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Some of the math in quantum physics kinda makes me angry to be honest. Like. Ok sure, linear operators don't always commute. Fine. That's completely okay. Sometimes a*b isn't the same as b*a. And yeah, sometimes you want to describe how they don't commute with the commutator [a,b] = a*b -b*a. Fine.

But the commutator having physical fucking meaning?? What the fuck??

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The programming/math thing of not getting a thing for hours or even days, then getting it and going "oh I'm dumb "

And immediately after having to explain to your friends you don't actually believe you're dumb and it's not self loathing it's just part of the process

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i like that "without loss of generality" aka wlog just gets written as Πin german

its a ligature of OE short for Ohne Einschränkung (without restriction) and i just think its a neat symbol<3

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Discuss: 0 is prime in any integral domain.

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Of course (0) is a prime ideal in any integral domain, since R/(0) is isomorphic to R. However, if you take 0 to be a prime element in a domain, it would also be irreducible (at least morally, since without counting 0 every prime element in a domain is irreducible). However, this provides problems when looking at factorization in any ring with more than two elements since 0 = 0*a = 0*b etc.

Conclusion: 0 is prime in Z/2Z and not prime otherwise qed.

Discuss: 0 is prime in any integral domain.

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Motivating example: Ok so latex complaints. When you type not equals and the empty set, the lines aren't parallel.

Lemma: This is fixable.

Proof: A silly little application of $\neq$ \rotatebox[origin=c]{22}{$\varnothing$}

Remark: There are other options. If you dare.

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The moving sofa problem has finally been solved!

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Sharing of Proof Between Friends

After spending over nearly eight hours of each day in a mathematics department for two years straight, I’m shocked that many high schoolers believe math is a solitary pursuit.

In reality, this community seems to be one of most welcoming and collaborative academic communities I’ve found. Let me share some moments:

“Hey! How’s it going?”

“Alright, been stuck on this interesting question my friend emailed me the other day… he said the first part of the proof is pretty easy, but I’ve been at it for 10 hours…”

“Never trust a mathematician who throws around the word ‘easy,’ c’mon let’s try it together at the board.”

Thirty minutes later, they had completed the proof, and sat back with wide smiles to admire their work. In truth, there was rarely a conversation that didn’t eventually turn to math in that department…

“Yeah, and I heard that the guy cheated with his best friend’s sister… wild right?”

“Yeah… not to interject, but I have this representation theory question… would you all be willing to take a look?”

The conversation took an immediate turn with collective enthusiasm. I have been lucky to have my own “collaborative math” moments since returning to my undergraduate studies, and do my best to share this part of “math culture” with younger students curious about the major.

“So that’s the proof that motivates our paper! It’s quite short, but there’s something about it I love.”

“Wait… but you’ve only done half, and this is a biconditional statement, let’s try it together.”

There was a reason my mentor never encouraged me to look at the other side of the proof… it was far more “ugly,” but tons of fun to piece together with a fiend. We looked back at our work after forty minutes with satisfaction before returning to our neglected problem sets…

And finally, I tried to assist a student with a calculus question using the “process of questioning” the research world had taught me:

“I need to find a closed-form equation for this geometric series… but I can’t seem to get the alteration sign?”

“Try writing out the first six terms, do you see anything that you could simplify? Look at the denominator specifically…”

“Well, they’re all multiples of three…”

“Try pulling that three out, any more similarities?”

“The numbers multiplying the threes are powers of two! But I still need that alternating sign…”

“Remind me, what happens when you raise a negative number to an odd/even power? Try it with (-1)^n”

“If it’s odd, the number stays negative, and positive if even… so if I add this to the denominator, the sign is alternating depending on the index n!”

“YES! This little (-1)^n trick comes up everywhere, it’s a nice ‘tool’ to hold on to if you decide to take more math.”

The exchange was wonderful… and motivated me to review the calculus I’d excitedly ran past when I was younger. I wish this type of discourse was taught more expansively.

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I'm on the nlab!

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Yesterday, someone showed me this picture... and yeah, I feel it.

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One of these is not like the others, and yet... it kinda fits in

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Top ten fandoms where fanon RUINED canon for me:

Catholicism

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