Want to learn applied category theory? You can now read my lectures here:
https://math.ucr.edu/home/baez/act_course/
There's a lot of 'em, but each one is bite-sized, and basically covers just one idea. They're self-contained, but you can also read them along with Fong and Spivak's free book, to get two outlooks on the same material.
Huge thanks to @be22abe6 for making these into nice web pages! But they still need work. If you see problems, please let me know.
I need to add more of my "Puzzles" to these lectures. Students in the original course also wrote up answers to all of these puzzles, and to many of Fong and Spivak's exercises. But it would take quite a bit of work to put all those into webpage form, so I can't promise to do that. 😢
https://media.mathstodon.xyz/media_attachments/files/111/136/575/238/572/251/original/aeb1c4a238bbdd04.jpg
@0c1a9324 - it's indeed obfuscated. I went to the website and chose a combo flu / COVID booster shot on the basis of preexisting conditions (type II diabetes), but my appointments reminders only mentioned the flu vaccine and only when getting the actual shots was it clear that I'd be able to get a COVID booster. The doc there was easily persuaded that I needed one.
@5fb5d869 - it does, thanks! I was overlooking the difference between SL and PSL.
I'm going to call 2n "N".
Now that I think about it, there's an "obvious" element of order N in SL(2,F) where F is the rationals with a primitive Nth root of unity adjoined, say
ζ = exp(2πi/N)
Namely, it's
ζ 0
0 1/ζ
People who think about rotations a lot know this is conjugate to the real matrix
cos θ -sin θ
sin θ cos θ
where θ = 2π/N. But then the less obvious part is that's it's also conjugate to
0 1
-1 2cos θ
I think there's something I still don't understand well enough, some very classical stuff about the relation between the field F and ℚ[cos θ].
@5fb5d869 - Neat! Among other things, you're finding an element of order n in SL(2,F) where the field F is the rationals with cos(π/n) adjoined. I guess this is a subfield of the 2nth cyclotomic field: the rationals with a primitive 2nth root of unity adjoined. I'm a bit confused about why we need the factor of 2 here, but the nth cyclotomic field would only have cos(2π/n) in it.
@913f1b00 - excellent! You might like my own explanation of the Weierstrass elliptic function, which is an attempt to cut through the nonsense and get to the point:
https://math.ucr.edu/home/baez/week13.html
Here is the first of my fall This Week's Finds lectures! In it, I explain the idea of categorification: taking math that uses sets, and replacing these sets by categories. Then I explain what happens when you categorify the idea of 'category'.
The quality of the video starts out being pretty bad - my poor pal Tom Leinster had to learn how to use a new camera on the fly: the old one broke due to a 'firmware update'. But apparently adjustments in the lighting made things a bit better after a while. Also someone found some white chalk instead of yellow chalk.
The talk didn't get as far as I wanted - among other things, someone started drilling in the room above, and Neil Turok had to run upstairs to tell them to knock it off.
Sigh.
https://www.youtube.com/watch?v=ZVecriTCBLU
@a547f8fb wrote: " I wonder how we can distinguish the black and yellow regions in your figure, since they are combined into the fundamental region of SL(2, Z)."
If you read my article you'll see the answer to this explained in detail:
https://johncarlosbaez.wordpress.com/2023/09/23/the-moduli-space-of-acute-triangles/
In brief the answer is GL(2,Z), a group which is 'twice as big' as SL(2,Z).
The history of the Earth is long. You always knew there had to be boring stretches here and there. But still it's funny that there's a billion-year time period called the Boring Billion.
Imagine you're a paleontologist and someone asks you what you're working on. "The Boring Billion". 🥱
But it's actually interesting that from 1.8 billion to 0.8 billion years ago the oceans may have been black- and milky-turquoise - full of purple bacteria that photosynthesize but produce sulfur instead of oxygen.
And maybe this is when modern cells - eukaryotes - first arose, and developed into multicellular organisms, and diversified into plants, animals, and fungi! That would be the very opposite of boring. But we have no direct fossil record of this, so we can only theorize.
Before the Boring Billion came the Oxygen Catastrophe, when oxygen created by photosynthesis increased to the point of threatening all existing life. That could be why eukaryotes developed. The atmosphere used to have a lot of methane in it - a powerful greenhouse gas - but the oxygen reacted with that, plunging the world into massive ice ages.
But eventually these subsided, and the Boring Billion began.
They ended with an even more extreme bout of ice ages: the Cryogenian. Scientists are still arguing about whether the whole earth froze over ("Snowball Earth"), or whether there were some liquid oceans left near the equator ("Slushball Earth").
So, as with human history, it seems paleontology naturally focuses on crises and tend to skims over the "boring" periods when good things are slowly developing.
For more, see:
https://en.wikipedia.org/wiki/Great_Oxidation_Eventhttps://en.wikipedia.org/wiki/Boring_Billionhttps://en.wikipedia.org/wiki/Cryogenianhttps://media.mathstodon.xyz/media_attachments/files/111/079/763/917/031/119/original/6bc3a5dcfb7812a6.jpg
@1b46a604 - Here's a sort of digressive question. If the Earth warms 2 or 2.5 °C above pre-industrial levels what is the *equilibrium* sea level rise that we expect? I'm reading some very different numbers. Is there any educated conventional wisdom about this?
