Fields Of Mathematics

ok now i might have some kind of super memory??

a week ago i played chess with bf and we didn't finish, so now i arranged the board as i remembered it and i got 13 out of 14 pieces correctly

i mean wow i didn't know i am capable of something like this

might be autism i suspect i might have

anyway now i want to know everything about human memory and take advantage of that

I know your thesis was about something to do with algebraic topology, may I ask what exactly it was about?

(and congrats to you getting your bachelors degree and into a masters program)

(thank you!)

my thesis was about an open question regarding a certain skein module of tangles on 2n nodes. the conjecture is that the module is free and in my thesis I constructed a generating set that is free for n=2,3 (direct calculation) but I have yet to prove that for a general n. if you are interested I can send you the paper in which the question was posed, all the details are explained there and would be hard to write down here without tex lol

just had a reflection about perfectionism. today I had an exam for which I was prepared very well, but my stupid brain happened and I didn't get the highest grade. my boyfriend was comforting me and he asked since when I want to ace everything, this question made me think

indeed, I don't want to ace everything. I am taking 4 courses this semester, one of which I don't care about enough to strive for the best grade, one of which is way too hard to aim that far, two of which I thought were achievable. and now I didn't achieve that. it feels different to set unrealistic goals and then never achieve them than to set very realistic ones and still fail, that's what I realized today

I am not a perfectionist. I used to be, years ago, and then I learned to set realistic goals. now I'm thinking, isn't perfectionism a kind of a coping mechanism? deep down you know your goals are impossible, so it's not really surprising when you fail. you are never satisfied, sure, but maybe it does feel more safe this way than to not know if you will be satisfied or disappointed. if that's the case then setting realistic goals is absolutely not the way to heal perfectionism

"numbers don't lie" the real numbers are literally a lie group

Mathematicians be like:

Def 1.1: A function f is fucked-up iff it is not Lebesgue measurable

Def 1.2: A function is evil iff its graph has non-integer Hausdorff dimension.

Exercise 1: Prove that there exist fucked-up and evil functions

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.

some more animation: how secant-teenager becomes tangent-adult

also! it illustrates a well-known inequality of a differentiable convex function and its tangents & the monotony of slopes

i’m starting a collection

13 IX 2022

my euclidean geometry journey will be over soon and the start of the semester is so close, it's kinda scary

recently I stumbled upon someone's post with a time-lapse video of their study session. I liked it so much that I decided to make mine

this is me learning about the snake lemma and excision

the excision theorem is the hardest one in homology so far btw, I spent about 4 hours on it and I am barely halfway through. I like the idea of the proof tho, it's very intuitive actually: start simple and tangible, then complicate with each step lmao

I realized two things recently. one of them is that deeply studying theorems is important and effective. effective, uh? in what way? in exams we don't need to cite the whole proof, it suffices to say "the assertion follows from the X theorem"

yeah right, but my goal is to be a researcher, not a good test-taker, researchers create their own proofs and what's better than studying how others did it if I am for now unable to produce original content in math?

the second things is that I learned how to pay attention. I know, it sounds crazy, but I've been trying another ✨adhd medication✨ and after a while I realized that paying attention is exhausting, but this is the only way to really learn something new, not just repeat what I already know. it made me see how much energy and effort it takes to make good progress and that it is necessary to invest so much

I am slowly learning to control my attention, which brings a lot of hope, as I believed that I had to rely on random bouts of hyperfocus, before I started treatment. I am becoming more aware or how much I am focusing at the given moment and I'm trying to work on optimizing those levels. for instance, when I'm reading a chapter in a textbook for the first time, it is necessary to remember every single detail, but wanting to do so consumes a lot of energy, because it means paying constant attention. it is ineffective because most likely I will have to repeat the process a few more times before I truly retain everything. being able to actually pay attention at will sure does feel good tho, as if I had a new part of my brain unlocked

I am solving more exercises for algebraic topology, procrastinating my lecture prep lmao. I am supposed to talk about the power of a point and radical axes, I have a week left and I can't force myself to start, because there is so much good stuff to do instead

I have a dream to produce some original results in my bachelor's thesis. it may be very difficult, because I hardly know anything, that's why I'm calling it a dream, not a goal. the plan is to start writing at the end of the semester, submit sometime in june

I spent last week at the seminar on analysis and oh boi, I will have to think twice next time someone asks if I like analysis. the lecturer who taught me at uni had a different approach than the "classic" one. we did a little bit of differential geometry, Lie groups and de Rham cohomology, those are the things I like. meanwhile at the seminar it was mostly about analytic methods of PDEs, the most boring shit I have ever seen

complex analysis will most likely be enjoyable tho, I'm taking the course this semester

for the next few days I need to force myself to prep that damn geometry lecture. other than that I plan to keep solving the AT exercises and maybe learn some more commutative algebra. I wish everyone a pleasant almost-autumn day 🍁

7-9 VIII 2021

did math and coding nothing special really

sleep: good

concentration: good

phone time: good

reading about measure theory. here is a great book:

everything is so well explained here. i wish i could do more math than i have time for but i guess it's fine, it's holidays, i will wreck my brain completely anyway when october comes

tomorrow more measure theory and topo