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My basil finally developed roots! I'm currently reading about quantum mechanics, ferrofluids, and language models.

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[08/09/2024:] Motivating Ladder Operators II
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Working on notes on the quantum mechanics, derivatives, and uploading my previous course notes onto this blog!

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Finally started a projects page! I've recently made some nice upgrades to my post component, so it looks pretty clean! ;)

🌊

I'm considering whether or not to continue this project using WebGL or Three.js.

I'm also researching methods for generating the 3D scenes I want for this project automatically.

In the meantime, I've decided to proceed with some preliminary prototypes of the other interactive parts of this project.

Orange Juice

I like orange juice. :)

Mlog

Homotopies

September 3, 2024
By Aathreya Kadambi
Expanding on Lectures by Professor Alexander Givental and Fomenko and Fuchs

It turns out that we are not just interested in spaces of continuous maps, but we are concerned with path-connected components of these spaces, or like things that are continuous but also can deform the thing.

Consider the maps $X \xrightarrow{f_1} Y$ and $X \xrightarrow{f_0} Y$ . We say $f_0 \sim f_1$ are homotopic if there is some $F : X \times I \rightarrow Y$ such that $F(x,0) = f_0$ , $F(x,1) = f_1$ .

\pi(X,Y) = C(X,Y)/\sim

we can consider homotopy classes of maps

X \rightarrow Y

. We get path-connected components or something (maybe homotopies parition any set into classes of path connected components or something).

The classic example to distinguish path connectedness and connectedness is the union of $\sin(1/x)$ and the $y$ -axis, which is supposedly connected but not path connected.

We say that $X$ and $Y$ are homotopy equivalent if $g \circ f \sim \text{id}_X$ (homotopic to the identity on $X$ ), and $f \circ g \sim \text{id}_Y$ .

We define a retraction $r : X \twoheadrightarrow A\subseteq X$ so that $r|_A = \text{id}_A$ . A retraction is deformational if $i \circ r \sim \text{id}_X$ . Strict if the homotopy is identical on $A$ , like for example, $A$ doesn’t rotate or something during the retraction and stuff. In other words, strict means that $A$ doesn’t move during the retraction. Apparently there are some pathological examples, for which one can refer to the book.

Proposition. Deformation retraction is a homotopy equivalence.

Proof.

$i \circ r \sim \text{id}_X$ , $r \circ i = \text{id}_A$ .

Now $X$ is called contractible if $x_0 \in X$ is a deformtaional retraction of $X$ .

Proposition. $X$ is contractible is equivalent to $X \sim \text{pt}$ .

Remark. There is some result that we will eventually discuss called Borsuk or something about extending retractions or something.

There is a classification of homotopies or something:

Theorem. TFAE (the following are equivalent):

$X \sim Y$
$\forall Z$ , $\exists \alpha^Z : \pi(X,Z) \rightarrow \pi(Y, Z)$ which is natural with respect to $Z$ , meaning that for any $\psi : Z \rightarrow W$ , then whenever we have a map from $X$ to $Z$ , we can compose it with a map from $Z$ to $W$ to get a map from $X$ to $W$ or something? More like in the sense that $\pi(X,Z) \rightarrow \pi(X,W)$ there is a map $\psi_*$ induced by $\psi$ . And there is also a map $\psi_*$ from $\pi(Y,Z)$ to $\pi(Y,W)$ . In between we have two bijections, and it commutes. Just naturality from category theory, but not sure how the naturality idea is motivated from anything right now.
$\forall Z$ , $\exists \beta_Z : \pi(Z,X) \rightarrow \pi(Z,Y)$ which is a bijection which is natural with respect to $Z$ .

Proof.

We start by showing that 1 implies 3. If $X \sim Y$ , then there exist maps $f : X \rightarrow Y$ and $g : Y \rightarrow X$ such that the compositions are identity. Then we get maps:

\pi(Z, X) \xrightarrow{f_*} \pi(Z,Y) \xrightarrow{g_*} \pi(Z,X)

and the composition is identity. You can also do this in the opposite order and get identity. Therefore, the maps are set-theoretic inverses and so these are bijections.

Strat. Many things in mathematics relating to associativity come down to associativity of maps, for example associativity of vector addition comes from that of maps by thinking of vectors as translation maps. Apparently that comes in here too. He drew a square with $\pi(Z,X)$ , $\pi(Z,Y)$ , $\pi(W,X)$ , and $\pi(Z,Y)$ .

Now we can do the direction that 3 implies 1. For this direction, he started by taking the image of the identity of $\pi(X,X)$ in $\pi(X,Y)$ under the map $\beta_X$ which was defined previously as $f_*$ I think (so that $\beta_X(id) = f$ ). Awesome proof using the square $\pi(X,X)$ , $\pi(X,Y)$ , $\pi(Y,X)$ , $\pi(Y,Y)$ basically by tracing where the elements $\text{id}_X$ and $\text{id}_Y$ go to throughout the diagram. We then do a similar square for the other way.

Remark. There is also a basepoint version of this, where we define $\pi_b(x,y)$ similarly to $\pi$ above. Another remark is

\pi_b(\Sigma X, Y) = \pi_b(X, \Omega Y)

We can essentially think of these

C

\pi_b

\pi

or whatever as functors, and so we see that

\Sigma

and

\Omega

are sort of adjoint. It is also important to note that they are not just functors to sets, they have some algebraic structure because loops can be adjoined, you can combine two loops to get another loop. I should note he put strikethroughs through the

b

s in

\pi_b

but I am not completely sure why. Something like this is in Fuchs Chapter 4. Apparently Fuchs makes two errors in each of his lectures.

Lemonade

A Sweet Homemade Drink Made in Many Parts of the World