Latest Posts by bsdndprplplld - Page 4

Quatrefoil Knot

when K ⊆ L is a finite extension by one element, say α with the minimal polynomial f, we can write 0 → (f) → K[x] → L → 0, where (f) is the kernel of evaluation at α. this is quite disappointing and very basic, but I haven't found anything better really. when there are finitely many intermediate fields between K and L for an extension L/K, L can be expressed as an extension by one element (Artin's theorem), which is still very specific

I didn't know about the group extensions, it makes the category of fields even more disgusting. I was hoping that the algebraic closure can be expressed as a colimit but of course not, not in the general case at least. but maybe at least some type of extensions can be realized as such? that's a nice thing to ponder. I'm pretty sure it will fail like every other request I had for this abomination of a category

I wonder what is typically done to make working with this category more pleasant. passing to Grp with the Galois group is one idea, the other I guess would be working with vector spaces or algebras? that would make sense considering that integral and finite ring maps are a thing and the field automorphisms play a role in the integral closure of ℤ in ℚ[√d]

on a sidenote, I laughed at the "lower body" and it reminds me how funny it is to talk about kernels in Polish. kernels and testicles are the same word

I've always thought 'splitting field' was a very cool sounding term. The Galois theorists did good with that one

The proof is left as an exercise to the IRS

are you a girl?

I am, but I thought that was obvious given that I have a picture of me in my icon.

stepped on a plum (overripe plum) (barefoot) it was on the driveway got out of the car and accidentally (didn't know it was there) stepped on the plum (warm) (on the ground) (it had fallen from the tree) barefoot (no shoes) wearing long pants (too long) (need to hem them) plum viscera got on them (the pants) unexpected plum on the driveway (hot plum) (97 degrees out) already super hungover (throwing up all morning) (should not have been driving at all) and I stepped out of the car (black car) (97 degrees out) and onto the plum (unexpected) (didn't know the plum was there) and it burst (plum nightmare on my only good pair of sweatpants) still we find ways to keep ourselves going from day to day

imo euclidean geometry kinda sucks, but if we mean geometry in a more general sense then algebraic geometry is the one

I've decided to start a fight

anyways geometry sucks algebra best math

yeah I say that I'm vegetarian because the texture of meat fucks with my sensory issues. I would know if the meat in my meal was real and I would probably throw up and have a meltdown, then proceed to have my day ruined. I can't believe someone actually could put me in such a position jfc

Look I clown veganism often enough but really, truly, don’t ever fucking feed somebody something without their knowledge or consent. It’s hugely fucked up and not OK.

meanwhile typical conversations between my friends:

– so what do you do in math?

– differential equations

– ugh I always hated differential equations

– you?

– general topology

– ugh I always hated topology

The curse of a mathematician is to work in a disliked field

from now I’ve got to start chasing regular discomfort, because otherwise comfort isn’t comfort but stagnation. you can’t have one without the other. to be challenged is to grow and it’s the only way to actually feel at peace

Me: I should write something

me : … or I could spent 78 hours straight making a miniature library with a working LED chandelier

I can relate to your undergrad experience! And I think it might be a good sign looking forward, because you've developed insights and ways of thinking and motivation to go beyond undergrad and seek out new spaces where you can do your own work. That's by no means common, I know many fellow undergrads who are a) as mystified when they retake a class as they were the first time round, and b) feel accomplished enough to have passed eg Introductory Analysis and have no drive to look onward. You seem to know very much what you're good at, what interests you and which areas you'd like to grow in. I'd argue that undergrad studies, which give you an introduction and overview of the field and teach basic reasoning skills while not expecting any really original problem solving aren't exactly made for people like you. Talking to your professors or Masters or even PhD students is a really good idea!

thank you for your input, it brings a huge relief!

I already talked to two of my professors and they said that there is nothing to worry about. my advisor said that in his opinion learning new concepts while working on some problem is the right way to learn and from his experience this is way more rewarding than learning for school or even "just to learn". he also said that if I'm interested in working more on open stuff then he will let me know when he finds some questions I could ponder. the other professor said that it's a good thing, because from his experience a lot of people tend to get discouraged when there is no way of knowing how long solving the problem will take or how much new theory is needed, and I seem to be the other way around, so the work I'll be doing in the future probably won't scare me as much

I talked to some of my friends who are about to finish undergrad like me, and there are people who feel the same way as I do. coincidentally, those are the people who had the same situation as mine, that is, they were lucky enough to find an advisor who gave them an open question to work on. other people I talked to seem to be fairly content with studying for the classes and completing homework assignments, and they didn't get to work on something open yet, so maybe it has something to do with getting the taste of the good stuff haha

I can see now that the future looks good and I'm motivated to go exploring. I am aware that I have so much more to learn, but having got the reassurance that I'm probably doing it right, it doesn't sound as scary anymore

I have just met you and I love you (via)

Venn diagrams that have a number of sets that AREN'T prime numbers (except 1, but it's trivial) cannot be rotationally symmetrical, so here are a couple real 6-set venn diagrams.

Someone PLEASE use one of these to make a diagram. I'm begging. i need to see it with my own eyes.

req'd by @strictly-script

sure we won't?

text: Abelian't

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

I've been thinking about how different math feels after three years of consistently doing it. it's a sad thought, because I used to get super excited about learning new things and solving problems, whereas now my standards seem to be higher..?

