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Nice! Thanks for this.

More details at: https://arxiv.org/abs/1008.4872

Their explanation for why their idea (SSD) might work - precision-exploration conflict hypothesis - is something adaptive decoding also tries to solve.

https://ai.meta.com/research/publications/adaptive-decoding-...

Thank you very much for the link.

This letter from Maxwell to Faraday: https://epsilon.ac.uk/view/faraday/letters/Faraday3354 is very surprising to me. Already they are thinking of gravity in field theoretic terms. Just to set context: Riemann's Habilitationsschrift was just 3 years earlier but not to be published till late 1860s.

Indeed. In fact, it is one of the most amusing aspect of the anglophone west (at least for the last few decades). Despite public perception (by public I mean those who have been to university since the 90s), Western historians of science and mathematics in general have never not acknowledged the previous works of the Persianate civilizations commensurate to their knowledge of them in their time. But somehow in the last few decades professional historians have had to waste time figuratively looking over their shoulders lest they be percieved as being Eurocentric. And, if they were to somehow find a way to show -- requiring whatever hermeneutical gymnastics -- that a prominent scientist was influenced (or even better, had stolen) from some other "cultures" than nothing better! (ex: Copernicus from the Maragha school as an example of interpretive gymnastics)

But, of course, this is one of the symptoms of the degeneration that now afflicts your particular civilization and is bringing about it's inevitable transformation to something else -- but better this than the fate of the Abassids or the Sung.

On the contrary, what is presented by the OP is one of the many reasons that worship of science's heroes, unfashionable for decades, a whiggish pablum, is justified. If great results were birthed fully-formed -- a view I've frankly never heard anyone profess who has bothered to consider such things even briefly -- they would hardly be any heroes. Even little children who reflexively chomp on every superhero film aeroplaned towards their face understand this.

If I'm being honest I really don't have a good recommendation for a text in intro abstract algebra. I learned it from Michael Artin's Algebra. Artin is a true master -- along with David Mumford, he was the main apostle for Grothendieck style AG in the US -- but his book was not very easy to learn from.

There is indeed a deep connection between what is going on behind Arnold's proof and the classical Galois theory. But it needs quite a bit of sophistication to flesh out properly (not apparent in his famous lectures given to high school kids). There is a Galois theory for Riemann surfaces over algebraic functions where the coverings behave like fields do in the classical correspondence. If any one is interested, check out chapter 3 of Khovanskii's Galois Theory, Coverings and Riemann Surfaces.

Let me take this opportunity to post one of the best texts on Galois Theory I have read -- and I had to go through quite a few while preparing for a class.

https://pages.uoregon.edu/koch/Galois.pdf

The subject is developed very naturally and every idea is beautifully motivated. It begins with a quick one chapter intro of Arnold's proof of Abel-Ruffini.

Richard Koch's home page (https://pages.uoregon.edu/koch/) has other examples of his fantastic pedagogy.

You know how in every zombie movie there's a bit where our intrepid protagonists must blend in to avoid capture. I think this is that sort of thing.

Atiyah is truly one of the giants of modern mathematics. I remember long ago I struggled through a reading course of his and Bott's Yang-Mills paper in graduate school. Like many great works of math it too had that paradoxical characteristic of transforming seemingly 'non-mathematics' into mathematics* by reversing the usual direction of application of one to the other, in this case, from physics to math. It would start a whole movement that'll produce much of modern geometries greatest hits like Donaldson's (his student) theorem in 4 manifolds to Witten's great papers.

* A reason I think modern LLM architecture as they currently stand with their underlying attention mechanisms will not produce interesting new mathematics. A few other ideas are going to be needed.

And why should I simply assume that "Education Economists"* really know the subject they purport to talk about? Because they are credentialed members of university departments with some label? Because a few of them won some Bank of Sweden award?

Just because a particular department or field of study exists in academia does not magically give them the imprimatur you think it does.

* Btw, I know for a fact that a few of them are not "education economists"

Claude is still better in my opinion.

There's a suite of code-related tasks -- covering a diversity of areas, including dev ops, media manipulation etc., derived from issues I have faced over the years -- I perform for every new release. No model has solved the set of issues solved in one go but Claude still remains the best.

An example of the sort of problems in the suite:

I have a special problematically encoded mp4 file with a subtle issue (something I ran into a couple of years ago while fixing a bug in a computer vision pipeline). In the question prompt I also pass the output of ffprobe and ask for the ffmpeg command that'll fix it. Only Claude has figured the real underlying issue out (after 4 interactions).

Thank you so much for this! I have read a fair amount of Lem but somehow completely missed this one.

Here is Frank Calegari's excellent talk on this work:

https://www.youtube.com/watch?v=znBdPEyDScY

Re: that dig at 4-manifolds

Are you aware of the book on [The Disc Embedding Theorem](https://academic.oup.com/book/43693) based on 12 lectures Freedman gave roughly a decade ago.

