@hallasurvivor@sunny.garden

Christina Grossack @hallasurvivor@sunny.garden

· 5mo ago

In case you haven't seen it, here's a SUPER cool video where someone builds puzzles based on sporadic groups. In particular, the M12 puzzle looks VERY clean!

I'm sure lots of people would find this interesting, but I think that @henryseg@mathstodon.xyz in particular would be REALLY interested!

There's one based on M24 as well, which I'll leave as a surprise for those who watch the video 👀

Turning Sporadic Groups into Puzzles

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

I have a friend visiting me from out of town and he gave me this fun puzzle which COMPLETELY broke my brain!

The quaternions are a "skew-field" in the sense that they satisfy all the axioms of a field, except they're not commutative.

In the quaternions, we have a crazy fact:
x²+1 has uncountably many roots!

Puzzle 1: Can you find uncountably many roots of x²+1 in the quaternions?

Puzzle 2: This means that in the proof of "a polynomial of degree n has at most n roots" we must have used commutativity somewhere... Can you find where? (I couldn't!)

I'll post solutions in a few days, but as always I'm curious to see how people approach this! Please reply with your thoughts, and boost if you think other people might be interested ^_^

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Christina Grossack @hallasurvivor@sunny.garden

· 5mo ago

Just learned that the theory of generating functions can be seen as harmonic analysis on Z?? Then the usual way we use generating functions to solve recurrence relations is a special case of Fourier transforms turning translation into multiplication, which turns the recurrence relation into an algebraic equation!

If you haven't seen this before, it's really fun to work out for yourself ^_^.

Whether or not you've seen this before, post your thoughts here! If you already knew this, I'd love to know where you learned it, and if there's any other applications of this observation! And if this is new to you I'd love to see your thoughts as you try to prove this yourself!

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

It's kind of funny, I remember learning algebraic geometry and wondering why people would care about the derived pushforward R*f. I don't think I ever got an answer to that specific question -- I just learned so many reasons to care about derived functors broadly that now R*f seems inevitable... BUT ☝️ I finally have a 𝑟𝑒𝑎𝑙𝑙𝑦 good reason that I think would have satisfied younger me!

If you're interested in sheaves then you're interested in their global sections, as well as the cohomology groups of your sheaf.
The problem is that computing cohomology is hard!

Say that f : X → Y is a map which we'll think of as taking a "complicated" space X to a "simple" space Y. Then for a sheaf ℱ you might hope to recover the cohomology H*(X,ℱ) in terms of the cohomology of the pushforward fℱ on the simpler space Y...

The 𝐋𝐞𝐫𝐚𝐲 𝐒𝐩𝐞𝐜𝐭𝐫𝐚𝐥 𝐒𝐞𝐪𝐮𝐞𝐧𝐜𝐞 tells you this dream is a reality! ~If~ you keep track of the higher pushforwards too!

Indeed, one can relate the the cohomology groups Hⁱ⁺ʲ(X,ℱ) to the cohomology with coefficients in the higher pushfowards Hⁱ(Y, Rʲfℱ)!

So as soon as you find yourself interested in computing sheaf cohomology, you ℎ𝑎𝑣𝑒 to be interested in derived pushforward in order to gain access to this powerful computational tool!

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

Just said "the situation is worse in the absence of inequality" in a meeting, lol.

I was talking about the theory of an object vs the theory of a 𝑑𝑒𝑐𝑖𝑑𝑎𝑏𝑙𝑒 object, but I had to chuckle at how that sounds out of context, haha

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

Replying to

@johncarlosbaez@mathstodon.xyz I don't agree that working with a proof assistant will reduce the chance that we'll come up with radical new ideas. It's not at all difficult for me to picture someone like Grothendieck, who also broke out of many existing formalisms, writing his own library from scratch in order to express his ideas -- In many ways this is exactly what he did! Though of course he (and his collaborators) wrote a long series of books rather than writing a long list of agda/lean/etc files. In fact, it's quite easy for me to picture someone like Grothendieck writing their own theorem prover! Perhaps in that world EGA/SGA would look much more like the currently-under-development synthetic algebraic geometry, formalized in a proof assistant that's custom made for arguments in the (big) zariski or etale topos. @andrejbauer@mathstodon.xyz @dougmerritt@mathstodon.xyz @MartinEscardo@mathstodon.xyz @JacquesC2@types.pl @pigworker@types.pl @xenaproject@mathstodon.xyz

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Christina Grossack @hallasurvivor@sunny.garden

· 5mo ago

Hmmm... Wondering if I should try and learn some condensed matter physics, lol

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Christina Grossack @hallasurvivor@sunny.garden

· 28mo ago

It's finally done!

Have a 3 part series on the topological topos

First, let's talk about some fundamentals. How do we think of objects in the topos? How do we compute with them? How do they relate to objects in "the real world"?
https://grossack.site/2024/07/03/life-in-johnstones-topological-topos

Then, since post 1 is loooong, a little palate cleanser. It's a short post about how we can use the topological topos to more effectively study algebraic structures equipped with a topology.
https://grossack.site/2024/07/03/topological-topos-2-algebras

Lastly, another long one. Let's talk seriously about the internal logic of the topological topos. What's true and why? And what does this truth mean in the real world? This one isn't for the faint of heart!
https://grossack.site/2024/07/03/topological-topos-3-bonus-axioms

---

Thanks for all the encouragement, everyone! This was easily the most effort I've ever put into... maybe any math project I've ever done, haha. But I think it turned out really well ^_^.

