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Happy New Year! A bunch of us are trying to develop "applied category theory" - and now we've started a new journal on this subject, called "Compositionality". It's free to publish in, and free to read. The first issue just came out! (1/n)https://johncarlosbaez.wordpress.com/2019/12/30/compositionality-first-issue/ …
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John Carlos Baez Retweeted
‘If They Hadn’t Tortured My Husband, He Wouldn’t Have Fallen Into a Coma’ Torture and death in China's concentration camps in East Turkestanhttps://www.rfa.org/english/women/coma-01302020130439.html …
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There was a typo in my first tweet: I meant to say SUPERFLUID QUASICRYSTAL! It's impossible to have a periodic crystal with 8-fold symmetry... so they got a quasicrystal. Superfluid crystals are older: they're also called "supersolids".https://phys.org/news/2017-03-mattersupersolid-crystalline-superfluid.html …
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Superfluid quasicrystals were first created in 2018. The holy grail, a "Bose glass", remains to be seen. It's a Bose-Einstein condensate that's also a glass. New forms of matter with strange properties - I love 'em!
(5/n, n = 5)https://phys.org/news/2018-04-solid-physicists-state.html …Show this threadThanks. Twitter will use this to make your timeline better. UndoUndo -
But when a Bose-Einstein condensate localizes, all the atoms get trapped in the *same place* - because they're all in exactly the same state! This was discovered at the University of Cambridge very recently: (4/n)https://twitter.com/UlrichCamPhy/status/1222833640801034240 …
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It's well-known that when the potential wells in a crystal get strong enough, its electrons "localize": instead of having spread-out wavefunctions, they get trapped in specific locations as shown here. This is called "Anderson localization". (3/n)pic.twitter.com/9krK11ymsm
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As you increase the intensity of the lasers, the Bose-Einstein condensate (in blue) suddenly collapses from a quasicrystal to a "localized" state where all the atoms sit in the same place! Here the gray curve is the potential formed by the lasers. (2/n)pic.twitter.com/Rw0f8DYNr0
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Condensed matter physics is so cool! Bounce 4 laser beams off mirrors to make an interference pattern with 8-fold symmetry. Put a Bose-Einstein condensate of potassium atoms into this "optical lattice" and you get a SUPERFLUID CRYSTAL! But that's not all... (1/n)pic.twitter.com/ba9AYC0Hxa
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Computer science and category theory blend in the theory of "optics": different systems for viewing and interacting with data. As part of the ACT2019 school, Emily Pillmore and Mario Román have written a great overview of these: https://golem.ph.utexas.edu/category/2020/01/profunctor_optics_the_categori.html … Lenses, prisms & more!pic.twitter.com/sLcko6gGur
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Mahowald and M. Mori showed that πₙ(S⁵) has more than one element for all n > 5. But that's where this business stops! If k > 5, πₙ of the k-sphere has just one element when n=k+4. I don't understand these computations: I merely admire them. (4/n, n = 4)
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It instantly follows that πₙ(S³) has more than one element except for n=0,1,2. (Oh, duh - πₙ(S²) has just one element when n = 0. I ignored that case in my first tweet.) Curtis showed earlier that πₙ(S⁴) has more than one element for all n > 3. (3/n)
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It's a hard calculation, and the hardest case is when n = 1 mod 8: https://arxiv.org/abs/1506.00952 Sergei O. Ivanov, Roman Mikhailov and Jie Wu wrote this in 2015 - and then realized that Brayton Gray could have known this result back in 1984. (2/n)
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One great thing about math is that you can still learn new things about the sphere - yes, the good old 2-sphere, S². There's more than one way to wrap an n-sphere around the S² except for n=1! More precisely: (1/n)pic.twitter.com/6BSDzf5E7z
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As you keep zooming in, you think: it's a point! No, it's a line! No, it's a rectangle! No, it's a bunch of points! No... It's a subset of the plane invented by Simon Willerton, which shows that your estimate of the dimension can keep changing as you zoom in more and more.pic.twitter.com/Qn8LhnToNK
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John Carlos Baez Retweeted
China detains Hui Muslim poet
@ismaelan over his online criticisms of mass atrocities in East Turkestan (Xinjiang)https://www.rfa.org/english/news/china/poet-01272020163336.html …Thanks. Twitter will use this to make your timeline better. UndoUndo -
You can turn a bundle into a presheaf, but you get a very nice kind of presheaf called a sheaf. You can turn a presheaf into a bundle, but you get a very nice kind of bundle called an etale space. Sheaves and etale spaces are secretly the same!https://johncarlosbaez.wordpress.com/2020/01/07/topos-theory-part-2/ …
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This is from "Excitations in strict 2-group higher gauge models of topological phases" by Bullivan and Delcamp. Physicists are starting to use 2-groups - categorified groups - to create mathematical models of new topological phases of matter. https://arxiv.org/abs/1909.07937 pic.twitter.com/4jeHTamOjU
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After this speech, all reasonable Republicans realize Trump needs to be removed.https://twitter.com/RepAdamSchiff/status/1220559375938609152 …
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So, only in the far future will the Universe cool down enough for large black holes to start slowly decaying via Hawking radiation. Entropy will continue to increase... going mainly into photons and gravitons! Do interesting things now - don't wait. (11/n, n = 11)
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Hawking predicted that black holes slowly radiate away their mass when they're in a cold enough environment. But the Universe is much too hot for supermassive black holes to be losing mass now. Instead, they *grow* slightly by eating the cosmic microwave background! (10/n)
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