@nilesjohnson@mathstodon.xyz
@nilesjohnson@mathstodon.xyz avatar

nilesjohnson

@nilesjohnson@mathstodon.xyz

Working in topology and category theory;
Assoc. Prof. at Ohio State, Newark

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nilesjohnson, to random
@nilesjohnson@mathstodon.xyz avatar

Apple crush commercial, but it's smashing cyber trucks, crypto mining rigs, llm servers, the boston dynamics robots, army drones, etc. Has anyone made this version yet?

nilesjohnson, to random
@nilesjohnson@mathstodon.xyz avatar

Do people have advice for how to read a math paper? I've read a few things around the web [1,2,3,4], and here's what I see so far:

  • Most advice emphasizes or at least suggests nonlinear reading (focus on main statements, skip certain details, etc.).

  • Some advice distinguishes between different reading goals (reading to know a result, reading for broad context, etc.).

  • There are some differences for people at different stages in their careers (students, faculty with their own students, etc.).

Before I spout my own opinions, I'm curious to hear what other folks use, or suggest, or have heard suggested to them. Feel free to check out some or none of these references!

[1] https://math.stackexchange.com/questions/13460/how-to-read-a-mathematical-paper
[2] https://eprints.maths.manchester.ac.uk/2484/1/reading.pdf
[3] https://www-users.york.ac.uk/~ar521/papers/roux2003b.pdf
[4] https://www.ams.org/journals/notices/202005/rnoti-p660.pdf

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

p.s. although I've wondered and thought about this off and on for years, this particular post is indeed related to the discussions @johncarlosbaez has been stirring up around writing math. I'm just sure the two practices are related ;)

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@johncarlosbaez ha ha, yeah, I've already seen good advice about the writing ;) It got me thinking about the connection between reading and writing.

ColinTheMathmo, to random
@ColinTheMathmo@mathstodon.xyz avatar

So, my discussion charting 'bot has been reported for the first time. No reason given, and I don't know who it was.

Most charts get multiple "Like"s, so clearly some people like it. Should we ignore the report, change the behaviour to close access, or should I kill the 'bot?

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@ColinTheMathmo I like the bot, please keep it!

Edent, (edited ) to random
@Edent@mastodon.social avatar

I need to store the hash of a file.

For various boring reasons, I only have 128 bits of space. This cannot be changed.

Is it better to use MD5, or truncate the output of SHA-256 / 512 ?

Please argue in the replies.

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@Edent This writeup by @retr0id might be what you're thinking of; it definitely stuck in my mind!

https://www.da.vidbuchanan.co.uk/blog/colliding-secure-hashes.html

Note that he does produce colliding hashes, but part of the process uses a birthday attack, so it's a search for a pair colliding hashes, not one that collides with a given hash.

The pair of hashes in this picture is the collision he finally finds. There are a lot more interesting details in the writeup; I really enjoyed it!

nilesjohnson, to random
@nilesjohnson@mathstodon.xyz avatar

Why are introductions so hard to write? I was thinking about this today, and I have three ideas!

  1. Maybe I don't really know (yet) what I'm introducing.
    (This is the easiest case: write the rest first!)

  2. Maybe I know too much about what I'm introducing, so I have too much to say. (This happens after I've done step 1.)

  3. Maybe I've forgotten what it's like to not know the content I'm introducing. (This is similar to 2, but can be more or less pronounced, depending on what the content is.)

In case 2, I think it's useful to distinguish between whether I need to compress material (summarize, glossing over details) or select material (state highlights). And, usually a combination of those two is needed, but it's useful to recognize the difference. Any other suggestions?!

In case 3, sometimes it helps to think of a single specific person, and write just to them. This is related to the more general advice of identifying the audience when writing, but it's weird how changing from "I'm writing for grad students" to "I'm writing to this specific student I know" can really change things! Any other suggestions for case 3?!

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@johncarlosbaez oh that's an excellent point about the first sentence!