If you have comments, I would love to hear what you think!

Life in Johnstone's Topological Topos 1 -- Fundamentals

grossack.site

Life in Johnstone's Topological Topos 1 -- Fundamentals

Chris Grossack's math blog and professional website.

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

Replying to

@MartinEscardo@mathstodon.xyz Sorry, let me adopt your convention to make my reply more clear -- but you can tell I was replying to John since it's his post that show up above mine when you click my post (at least on my instance) @johncarlosbaez@mathstodon.xyz @andrejbauer@mathstodon.xyz @dougmerritt@mathstodon.xyz @JacquesC2@types.pl @pigworker@types.pl @xenaproject@mathstodon.xyz

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

Replying to

@francisbrb This is what I had in mind! I think the more conceptual phrasing is that R[x] is the free R-algebra on one *commutative* generator, and so the "evaluation at a" map R[x] --> R is only a ring hom when a is central. The reason for this is essentially your argument about multiplying terms.

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

Replying to

@ncf Thanks for the puzzle and the solution!! That's REALLY clever, and I don't think I've seen the trick before. Perfectly sized to think about over a late breakfast 💖

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

Replying to

@highergeometer @MartinEscardo I would also like to read the paper (actually both papers) if they're available

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Christina Grossack @hallasurvivor@sunny.garden

· 7mo ago

Replying to

@cbaberle I'm glad I finally found the time to read this! I really enjoyed it ^_^. Thanks for sharing!

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Christina Grossack @hallasurvivor@sunny.garden

· 28mo ago

Replying to

@leemph It looks like I was combining two different mathoverflow posts in my mind. First, the comment from Peter Scholze on his answer here, saying that in his world "topology" is either anima, topoi, or condensed sets: https://mathoverflow.net/q/441610/145247 Then an answer here by Maxime Ramzi that says something similar: https://mathoverflow.net/q/459101/145247 As for what role plays the concept of "topos", remember that there's a very close connection between topoi and locales. Indeed, there's a fully faithful 2-embedding from locales to topoi, and this embedding admits a left adjoint ("localic reflection").

mathoverflow.net

Properties of pyknotic sets

In Peter Johnstone's 1979 paper On a topological topos, he proposed the topos of sheaves on the full subcategory of topological spaces spanned by the single object $\mathbb{N}_\infty$, the one-point

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Christina Grossack @hallasurvivor@sunny.garden

· 28mo ago

Replying to

@johncarlosbaez Anyways, it's funny that I made this mistake, since in a lot of ways the topological topos solves the same problems as condensed sets, so I should probably have profinite topologies closer to the front of my mind when thinking about examples!

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Christina Grossack @hallasurvivor@sunny.garden

· 7mo ago

Replying to

@francisbrb @highergeometer I agree with this! Ryan O'Donnell used to do this for some of his papers (see here: https://www.youtube.com/@RyanODonnellTeaching/videos) and I loved watching them! Also, if you recorded some videos explaining the basics of higher geometry I would ABSOLUTELY watch them (what are gerbes? what problem do they solve? how do you compute with them? etc. of course, for any topic you like -- gerbes are just an example).

youtube.com

Ryan O'Donnell

Computer Science and Mathematics lectures by Ryan O'Donnell // Online research talks, plus full course lecture videos for: 1. CS Theory Toolkit (grad course, assumes basic undergrad CS Theory) 2. Quantum Computer Programming in 100 Easy Lessons (undergrad & below course, assumes basic programming and probability) 3. Quantum Computing and Information (grad course, assumes basic undergrad linear algebra and CS Theory) 4. Undergraduate Complexity Theory (corresponds to Part III of Sipser'

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Christina Grossack @hallasurvivor@sunny.garden

· 6mo ago

Replying to

@trebor this is exactly the issue! Nice ^_^

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Christina Grossack @hallasurvivor@sunny.garden

· 28mo ago

Replying to

@johncarlosbaez As I understand, it's literally a synonym for a homotopy type. It's supposed to be something like the "soul" or "true essence" of a topological space (at least if you're a homotopy theorist) which explains the fun name. I'm a fan of whimsy in math where I can find it, so maybe I should switch over to saying "animus". If I ever teach a class on this stuff I'll have to wear like wizard robes and a pointy hat when I introduce it :P

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Christina Grossack @hallasurvivor@sunny.garden

· 5mo ago

Replying to

@MaloTarpin yeah, that's exactly what I'm thinking of! I've never seen it by this name! What does the Z-transform get used for in the signal processing world?

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Christina Grossack @hallasurvivor@sunny.garden

· 5mo ago

Replying to

@MaloTarpin Thanks! Super interesting ^_^

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Christina Grossack @hallasurvivor@sunny.garden

· 5mo ago

Replying to

@MaloTarpin@mathstodon.xyz The cold kind, haha. What I'm really interested in thinking about is topological quantum computing, but if I'm going to do that I'd like to know at least a little bit about how these might actually be built in the real world

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