All Drafts, Unorganized, Unfinished

Drafts

You’ve come to the right place to see how Sridhar thinks about drafts.

Golay codes

Assorted results about Golay codes:

Proofs of Fermat's Little Theorem

What I’d like to say about proofs of Fermat’s little theorem (or the Euler-Fermat theorem more generally) is going to be annoying to write on Twitter with its character count restrictions, lack of editing, etc, but here is the gist of it:

Zorn's Lemma

Here are two framings and proofs of Zorn’s Lemma:

Commas

Various random facts about comma categories I wished to record somewhere:

Thoughts on what calculus is about

Calculus is about certain kinds of weighted sums.

Associativity and the Associahedron

The mean number of points in cycles of length 𝑘 over all permutations of an 𝑛-element set is 1. Why is this?

Gram-Euler Theorem

The Gram-Euler theorem says that if you have a contractible polytope with straight sides, the sum of the interior solid angles at each cell (where this means the fraction of an infinitesimal sphere at that cell which is contained in the inside of the polytope), counted positively for even-dimensional cells and negatively for odd-dimensional cells, adds up to 0 (when we include all cells of dimension ≥ 0, such that the similar calculation of Euler characteristic (as though they all had solid angle 100%) would be 1), or to 1 (when we also include a unique cell of dimension -1, with angle 100%, such that the reduced Euler characteristic including this would be 0).

Gaussian Functions

Some facts about Gaussian functions (exponentiated quadratics):

The Typed Lambda Calculus Is Normalizing

UNPOLISHED: Some proofs of some facts about normalization in the typed lambda calculus (terms in the simply typed lambda calculus with products, possibly with NNO as well, are normalizing, or possibly strongly normalizing, with unique normal form):

The Fourier Transform

UNPOLISHED: Eventually I will record all my thoughts on the Fourier transform here. For now, I just want to record some thoughts about the Fourier inversion theorem.

Quiver test

Testing including Quiver diagrams.

Supply chain stuff to write up

Supply chain phenomena to write about:

Assorted facts about monads to write up somewhere

If a monad has a left adjoint, then this left adjoint is a comonad with the same Kleisli category and same inclusion of the pure category into the Kleisli category. If a monad has a right adjoint, then this right adjoint is a comonad with the same Eilenberg-Moore category and the same forgetful functor out of the Eilenberg-Moore category. ** More assorted facts about monads: A monoid in the general sense in a 2-category always induces a corresponding monad (monoid in the 2-category of categories), via the appropriate Hom(*, -) functor. This takes the monoid M with respect to monoidal structure x to the monad M x -. Dually, a comonoid is taken to a comonad, of course.

Bell's Inequality

Salvage the following posts from SDMB:

Free Theory With An Internal X

Consider the free theory with an internal X extending x. When we externalize its internal X, we get some X extending x. But is it x on the dot, or might it be changed from x to have more in it? It seems obvious but not obviously obvious that, when Set itself is an X, we should get back x on the dot. Here, I want to write out the reasoning, to make it obviously obvoius.

Functorial Semantics

TODO: Scratch notes for a post on Lawvere theories, monads, functorial semantics, categorical logic, etc. Taken from a Twitter DM conversation.

Quadratic Forms

There are lots of things to say about quadratic forms (or equivalently when division by 2 is available, symmetric bilinear forms, or the symmetric component of arbitrary bilinear forms), a notion which comes up over and over in math. Particularly positive-definite ones (which are often studied under the name “geometry”, as this is what the theory of geometry amounts to; an inner product space is a positive-definite quadratic/bilinear form).

Associativity and the Associahedron

The order $N$ associahedron is an abstract polytope whose vertices are all binary trees with $N$ many leaf nodes (where a binary tree is either a leaf node or an ordered pair of two child binary trees). One can describe this same thing in many different words, as having to do with parenthesization or such things. It’s all the same.

Sophomore's Dream

Here are a couple little curiosities in calculus (known since at least 1697 by Johan Bernoulli). They are sometimes called “sophomore’s dream”, because they feel too good to be true, like the kind of glib substitution of one thing by another somewhat similar thing that a naive student might make, and because something else already took the name freshman’s dream [TODO: Write about and link to freshman’s dream].

Clifford Algebra

I don’t really care for Clifford algebra, but here’s my attempt to understand the parts of it that aren’t obvious anyway:

The Central Limit Theorem

Brownian

Extracted from Facebook comments:

Boolean Algebras

TODO: To be written out into a full thing on how Boolean algebra is the finitary theory of finite sets, of 2, etc. Limit vs. product theory perspective on Boolean algebra, etc.

The Fast Fourier Transform

Let \(B\) be a value such that \(B^N = 1\) and define the \(K\)-weighted base-\(B\) transform of order \(N\) of a function \(f\) to be \(s \mapsto K \sum_{t = 0}^{N - 1} f(t) B^{st}\). [The domains of these functions should be thought of as the integers modulo \(N\), so that these function as bidirectionally infinite sequences]

Blind Bartender Problem

Suppose you play a game: You have a bidirectionally infinite sequence of coins, heads or tails. Your goal is to make this into some particular target sequence. At any moment, you announce some set of positions that you wish to flip. Your opponent, however, can respond by translating your set by some amount before it takes effect; e.g., if you wanted to flip positions -9, 8, and 20, your opponent can turn that into -4, 13, and 25. You keep going like this in turns. From what starting positions can you force a win, how do you do so, and how does your opponent stymie you otherwise?

Adjunctions

[TODO: Equivalence between hom-set definition of adjunctions (including as abstracted to any 2-category) and unit/counit/triangle identities definition. Automatic equivalence also to the “backwards” definition: An adjunction between functors F : C → D (the left adjoint) and G : D → C (the right adjoint) is a bijective correspondence between Hom(xG, y) and Hom(x, yF), natural in x and y. Automatic because the unit/counit/triangle identities definition is symmetric. Note also the concept of an adjunction up to adjunction, aka “quasi-adjunction”, where we don’t get a bijection but only an adjoint correspondence in a higher category. Note how adjunction are like half-inverses, just enough to make the proof of unique inverses still go through.]

Shapey Algebra

Key concepts: Preorders, (higher-)groupoids. As their intersection, setoids. As their union, (higher-)categories.

Adjunction Factorization

[There are a million things to write about adjunctions. For now, I’m just writing some motley scratch notes for myself and will shape this up into an introductory post later.]

Birkhoff’s HSP Theorem and Beck’s Monadicity Theorem

This is all TODO notes for a post not yet actually written, but scratch thoughts I want to keep archived for myself to write with.