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@MartinEscardo ha ha, actually I went to check some of my recent papers and I also have a mix :/

@johncarlosbaez

nilesjohnson, to random
@nilesjohnson@mathstodon.xyz avatar

Well, 2024 really is happening isn't it. Happy new year everyone :)

thosgood, to random
@thosgood@mathstodon.xyz avatar

we've been able to run TikZ entirely the browser for 5 years, and somehow I have never come across this before despite all the time I've spent looking around for exactly this???

https://tikzjax-demo.glitch.me

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@christianp @thosgood haha, I thought "didn't @kisonecat work on something like that?" and, yeah!

0xabad1dea, to random

of all the sites I’d expect to have a silly time-gated Halloween skin, CTAN was not high on the list…

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@0xabad1dea lol, and yet, somehow it makes perfect sense

futurebird, to random
@futurebird@sauropods.win avatar

This is a subtle math education question.
Consider this sequence of algebra steps:
a+b=c given
a=d given
d+b=c substitute

This is substitution. And feels natural to me. Replace a with d. But, I have many students who write:

a+b=c given
a=d given
b+d=c substitute

Still correct but I don’t understand why they swapped the order. What are they thinking? If it was just one or two students I’d call it “the noise of young minds” but it’s six of them. Why do they write it like this?

nilesjohnson, (edited )
@nilesjohnson@mathstodon.xyz avatar

@futurebird Since it's with all kinds of symbols, I wonder if it's something like the "temporal" or "process" order suggested here. Like, they're thinking that they have b, and add d to it, so that's the order they write. I wonder how one could test that guess.

Edit: maybe see what happens in examples like
a - b = c
x - a + b = c
x - (a+b) - y = c

@briankung

ZachWeinersmith, to random
@ZachWeinersmith@mastodon.social avatar

Hey mathstodon, help me understand something. What does Hilbert mean when he argues that to prove the existence of a mathematical concept, you only have to prove that if it existed you arrive at no contradiction.

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@ZachWeinersmith I'm a mathematician, although not an expert in the philosophy or history side. The way I'd understand Hilbert's meaning is that it's a definition of "exist". Your two comments "exists in a sense [that is similar to the plain English meaning]" and "doesn't not exist [as in, doesn't create a contradiction]", would be justifications for his choice of terminology.

(I don't know if what he proposed is a good definition of existence, or even whether that's how he would have understood his statement. But I don't think I'm in any way unusual in reading his statement that way nowadays.)

system76, (edited ) to random
@system76@fosstodon.org avatar

This shortcut is fun to know if someone walks away from their desk... watching them figure out how to turn it off while the computer is yelling at them is...priceless

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@system76 Two things: (a) I didn't know about this, and I tried to get a screen reader working before but failed. So, I'm glad to know about it! (b) The joke context here is not great. As I read it, the punchline is either "This accessibility tool is funny!" (yells at you, not something that is actually useful); or it's "Coworkers won't know about this!" (because, implicitly, someone who relies on a screen reader wouldn't be your coworker).

I'm sure you can think of a better way to let people know about accessibility features.

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@system76 Oh, I realized there's no alt text for the image (lol, of course). Here's a suggestion:

Pop!_OS tip
Toggle Screen reader on/off
Super + Alt + S

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@system76 thanks! You know, mastodon also has an edit feature for post content... up to you.

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@system76 what I mean is, you could just replace the "joke" with something different, if you want. Especially now that people who use screen readers will be able to tell which keyboard shortcut you're referring to...

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

Who drew this picture for me? - without the purple and yellow. Now I want to use it, and I forget who to credit!

Recently Quanta came out with an article explaining modular forms:

https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/

It does a heroically good job. One big thing it doesn't do is explain the funny looking 'fundamental domains' in the upper half-plane.

By sheer coincidence, I just wrote a little article explaining the concept of 'moduli space' through an example which does touch on these fundamental domains:

https://johncarlosbaez.wordpress.com/2023/09/23/the-moduli-space-of-acute-triangles/

It's due October 1st so I'd really appreciate it if you folks could take a look and see if it's clear enough. It's really short, and it's written for people who know more math than your typical Quanta reader, but not necessarily anything about moduli spaces.

The cool part is the connection between the moduli space of acute triangles — that is, the space of all shapes an acute triangle can have — and the more famous moduli space of elliptic curves!

