We Need Books That Affect Us Like A Disaster, That Grieve Us Deeply, Like Being Banished Into Forests

parents got a new cat they named lord montague and this morning i heard my dad in the other room say "i would have to advise against that decision, my lord" followed by a crashing sound

I'm reblogging this to compare it later with 1.A from Hatcher's Algebraic Topology. in that chapter he defines the topology on a graph if anyone else wants to check it out

Intuitively, it seems to me that graphs should be some sort of finite topological space. I mean, topology studies "how spaces are connected to themselves", and a graph represents a finite space of points with all the internal connections mapped out. That sounds topological to me! And of course many people consider the Seven Bridges of Königsberg problem to be the "beginning" of topology, and that's a graph theory problem. So graphs should be topological spaces.

Now, I vaguely remember searching for this before and finding out that they aren't, but I decided to investigate for myself. After a bit of thought, it turns out that graphs can't be topological spaces while preserving properties that we would intuitively want. Here's (at least one of the reasons) why:

We want to put some topology on the vertices of our graph such that graph-theoretic properties and topological properties line up—of particular relevance here, we want graph-theoretic connectedness to line up with topological connectedness. But consider the following pair of graphs on four vertices:

On the left is the co-paw graph, and on the right is the cycle graph C_4.

Graph theoretically, the co-paw graph has two connected components, and C_4 has only one. Now consider the subgraph {A, D} of the co-paw graph. Graph theoretically, it is disconnected, and if we want it to also be topologically disconnected, it must by definition be the union of two disjoint open sets. Therefore, in whatever topology we put on this graph, {A} and {D} must be open. The same argument shows that {B} and {C} must be open as well. Therefore the topology on the co-paw graph must be the discrete topology.

Now consider the subgraph {B, D} of C_4. It is disconnected, so again {B} and {D} must be open. Since {A, C} is also disconnected, {A} and {C} must be open. So the topology on C_4 must again be the discrete topology.

But these graphs aren't isomorphic! So they definitely shouldn't have the same topology.

It is therefore impossible to put a topology on the points of a graph such that its graph-theoretic properties line up with its topological properties.

Kind of disappointing TBH.

25 XII 2022

this chunk of the semester is finally over, sweet jesus I'm so exhausted. I'm getting the well-deserved rest and later catching up with all the things I put on my to-do list that I kinda learned but not really

the test I had last week went fine. frankly I expected more from it after solving more than 50 problems during my prep, but I scored 74%, which is objectively great and more than I predicted after submitting my solutions

here is my math plan for the break:

in algebraic methods I started falling behind a few weeks ago when I missed two lectures while being sick. they were about resolutions, derived functors and group homology and afterwards I wasn't really able to stay on top of my game like before. high time to get back on track. in commutative algebra I was doing ok, but there are some topics I neglected: finite and integral maps and Noether's normalization. for complex analysis everything is great until we introduced the order of growth and recently we've been doing some algebraic number theory, which btw is a huge disappointment. don't get me wrong, I understand the significance of Riemann's ζ, but the problems we did all consisted of subtle inequalities and a lot of technical details. I am doing mainly algebraic stuff to avoid these kind of things lol

when we were doing simplicial sets I stumbled upon some formulas for the simplicial set functor and its geometric realization and I thought it to be a nice exercise to probe them, so here it is:

I won't know if this proof actually works until I attend office hours to find out, but I am satisfied with the work I put into it

I already started making some notes on the derived functors

other than that I have this nice book that will help me prepare for writing my thesis, so I'd like to take a look at that too

as for the non-math plans, I am rewatching good doctor. my brain has this nice property that after a year has passed since finishing a show I no longer remember anything, the exponential distribution is relatable like that. this allows endless recycling of my favourite series, I just need to wait

I wish you all a pleasant break and I hope everyone is getting some rest like I am

Thinking about how when my oldest brother took Japanese classes his professor was like your pronunciation is really good 😊 but you need to watch movies that aren't about the Yakuza because you sound like a criminal

you really can't

wha?

please explain what this message means or else the curiosioty will kill me

Nothing but respect for this mathematician's webpage

funfact: in poland nobody really cares about eye contact, maybe other than people who want to have an intimate conversation with you like you'd have during a date or something

I was genuinely suprised when I learned that avoiding eye contact is a symptom of autism, because I didn't notice anybody ever trying to make it. I started paying attention to this whole thing after my diagnosis, where the doctor asked if I always look at the walls while talking to people. it turns out that people indeed are trying to look into my eyes even during the most mundane and routine interactions, but nobody (other than my now ex boyfriend who was so sad when he found out that I perceive eye contact as a threat) ever pointed it out as something that I should do. but then I see (presumably american or just non-polish) people talking about being offended by someone not making eye contact and I experience a massive cultural shock lol

girl i am not looking at your tits i prommy i just hate eye contact

Please fund my research in finding fewer applications of mathematics. I'm going to start my project with trying to find fewer uses of trigonometry, so that ideally we can eliminate the need for remembering trigonometric identities. Then I'm going to move on to researching fewer uses for integration by parts, because that tends to get real tedious real fast. With your unending financial support, I believe I can return mathematics to the purity and simplicity it has always yearned for.

30 VII 2021

did some stuff today. found out my cat would die tomorrow if it wasn't for an operation he had today, that didn't feel good but also oh god was he lucky

sleep: better but still trash. yesterday fell asleep between 3 and 4, today planning to go to bed at 3 so in a few minutes

concentration: not as great. couldn't focus because my cat was fucking dying

bo phone time: decent

did some topo today, i think i managed to understand the idea behind the quotient spaces and i really liked it. can't wait to dive deeper tomorrow. other than that i wrote a method that takes a parametric function and provides a partition dividing the curve into k intervals. also i'm almost done with the art comission

sooo tomorrow i plan to finish the code, i want to achieve the functionality that takes a parametric function and draws it with a dashed line. that's why i needed the partition, it's gonna look fucking beautiful. and i need to complete the comission. i hope there will be some time left for quotient spaces, i am very hyped. oh and i forgot i'm drinking tomorrow. so i guess no topo for that gal. eh

One of my favorite thing I’ve learned about animals studies is that you should avoid using colorful leg bands when you’re banding birds because you can accidentally completely skew the data because female birds prefer males with colorful bands

Apparently if you put a red band on a male red wing blackbird his harem size can double

So like you can completely frick up the natural reproduction of a group of birds by giving a guy a bracelet so stylish that females CANNOT resist him