I think you've simply redefined genius. Many years ago I read an article on youth football, if I remember correctly, and in it there was a bit about the writers visit to the Ajax Youth Academy. In it he writes of a moment during practice when a plane flies over and all the 7(?) year olds on the pitch look up to see it except for one kid who keeps his eyes on the ball. That kid (of course) grows up to be a very good midfielder for Real (I'm forgetting the exact details, I think its Wesley Sniejder?). My point is: whatever that motive energy is that manifests as the single minded pre-occupation with math at an age when everybody else's attention is all over the place is that inherent thing that people call genius. I have read many of Thurston's non-mathematical writings about himself and in it this sort of singular pre-occupation is also clear -- which is why he developed his preternatural geometric vision.

Maybe I am misinterpreting but this is a beautifully crafted bit of wry humour. :D

You must interact with more interesting people than I because to me LLMs have demonstrated as much "common sense, insight, creativity, self awareness, humor" as the average person I run into (actually maybe more but that makes me sound crazy to myself).

As a first pass, check out the pop history book called Fearless Symmetry by Ash and Gross.

Next, and perhaps I shouldn't be suggesting it, Serre's A Course in Arithmetic. Despite it's reputation of terseness it is a great book by one of history's great mathematicians and worth every sweat and tear spent over its short number of pages.

Yet another way is to see the lectures by Richard Borcherds (who won the fields for the moonshine conjecture) on youtube.

Finally, since this hacker news check out William Stein's Modular Forms: A Computational Approach (pdf online)

This guy's other notes are also well thought through and written. Thanks for the link.

The guy being interviewed (Daniel Litt) has a great expository video on Hilbert's third problem at Numberphile. (Here)[https://www.youtube.com/watch?v=eYfpSAxGakI]

Hilbert's Third was the first of his 23 problems to be solved. It asks whether volume is enough to determine whether one can transform one polyhedron into another via cutting into finite pieces and re-arranging with rotation and translation (such polyhedrons are called scissor congruent). The situation is true in 2 dimensions.

Dehn showed that's not the case for polyhedron. He created his eponymous invariant in order to solve the problem: two polyhedrons with the same volume but differing dehn invariants are not scissor congruent.

I love the result because it was one of the first bits of serious math I read as an undergrad and it was also the first time I got a glimpse into how tensor products could be used to package information.

A pretty clear account can be found in Harthshorne's Geometry: Euclid and Beyond.

Thanks for the reference. I am indeed aware of The Renaissance Mathematicus and I do occasionally read the blog. Unfortunately I find his posts are often lighter on mathematical detail. Maybe the situation is different with his writing on Kepler.

Surprised to see this on HN. He had lived a long full life.

This brings to mind the depressing fact that Serre too will pass away one of these days. And we will ever see that kind of mathematician ever again.

This is the first part of a two part series on Kepler's work. It's a wonderful work of pedagogy striking the pretty difficult balance between mathematical detail and pop science story telling.

The videos go into a surprising amount of detail; the second part has a lot of the key details of Kepler's New Astronomy -- something I haven't even found in standard textbook treatments (including the first part of the great Celestial Mechanics monograph by Shlomo Sternberg).

In the fact, this person's channel is truly a gem and very underrated. I find a lot of his videos match or even surpass other such channels like 3b1b or vertasium.

This is not correct since you can claim the above for any number of votes actually obtained (if you asked someone to pick a number from 1 to 10 million, any number the person picks (assuming iid picks) will be by definition 10^-7).

The problem is more subtle.

There are around 10,000 integers n such that n/10058774 when rounded to 3 decimal places gives 0.512. Of those 10,000 this particular one has the smallest rounding error. That's what gives one the sense that probably they started with the clean fraction 0.512 and then worked their way to the tally.

The only solution here is mandatory re-education and/or limiting who can vote, or hopefully waiting for the oldest and most stubborn conservatives to die out so alphas and gen-z can vote with a little more heart and brain.

Goodness.

Thank you. It's a bit surprising to see Hegel here, I was thinking more on the lines of the analytic philosophy. Of course many early 20th c. mathematicians who were interested in philosophy such as Weyl or Gian-Carlo Rota would not have thought much of such a distinction.

Well, if concepts turn out to be simplicial (or cellular) complexes maybe philosophy can be made into applied algebraic topology.

You scots are the most extreme combination of a people who have made vast parts of the modern world and people who know nothing about the achievement of their own people. I spent a year doing my postdoc in your country and have to say you guys are my favourite lot on this otherwise wretched rock.

For example, we can compare variance in song energy and danceability for U2, ABBA, and Katy Perry, then rank these artists against each other. According to this subset of variables, ABBA's discography offers greater sonic variety than U2's or Katy Perry's.

Stopped reading here