rather than position and there are mixtures like known momentum at the start and known position at the end. It's all very analogous to switching from position to momentum in QM by taking the Fourier transform.
I know of one paper that does something like this. It's in an optics journal and could take a long time to find again...

2025-04-30 19:39:52 UTC

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But even in that case what I say applies - the T term is being written in an unconventional way that gives the illusion of a contrary minus sign. This adjointness corresponds precisely to the sign flip you get when you do Euler-Lagrange for fields.

2025-04-30 18:12:26 UTC

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Part of the story comes from the fact that the derivative operation is anti-self dual in the sense that the adjoint ∂ᵗ of ∂ is -∂. (This is the true meaning of integration by parts :)
If you're minimising a quadratic function of a vector it's standard to write it as a quadratic form xᵗQx. Eg. if we're minimising (Mx)² we can convert it to standard form as x·MᵗMx. When we write the Lagrangian, which is really a quadratic form, in the usual way we write Lagrangians, we're not writing it in the usual way for quadratic forms.
We when we write T = ½(∂x)·(∂x) we've flipped the sign from the standard form T = -½x·∂ᵗ∂x. This means we write Lagrangians less symmetrically.
This may be more of a mathematical answer and what you want is more physical.

2025-04-16 19:16:05 UTC

Before you plug in your new Timex/Sinclair 1000, please take a moment to think about this exciting new adventure. We want to assure you that:
1. You will enjoy computing.
2. You will find it easy as well as enjoyable.
3. You shouldn’t be afraid of the computer. You are smarter than it is. So is your parakeet, for that matter.
4. You will make mistakes as you learn. The computer will not laugh at you.
5. Your mistakes will not do any harm to the computer. You can’t break it by pushing the “wrong” button.
6. You are about to take a giant step into the future. Everyone will soon be using computers in every part of their daily lives, and you will have a head start.
-- Steve Vickers, Timex/Sinclair 1000 Instruction Manual (aka the US ZX81 manual)
https://media.mathstodon.xyz/media_attachments/files/114/349/237/048/109/572/original/afe692984d9c5c53.png

2025-04-14 17:21:13 UTC

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I think this stackexchange question can be read as someone wondering about precisely your "quote", though it takes a little work to make a good choice of B.
https://math.stackexchange.com/questions/3333241/motivation-for-defining-a-tangent-vector-as-an-equivalence-class-of-curves

2025-04-14 16:45:53 UTC

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If I had to guess I'd say that someone probably tried formalizing vector fields as equivalence classes of paths through a point and then realised that the equivalence was basically A1 is equivalent to A2 if A1 and A2 have the same B at which point you can discard the A1 and A2 and just keep the B even if the A1 and A2 provide better mental imagery.

2025-04-14 13:51:35 UTC

Who originally came up with the idea that a vector field is a type of derivation? I remember being amazed by this idea when I learnt it as an undergrad. Probably the first technical construction that I was impressed with as a technical construction.
https://en.wikipedia.org/wiki/Derivation_(differential_algebra)

2025-04-09 20:12:30 UTC

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Yeah, I was thinking you could do this by finding a family of small tree surgeries that allow you to turn any tree into another.

2025-04-09 20:02:23 UTC

Woah!
https://arxiv.org/abs/2504.05337
(Is this connected to Leinster's magnitude?)
https://media.mathstodon.xyz/media_attachments/files/114/309/783/082/401/972/original/dfb8ae0f4c9f660c.png

2025-04-09 16:31:13 UTC

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Sorry that was unclear. I just wanted to mention the final result without giving the full argument, but at least leave some breadcrumbs for someone else to follow if they did want to derive the full argument.

2025-04-08 16:34:52 UTC

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Yeah, everyone knows we have ℘: ℂ/Λ → ℂ, but I like the fact that we can derive the differential equation it must satisfy directly from the group law and the definition of exp. Anyway, that was the missing piece I kept putting off :)

2025-04-08 15:14:59 UTC

The group law for an a elliptic curve over ℂ makes it a differentiable manifold with a differentiable group structure so it's a Lie group with a Lie algebra and an exponential map. This exponential map is in fact the Weierstrass ℘-function and the corresponding "logarithm" is the Abel-Jacobi map. For some reason no book I've read on elliptic curves mentions this fact.
I've been meaning to do this calculation for over 30 years and I kept putting it off - but I finally got around to deriving the differential equation for the exponential map and was actually kinda excited to see it's precisely the equation satisfied by ℘. (IIRC as a student I studied both subjects at the same time in different courses.)
I don't know why this isn't in all the text books. You can literally just read off the Abel-Jacobi map which is now well motivated rather than being this thing you can concoct and whose properties you discover after the fact.
(Maybe at some point I'll write up in more detail what's going on here.)
https://media.mathstodon.xyz/media_attachments/files/114/302/986/586/462/149/original/5bcdab22a19b11dc.png

2025-04-05 22:05:50 UTC

I really need to write my own PDF viewer. I miss being able to see slightly through paper pages to the next page and it's easily implemented.

2025-04-03 16:14:37 UTC

The name of this article caught my eye - but actually the content looks really interesting.
"Infinite-order combinatorial Transverse Intersection Algebra TIA via the probabilistic wiggling model"
(You may remember co-author Ruth Lawrence - she scored a 1st class degree from Oxford at age 13.)
https://arxiv.org/abs/2504.01021

2025-03-20 22:58:06 UTC

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I left Twitter and sold my TSLA holdings because I wanted to reduce the value of ksuM's assets. He bought the US so that's the only way I can think of to stop him. And I get the impression he's highly leveraged, and TSLA is overvalued, so people doing this probably make a difference.

2025-03-19 20:37:12 UTC

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There's more than a lack of preparedness - there's anti-preparedness. By and large Americans from any part of the political spectrum are brainwashed from birth and are incapable of having thoughts contradictory to the constitution. So even though it's trivially obvious that there is no magic to separation of powers, and anyone with eyes could see the Republicans have been stacking the Supreme Court, it has still taken people by surprise. Except that's not true - most Americans aren't even aware enough, or care enough, to be surprised. And among the ones who do care, many are cheering it along.

2025-03-18 23:33:29 UTC

There is a bias in the orientation of rotation of galaxies in the JWST Advanced Deep Extragalactic Survey
https://academic.oup.com/mnras/article/538/1/76/8019798
Reminds me of the old Cosmic Axis of Evil
https://en.wikipedia.org/wiki/Axis_of_evil_(cosmology)
and the alignment of cows.
https://media.mathstodon.xyz/media_attachments/files/114/186/038/885/485/114/original/ddcb032cdd6ee198.jpeg

2025-03-18 17:05:11 UTC

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I was just reading about Jean Leray who was a prisoner of war under the Nazis. I originally thought he was in a concentration camp but he was actually in a camp with sufficient freedom to be able to organise a "university". He knew hydrodynamics, which had military applications, so he lied and said he was a topologist. So he started bluffing as a topologist, and eventually made major contributions to the subject like inventing sheaves and spectral sequences!
https://mathshistory.st-andrews.ac.uk/Biographies/Leray/

2025-03-17 15:00:24 UTC

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Plenty of more rigorous work does eventually emerge, though I’m no longer up to date. Eg. the work of Witten, Borcherds and Connes. Seems to me that there are lots of opportunities for mathematicians here to pick through looking for theorems. That’s what I did for my PhD.

2025-03-17 02:21:55 UTC

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It's an elitist conceit that mathematics always has to be completely rigorous. I would add that by this standard there is very little mathematics in this world and certainly almost none before a few decades ago.