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@johncarlosbaez This looks like a lovely topic, and I haven't seen it before. Since you're asking for feedback, I got stuck on this sentence: "If we reflect a labeled triangle corresponding to a point in a yellow region we get a triangle corresponding to a point in a purple region, and vice versa." I can't tell what you mean by "reflect", even after reading the next two sentences.

"Points on the boundary between two regions correspond to isosceles triangles. All 6 regions meet at the point that corresponds to an equilateral triangle."

The immediate context here is about the S₃ action on triangle vertices. So, do you mean transposition of vertices, or reflection across a triangle edge (followed by repositioning that I can't do in my head), or something else? I vaguely recall that the yellow/purple regions are reflections across the indicated circles; is that right, and is that what you mean?

As a related but separate point, I think those two sentences about isosceles triangles and the equilateral triangle could be helpful before you talk about the S₃ action. They could go at the end of the previous paragraph, where you're explaining how how to read the picture.

Lastly, and I know you'll need another volunteer for this, a version of this picture with a few example triangles drawn on top would probably make what you're trying to say much clearer. A second version, with some of those parallelograms you mention, could also be useful, if someone is inspired to make it!

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@johncarlosbaez In the sentence "The colored part of the picture...", that second instance of GL(2,ℤ) should be SL(2,ℤ) ! @BartoszMilewski

ColinTheMathmo, (edited ) to random
@ColinTheMathmo@mathstodon.xyz avatar

A question for teachers, but I'd welcome thoughts from others.

In teaching, should one talk about a quadratic equation such as x^2+10x+25=0 as having one solution or a repeated solution, and do you think it matters?

If you're at all interested, or think some of your followers might be, then I'd be delighted if you would boost this for reach ... thank you.

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@ColinTheMathmo As coincidence would have it, I was just talking with a student about this exact issue yesterday. I think both perspectives are important! We were doing curve sketching in calc 1, and one of the observations was that x=-5 being a "repeated root" of f can be expressed technically by saying that it's a root of both f and f'. So, I think it's a nice example of how learning new mathematics can help one express precisely a phenomenon that isn't so clearly expressible if you haven't yet learned it.

nilesjohnson, (edited )
@nilesjohnson@mathstodon.xyz avatar

@ColinTheMathmo Oh, I wrote a bit below, but I hope I can be more clear: I think it's important to acknowledge that x=-5 isn't "two roots". And if you ask for the set of all roots of f, that set has one element. [EDIT: So, I read "one repeated root" to mean "one root, which is repeated", and that's the kind of answer I'd want to give.]

I asked my student this way: If you have the number -5, twice, how many numbers do you have? (We want to say "two", but really, it's just one number; f has just one root.) I think it's important to acknowledge the tension, because what I think is really cool (mentioned in my other post https://mathstodon.xyz/@nilesjohnson/111115213652844335) is that this is a case where some more advanced mathematics gives the language with which to express the idea precisely.

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@ColinTheMathmo Yeah! The comment from @feonixrift (https://x0r.be/@feonixrift/111114278098338424) is, I think, similar to mine in that "true for two reasons" can be expressed later, with some additional theory. I guess these points were probably both made in the earlier discussion.

@faelif

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

Another example to consider along these lines is how to describe the root of g(x) = cos(x) - 1 at x=0. There's no hope of factoring this, so I think the multiset description of roots doesn't work here either (but I'm happy to be corrected).

Students still know that there's something "repeated" about this root; but what is it? Is there any non-calculus way of expressing that? One could think of g(x,t) = cos(x) - t, for the perturbation approach, but maybe people know of others? I bet any such approach turns into a nice example of developing more theory in order to make precise statements!

CC: @boarders @ColinTheMathmo @faelif

nilesjohnson,
@nilesjohnson@mathstodon.xyz avatar

@boarders @ColinTheMathmo this is nicely put, but I'm curious, genuinely, what part of this you'd try to say in the specific context of the OP: teaching students about roots of quadratic polynomials. Having basic definitions be "subject to further revision" is something that can really frustrate young students, and I think that's an interesting aspect of the original question: how can one prepare students for that, knowing that we can't tell them everything at once?

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