spectrallysequenced - Tumblr blog

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That feeling when you have a space but that space is actually a functor but it's actually a space.

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There he is. There’s my boy!!!

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I started a blog series (Which will be accompanied by a video series soon) on schemes. The first chapter is on classical algebraic geometry, and the first notions of a generalisation to arbitrary rings. Unfortunately, since tumblr doesn't have LaTeX integration yet, I decided to do this off-site on Functor Network, so that means you will have to go through the mental anguish of clicking a link:

I hope it's worth it.

Edit: Associated video going live tomorrow on youtube:

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Just finished the second episode in my noncommutative algebra series!

Please check it out!

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Use of Hilbert's Hotel in Set Theory

I'm helping out some freshmen in their mathematics fundamentals course, and this wonderful argument came up:

Suppose that I have an infinite set, where the definition of infinite we are using is that contains a countable (i.e. bijective to the naturals) subset. This definition assumes the axiom of choice (don't worry about it), but if you'd like, replace the general set in the rest of this post by the real numbers, which we know contain the natural numbers as a countable subset. Let X be an infinite set, and {x_0,x_1,x_2,....} be a countable subset. Suppose we want to add a point to x. The context that this came up in was constructing a bijection from the half- closed ray [-2,inf) to the real numbers. You might know that if this were an open interval (-2,inf) then a bijection would easily be given by R->(-2,inf), x|-> (e^x)-2. Its inverse is log(x+2). We want to "hide" the point, get rid of it somehow. In other words, we add a point * to our set which is disjoint from it, and we want to construct a bijection X u {*} -> X, thereby hiding the point. In the case of the problem, this is a bijection [-2,inf)->(-2,inf), and our point * would be -2. You might know of Hilbert's Hotel, which has infinitely many rooms numbered 1,2,3,.... and is fully occupied. A new guest requests a room, so to accomodate them, the guest at room 1 moves to room 2, the guest at room 2 to room 3, and so on, so all the guests still have a room, but now we have room 1 for our new guest! We employ the same idea here, defining a function by:

f(x) = x_0 if x=*

x_{n+1} if x=x_n,

x otherwise

Then f is a bijection X u {*} -> X (Check!). We "made room" for our new point * by moving our countable set up an index, just as in Hilbert's Hotel!

In the case of our bijection [-2,inf)->(2,inf), we use the natural numbers, so we map -2 to 0, 0 to 1, 1 to 2 etc, and all other points to themselves.

Now we can compose our bijection with the bijection (-2,inf)->R we described earlier to show that [-2,inf) is in bijection with R.

Even basic set theory is cool

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Fun fact I actually went to a lecture about MOND from the inventor himself in highschool at Weizmann Institute and even back then we all thought he was kind of a crank. Speaking with grad students they said it's all he talks about, and he adamantly rejects any criticisms.

Dark matter skeptics like to make claims about parsimony–if we're only trying to explain discrepancies in gravitational observations, why posit the existence of entire new, magical particles when we could just tweak gravity directly instead?–but this argument makes absolutely no sense to me. Our modern understanding of physics puts relatively few constraints on the number and kinds of particles that might exist in the universe. General relativity, on the other hand, has a much more rigid theoretical structure and far fewer free parameters. Even leaving aside the shaky experimental status of MOND (and yes, it is shaky, especially in the last few years), a new particle is a much less drastic and more parsimonious addition to our theories than an ad-hoc correction to the fundamental laws of gravity. Reconciling that with current fundamental theory is a massive task.

As far as I can tell, the actual primary source of skepticism is just "if they were real, we should've seen them by now". But why would that be true? There's no law of the universe that says particles have to be easy for humans to detect. That belief comes from either an attachment to human senses as the arbiters of reality displaced onto only the forms of scientific observation that feel most tangible, or an unjustified blind faith in the ultimate adequacy of current experimental methods.

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Can't believe the father of stochastic analysis is named fucking Norbert Wiener that wouldn't be out of place in a diary of a wimpy kid.

And he looks like this

I love him

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It's 5am and I keep having to draw pairs of pants because I have to give a talk tmrw.

Real ones will understand

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Personally I view this as math showing us that cohomology is a more natural PoV. I mean yes, singular cohomology is quite artificial bit cohomology just ends up being better so often.

In fact sometimes you can't even define homology in a reasonable manner, but you can define cohomology (sheaf cohomology as an example, not enough projectives).

Homology turns out to work in the case of topological spaces, but this is more of a special case than indicative of a larger picture.

Math people, reblog with your fav theorem and why.

I'll start, the Wedderburn-Artin theorem is a beautiful structure theorem on semisimple rings which says they decompose uniquely as a product of matrix rings over division rings. This is a beautiful result but it also underlies a lot of very cool theory like Brauer Theory, Galois Cohomology and the theory of Galois and Étale Algebras.

What's yours?

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The Gelfand-Naimark theorem is so cool. You're telling me C* algebras are a natural model for Noncommutative Geometry? Hell yeah

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Mathblr I think @mathsuggestions is fucking hilarious why are we not blowing up their posts

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New years resolution: going to strike up a conversation with anyone who boops me. Urge ya'll to do the same. You wanted my attention, face the consequences.

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My favourite "trivial" theorem is probably schur's lemma. It's unreasonably useful and painfully obvious.

Math people, reblog with your fav theorem and why.

I'll start, the Wedderburn-Artin theorem is a beautiful structure theorem on semisimple rings which says they decompose uniquely as a product of matrix rings over division rings. This is a beautiful result but it also underlies a lot of very cool theory like Brauer Theory, Galois Cohomology and the theory of Galois and Étale Algebras.

What's yours?

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Math people, reblog with your fav theorem and why.

I'll start, the Wedderburn-Artin theorem is a beautiful structure theorem on semisimple rings which says they decompose uniquely as a product of matrix rings over division rings. This is a beautiful result but it also underlies a lot of very cool theory like Brauer Theory, Galois Cohomology and the theory of Galois and Étale Algebras.

What's yours?

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Math people, reblog with your fav theorem and why.

I'll start, the Wedderburn-Artin theorem is a beautiful structure theorem on semisimple rings which says they decompose uniquely as a product of matrix rings over division rings. This is a beautiful result but it also underlies a lot of very cool theory like Brauer Theory, Galois Cohomology and the theory of Galois and Étale Algebras.

What's yours?

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IMPORTANT MATH PSA

You know how people always ask if pi contains all strings of integers and then people answer "no pi is not normal". So FIRST OF ALL normality is wayyyy too strong of a property here. A number in which all finite sequences appear is called *disjunctive*. A normal number is one in which all integer sequences are uniformly distributed, much stronger property.

That being said, it is not known (but strongly conjectured) that pi is disjunctive. In fact, every sequence of eleven digits or less appears in the first 2.7 trillion digits of pi.

In conclusion:

π knows your SSN

Find your SSN in the digits of pi here:

https://bellard.org/pi/pi2700e9/pidigits.html

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Hi please listen to me rant about some cool algebra/geometry I saw this week (Field Patching, Noncommutative Geometry)