@johncarlosbaez

I do math, physics, and applied category theory. Currently at the @ToposInstitute.

John Carlos Baez
@johncarlosbaez

In physics, "symplectic geometry" describes how position and momentum are related. But we can also use it to describe how entropy and temperature are related - or volume and pressure! Here I explain how this works, as a warmup for some newer ideas: https://t.co/xsNEkSbxi4

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John Carlos Baez
@johncarlosbaez

Some mathematicians like to joke around. This paper is not about the aerodynamics of flying saucers. It's about the space of all ways to put an infinitesimal disk in a 3-dimensional manifold. This space has a lot of interesting geometrical structure! https://t.co/e7TzI8pwl5

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John Carlos Baez
@johncarlosbaez

His work was not completely unrecognized: in 1976 he got the Max Planck medal. But physics would have advanced faster if more people had paid attention to him! It's a good lesson: don't just pay attention to the bigshots in your field. Look for good ideas. (6/n, n = 6)

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John Carlos Baez
@johncarlosbaez

In 1938 he came up with a primitive version of the Higgs mechanism. He also discovered conservation of baryon number - an important conservation law. In 1941 he proposed that positrons are electrons traveling backward in time... an idea Feynman also had, later. (4/n)

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John Carlos Baez
@johncarlosbaez

Here are some things he did: In 1934 he devised a fully Lorentz-invariant perturbation theory for quantum fields. (A big deal.) In 1935 he developed a theory of mesons carrying the nuclear force. (Yukawa won the Nobel for this in 1949.) (3/n)

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John Carlos Baez
@johncarlosbaez

His full name was Baron Ernst Carl Gerlach Stueckelberg von Breidenbach zu Breidenstein und Melsbach. Born in 1905, he was a master of quantum field theory, the first to do many things... but he published in minor journals, so few recognized his greatness until the 1990s. (2/n)

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John Carlos Baez
@johncarlosbaez

Some physicists win Nobels, while others just have great dogs. Who was Stueckelberg, and why did Feynman think he deserved a Nobel? What did he do that was so great? (1/n)

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John Carlos Baez
@johncarlosbaez

Four lessons from Weinberg. It's good to read the whole one-page article. The idea of seeking out "rough water" resonates with me. In science, by the time things are nice and neat, most of the really interesting work has already been done. https://t.co/ilDhDQoKId

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John Carlos Baez
@johncarlosbaez

Once I was trying to find the right adjoint of a functor L. I found a functor R and constructed a natural-looking isomorphism hom(La,b) ≅ hom(a,Rb) So I thought R was the right adjoint! But it turned out the isomorphism wasn't really natural. So R was the 𝘄𝗿𝗼𝗻𝗴 adjoint.

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John Carlos Baez
@johncarlosbaez

Can someone confirm or disconfirm this news? Before I posted my tweet about this based on Lawrence Wright's tweet, I checked Wikipedia. It listed Weinberg as having died on July 23rd: https://t.co/7kTZToKXFI Now it does not.

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John Carlos Baez
@johncarlosbaez

For beginners, I recommend Weinberg's book 𝘛𝘩𝘦 𝘍𝘪𝘳𝘴𝘵 𝘛𝘩𝘳𝘦𝘦 𝘔𝘪𝘯𝘶𝘵𝘦𝘴. For those who know particle physics, I recommend his 1979 Nobel lecture: https://t.co/s4g2zAjRuD It's striking how little more we know of fundamental physics now than then! (4/n, n = 4)

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John Carlos Baez
@johncarlosbaez

While curious, the Glashow-Salam-Weinberg theory predicted the W, Z and Higgs bosons with the properties we now see - along with all other details of the electromagnetic and weak forces. It is a magnificent success, opening the door to deeper mysteries. (3/n)

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John Carlos Baez
@johncarlosbaez

His "unification" of the electromagnetic and weak nuclear forces is curious, since it starts with two other kinds of force and uses another particle, now called the Higgs boson, to split them a different way, into the forces we now see... while giving particles mass! (2/n)

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John Carlos Baez
@johncarlosbaez

Steven Weinberg died! For all the talk of unification, there are few examples. Newton unified terrestrial and celestial gravity - apples and planets. Maxwell unified electricity and magnetism. Weinberg, Glashow and Salam unified electromagnetism and the weak force. (1/n)

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John Carlos Baez
@johncarlosbaez

Does anyone have access to 𝘎𝘦𝘰𝘮𝘦𝘵𝘳𝘺, 𝘗𝘩𝘺𝘴𝘪𝘤𝘴, 𝘢𝘯𝘥 𝘚𝘺𝘴𝘵𝘦𝘮𝘴 by Robert Hermann? It seems to explain how statistical mechanics is related to contact geometry, but I can't find it anywhere! There's nothing more enticing than an inaccessible book.