Starting on September 21st I'll be giving a series of talks on Thursdays at 3 pm UK time in Room 6206 of the James Clerk Maxwell Building at the University of Edinburgh. Here's the topic of the first couple talks:
Categorification and the periodic table of n-categories
Categorification is a not-completely-systematic process of taking known math and replacing sets by categories, functions by functors, and equations by natural isomorphisms. Often categorifying simple results in math leads to deeper, more interesting results. When we iterate categorification we're pushed into higher categories. Higher categories exhibit striking patterns visible in the "periodic table" of n-categories, such as the Stabilization Hypothesis, Tangle Hypothesis and Cobordism Hypothesis. I'll concentrate on sketching the basic ideas, since the evidence is much easier to explain than the rigorous proofs. This will be an elementary introduction to higher categories.
Then I'll talk for a while about how combinatorics is categorified ring theory... and then other stuff.
The first talk is Thursday September 21st and the last on Thursday November 30th. I’ll skip October 19th and 27th… and any days there are strikes.
We'll make the talks hybrid on Zoom so you can join online:
https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: XXXXXX36
Here the X’s stand for the name of that famous lemma in category theory.
We'll also record them and make them available here: https://www.youtube.com/channel/UCy9yxdnj0yfOiMWvMa47PNw
If you join the Category Theory Community Server, you can discuss the talks here:
https://categorytheory.zulipchat.com/#narrow/stream/229141-general.3A-events/topic/This.20Week's.20Finds.20seminar
And I'll put notes here: https://math.ucr.edu/home/baez/twf/
This stone circle in the Outer Hebrides has he most metal name possible: Cnoc Filibhir Bheag. It's smaller than another stone circle nearby, but maybe that's why its name is better.
I imagine some long-bearded Druid, feeling a bit outclassed, slamming his staff on the ground and saying
"I hereby dub this stone circle CNOC FILIBHIR BHEAG!!!"
and everyone saying "Whoa, okay, that's cool".
https://media.mathstodon.xyz/media_attachments/files/111/052/471/911/428/269/original/78db3a385c8776c9.jpg
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Notes by John Carlos Baez | export
The history of the Earth is long. You always knew there had to be boring stretches here and there. But still it's funny that there's a billion-year time period called the Boring Billion. Imagine you're a paleontologist and someone asks you what you're working on. "The Boring Billion". 🥱 But it's actually interesting that from 1.8 billion to 0.8 billion years ago the oceans may have been black- and milky-turquoise - full of purple bacteria that photosynthesize but produce sulfur instead of oxygen. And maybe this is when modern cells - eukaryotes - first arose, and developed into multicellular organisms, and diversified into plants, animals, and fungi! That would be the very opposite of boring. But we have no direct fossil record of this, so we can only theorize. Before the Boring Billion came the Oxygen Catastrophe, when oxygen created by photosynthesis increased to the point of threatening all existing life. That could be why eukaryotes developed. The atmosphere used to have a lot of methane in it - a powerful greenhouse gas - but the oxygen reacted with that, plunging the world into massive ice ages. But eventually these subsided, and the Boring Billion began. They ended with an even more extreme bout of ice ages: the Cryogenian. Scientists are still arguing about whether the whole earth froze over ("Snowball Earth"), or whether there were some liquid oceans left near the equator ("Slushball Earth"). So, as with human history, it seems paleontology naturally focuses on crises and tend to skims over the "boring" periods when good things are slowly developing. For more, see: https://en.wikipedia.org/wiki/Great_Oxidation_Event https://en.wikipedia.org/wiki/Boring_Billion https://en.wikipedia.org/wiki/Cryogenian https://media.mathstodon.xyz/media_attachments/files/111/079/763/917/031/119/original/6bc3a5dcfb7812a6.jpgStarting on September 21st I'll be giving a series of talks on Thursdays at 3 pm UK time in Room 6206 of the James Clerk Maxwell Building at the University of Edinburgh. Here's the topic of the first couple talks: Categorification and the periodic table of n-categories Categorification is a not-completely-systematic process of taking known math and replacing sets by categories, functions by functors, and equations by natural isomorphisms. Often categorifying simple results in math leads to deeper, more interesting results. When we iterate categorification we're pushed into higher categories. Higher categories exhibit striking patterns visible in the "periodic table" of n-categories, such as the Stabilization Hypothesis, Tangle Hypothesis and Cobordism Hypothesis. I'll concentrate on sketching the basic ideas, since the evidence is much easier to explain than the rigorous proofs. This will be an elementary introduction to higher categories. Then I'll talk for a while about how combinatorics is categorified ring theory... and then other stuff. The first talk is Thursday September 21st and the last on Thursday November 30th. I’ll skip October 19th and 27th… and any days there are strikes. We'll make the talks hybrid on Zoom so you can join online: https://ed-ac-uk.zoom.us/j/82270325098 Meeting ID: 822 7032 5098 Passcode: XXXXXX36 Here the X’s stand for the name of that famous lemma in category theory. We'll also record them and make them available here: https://www.youtube.com/channel/UCy9yxdnj0yfOiMWvMa47PNw If you join the Category Theory Community Server, you can discuss the talks here: https://categorytheory.zulipchat.com/#narrow/stream/229141-general.3A-events/topic/This.20Week's.20Finds.20seminar And I'll put notes here: https://math.ucr.edu/home/baez/twf/