Niche/abstract topics rarely get highly illustrated explanations, much less in the form of a bedtime story.

A former prof of mine, Ravi Vakil, made this gem capturing how he thinks about exact sequences, cohomology, etc., which he kindly let me repost.

🧵
You can find the full story here: 3blue1brown.com/blog/exact-seq…

He goes on to talk about exact sequences, cohomology, and more. Admittedly, the audience is relatively niche, since it assumes you already know about those topics, but for the right person it certainly scratches an itch!

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More from @3blue1brown

23 Oct
The announcement for SoME1 winners is now out!

Links to winners in the thread below (in no particular order)

That weird light at the bottom of a mug — ENVELOPES

The Beauty of Bézier Curves, by @FreyaHolmer

Read 7 tweets
23 Apr
James Schloss (@LeiosOS) and I decided to organize a casual discord server for what we're calling the first Summer of Math Exposition (SoME1).

It's a space for people interested in producing online math/physics/cs explanations to self-organized.

discord.gg/Q3cC2GXN2f
There are so many people who have some great idea for a video or blog post or interactive experience that might help others learn a tricky topic, but who aren't sure where to start.
Going by the philosophy that the best way to start is to just start, this is a place to trade tips and ask questions of others in the same boat, and potentially find collaborators for what you're working on.
Read 4 tweets
13 Dec 20
It's a very good question! What follows are many tweets attempting to answer in a possibly way-too-verbose manner. No pictures (sorry), but I'll trust in the readers' mind's eye.
Commentary from mathematicians is more than welcome at the bottom of the thread, which includes scattered thoughts on whether there's a purely geometric reason to expect any rotation to have complex eigenvalues, i.e. one that doesn't appeal to the fundamental theorem of algebra.
First, just answering the question, let's review the basics: What it means for an operator to have an eigenvector with eigenvalue λ is that the way this operator acts on that vector is to simply scale it by λ.
Read 54 tweets
19 Jul 20
Okay, follow through with me on this one and I guarantee you're in for a fun surprise.

We'll start forming rows of numbers, kind of like Pascal's triangle, but with a different rule.

The first row is simple "1, 1"
For row 2, copy the first row, but insert a 2 between any elements that add to 2.

In this case that just means sticking a 2 between the two 1's.
For row 3, copy row 2, but insert a 3 between every pair of elements adding to 3.
Read 19 tweets
7 Jul 20
In light of the very sad news about Ron Graham's passing, I thought I'd share an interesting tidbit about his famous constant which I only learned recently (thread).
This number has captured the imaginations of many people with its unfathomable size, myself included. I distinctly remember when I was a kid first learning about it spending hours trying to wrap my mind around it and using it as inspiration for writing ever-larger numbers.
The existence of huge numbers is not in and of itself interesting, but what captivated me was how abstraction and recursion even let us _describe_ such numbers. Honestly, it was one of the first times I appreciated how powerful good definitions could be.
Read 11 tweets
30 Jun 20
Of all the videos I've made, one of my favorites topics to have covered was a proof of the inscribed rectangle problem by H. Vaughan using a Mobius strip.

But now there's a new result!
@QuantaMagazine recently did a great article about recent work by Greene and Lobb using a beefed-up version of the same idea, letting a Mobius strip encode geometric properties of the curve to solve a more general result, check it out:

quantamagazine.org/new-geometric-…
From recent comments on the video above, it looks there's a little confusion where some people thought this means the inscribed square problem (i.e. the Toeplitz conjecture) has been solved, but that's not quite the case.
Read 5 tweets

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