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John Carlos Baez
@johncarlosbaez

I really like how she cites earlier, related work - including stuff that I put on the nLab but never published. One reason is that everyone really likes being cited. But the other is that some people don't cite blog articles, the nLab, etc. That's gotta end! (5/n, n = 5)

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John Carlos Baez
@johncarlosbaez

If you know operads already, you'll say GREAT. If you don't, you may say SO WHAT? Tai-Danae shows that entropy arises naturally from a concept in operad theory - applied to the probability operad. So she gives a new outlook on the meaning of entropy. (4/n)

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John Carlos Baez
@johncarlosbaez

For example, suppose I say "there's a 40% chance I'll eat out, and if I do there's a 30% chance I'll order a salad and 70% chance I'll order a sandwich." What's the chance I eat out and order a salad? It's an easy calculation. That's the probability operad! (3/n)

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John Carlos Baez
@johncarlosbaez

The n-ary operations in the "probability operad" are just probability distributions on an n-element set. The cool part is how you plug one probability distribution into another. It's something we often do in real life! Tai-Danae explains the idea here. (3/n)

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John Carlos Baez
@johncarlosbaez

An operad has a bunch of "n-ary operations" for n = 0,1,2,3,.... You can plug the output of n-ary operation into any of the inputs of an m-ary operation and get an (n+m-1)-ary operation. The idea becomes clear in this picture shamelessly lifted from Tai-Danae. (2/n)

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John Carlos Baez
@johncarlosbaez

Tai-Danae Bradley has a new paper that explains how entropy arises naturally from what I'll call the "probability operad". And she has a blog article that sketches some of the background, with pictures. Since I love this stuff, I can't resist explaining a bit of it. (1/n)

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John Carlos Baez
@johncarlosbaez

I'm talking to them and @Joe_DoesMath about open systems in thermodynamics. You can read more about their projects at the Topos Institute blog. It's a lively place! (Left to right: me, @c0b1w2, Owen Lynch, Brendan Fong, Sophie Libkind.) (2/n, n = 2)

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John Carlos Baez
@johncarlosbaez

Summer research associates here at @ToposInstitute: Sophie Libkind, David Jaz Myers, Nelson Niu & Owen Lynch. Sophie and Owen are developing software in AlgebraicJulia. Nelson is writing a book on polynomial functors. David is working on generalized lenses. (1/n)

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John Carlos Baez
@johncarlosbaez

The connection between thermodynamics and 1-forms turns out to be very important! For a bit more of an explanation, see the question here, and the first answer. (2/n, n = 2) https://t.co/122u6VEV73

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John Carlos Baez
@johncarlosbaez

A 1-form is something you integrate along a path: for example the "f(x) dx" you see inside an integral is a 1-form. Heat is not a function: it's a 1-form. This confused everyone for a while, and it still confuses plenty of people now. (1/n)

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John Carlos Baez
@johncarlosbaez

Air is mainly made of diatomic gases: nitrogen and oxygen. So the pressure of an insulated cylinder of air is inversely proportional to its volume to the 7/5... just like a monatomic gas in 5 dimensions! It's fun to see how weird numbers come from basic physics. (5/n, n = 5)

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John Carlos Baez
@johncarlosbaez

We don't need to go to other dimensions to see other rules for the expansion of an insulated cylinder of gas! For a diatomic gas like oxygen, each molecule has 5 ways to wiggle so its energy is (5/2)kT. Its pressure is inversely proportional to its volume to the 7/5. (4/n)

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John Carlos Baez
@johncarlosbaez

The pressure of helium drops in funny way when you make it expand while preventing heat from flowing in. Pressure is inversely proportional to volume to the 5/3 power. Weird number! If space were 4-dimensional this would be 6/4, and in 5d it would be 7/5. (1/n)

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John Carlos Baez
@johncarlosbaez

So, Fisher's fundamental theorem of natural selection can be saved using Fisher information! I wonder what he'd think of this. For more details, you can read my paper or my short series of blog articles, starting here: (7/n, n = 7) https://t.co/d7LKKjG1EM

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John Carlos Baez
@johncarlosbaez

Under quite general conditions, I prove the variance of fitness equals the speed of at which this probability distribution changes! This speed can be understood as 'the rate at which information is updated'. (6/n)

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John Carlos Baez
@johncarlosbaez

There's a probability distribution saying what fraction of self-replicating entities there are of each type. We can measure the speed at which this distribution changes using the 'Fisher metric' on the space of probability distributions. It looks round, like this. (5/n)

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John Carlos Baez
@johncarlosbaez

It turns out there's a much better theorem. Variance in fitness may not cause progress in the sense of increased mean fitness, but it does cause change! And this change can be quantified using 'Fisher information' - a concept from information theory that Fisher invented! (4/n)

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John Carlos Baez
@johncarlosbaez

But that's false, even in simplified models of natural - except in special conditions. So a big argument broke out. Finally people figured out what Fisher must have really meant: a true theorem, but much less interesting. I explain it. (3/n)

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John Carlos Baez
@johncarlosbaez

My new paper fixes "Fisher's fundamental theorem of natural selection" - a result with a long and tortured history - using information theory. Briefly: the rate at which information in a population is updated equals the variance in fitness! (1/n) https://t.co/3MHfPGiIWH

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John Carlos Baez
@johncarlosbaez

Check out what Valeria and Brendan have done so far: https://t.co/jV1K1WD8Gs You can help annotate abstracts of papers to improve their ontology of mathematics! You can leave suggestions and questions on their blog. Or maybe you can even join the project! (3/n, n = 3)

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