I spent the day doing exercises from galois theory and statistics, in preparation for the tests I have soon. it felt like a chore. sure, the exercises were easy and uninteresting, I decided to start from the basics, so there is that. however, in general practicing like this became a routine and there used to be a sense of mystery around it that is now gone

when I don't have any deadlines but feel like doing some math the obvious choice is to learn something that will be useful in the future. more homological algebra, algebraic geometry, K-theory, or digging deeper into the topics I already am familiar with. all of those are good candidates and I used to be very motivated to just learn something new. but here comes to paradox of choice, where every option is good, but there isn't a great one

I think I might be annoyed with always learning the prerequisites for something not yet defined. it did feel exciting when I was studying the modules of tangles so that I could answer an open question, it doesn't feel as exciting to learn about the galois theory to pass a test. a metaphor comes to mind. doing math without a fulfilling goal feels like taking a walk – it's rather nice, I enjoy going on walks. with a fulfilling goal it feels like walking towards a destination such that the walk itself is a pleasant activity, but I really want to get to said destination. by that I mean that I still enjoy simply learning new stuff and working on exercises, but it doesn't feel as fulfilling as it used to, how much walking without getting anywhere can you do in three years? you can do the same thing in prison

three years is nothing compared to how much knowledge and experience is necessary to do actual research, I know that. I fail to feel it, but I know it. when I am asking myself what state of mind is the most fulfilling I'd say exploration, discovery, getting an idea that is new to me and seemingly comes from nowhere, not just an obvious corollary of what I've seen in lectures, an insight, an act of creating. I suppose all those things are to be found in the future, but god how long do I have to wait

on a more pragmatic and realistic note, I think I'll talk to my professors about what I can do to speed up that process. I'll ask them how the actual research feels and how they went from being a student learning basic concepts to where they are now

a question to those of you who are more experienced than me: does this even sound familiar at all? what were you like as a student and what took you to where you are now? how does math feel after 3, 5, 10 years?

i’m starting a collection

(I'm reblogging this for later to really look into all the amazing accounts I follow)

Let's get a new mathblr roll call going! There's an older version but it's got a decent number of inactive people on it so let's start fresh. Reblog and/or reply tagging yourself and any other good math accounts!

Math shitposters! Math academia aesthetic blogs! Math studyblrs! Unthemed blogs owned by people who happen to be math fans! CS, stats, physics and other math-adjacent dorks too if they like hanging out with the math crowd! I want them all!

2 IV 2023

oh god the programming task for today was so annoying. I was supposed to process the MIT database with ECG records, and the annotation part of it was hell. after three hours I finally did it but the anger I felt at that time put me seconds away from throwing my laptop out of the window lmao

a recent success is that I calculated the rank of the module that I am working with, the problem is almost solved! when I told my advisor about it he looked so happy, he said that maybe he should start looking for another problem for me to ponder, it was so satisfying. I have a thing for mentors. at each point in my life for which I had a mentor who would teach me my special interest the progress I was making improved significantly and those were always the happiest times of my life. I am not sure if my advisor will stay with me to further show me a way into the research, but it certainly feels like a possibility

today I did some algebraic topology and differential geometry, I'm trying not to fall behind with the material even when I don't feel like studying

next week the easter starts, so I will probably have to visit my family. it's an interesting feeling to see my sister all grown up, there is still the image in my head of when she was barely a teenager and we didn't have much to talk about. now she is almost 18 and the significance of the age difference is nearly gone. when she start university it will be even less noticeable as she will understand what I mean by "fuck my life it's exam session season" lol

for about a week I've been trying to eat more healthy food, it's going fine so far. my biggest problem is that I'm eating way too much sugar but undereating in the general sense at the same time. I'm trying to incorporate more fruits and vegetables into my diet, as well as different kinds of nuts. it's so important to be properly nourished for math and yet I neglect it so much

yesterday I had a conversation with my friend and he said that his vision for doing math is working on some huge open problem such as RH. obviously you do you, but this sounds like such a depressive idea to me lol. chances of solving something like this are almost non-existent, that's such a waste of time to work on something like this for 10, 20, 50 years and make no progress. I mean, it certainly would feel nice to prove or disprove something like RH, but I'm perfectly fine with reading papers and answering all the questions I can anwer, which might not be huge and famous but I'm pretty sure creating those small pieces of theory will be useful to somebody one day

free recall

here I am sitting and trying to learn something from a textbook by making notes and ugh I don't think this is gonna work

what I'm writing down will probably leave my head the second I switch tasks

today I found a cool video about taking notes during lectures and a method called free recall is mentioned there:

to summarize: taking notes during the lecture is ineffective, because it requires dividing attention into writing and processing the auditory input. instead of doing that one should just listen and then try to write down the contents of the lecture from memory. I can believe that – this is how I studied for my commutative algebra exam and the whole process went really fast. I highly recommens this guy's channel, he is a neuroscientist and bases his videos off of research findings

I will try to do this with textbooks and after a while I'll share how it felt and if I plan to keep doing it. the immediate advantage of this approach is that it gives raw information for what needs the most work and what can be skipped, which is often hard to see when trying to evaluate one's knowledge just by thinking about it. another thing that comes to mind is the accountability component – it is much easier to focus on the text while knowing that one is supposed to write down as much as possible after. kinda like the "gamify" trick I saw in the context of surviving boring tasks with adhd

I'll use this method to study differential geometry, algebraic topology, galois theory and statistics. let's see how it goes

The only thing that this university computer-science in-class-use file is dangerous to is my mental health, and the only thing it harms is my soul, but thanks for looking out for me google :)

26 III 2023

I had a lot of headaches recently, idk why. probably something to do with muscle tension, because my back, neck and jaw just lock up sometimes to the point that every movement hurts. I need to see a doctor about it, maybe I injured something or there is some other underlying cause

I wasn't very strict with studying this week, because a lot of stuff we did was a review of what I already knew but obviously it needs a refresher. if I keep ignoring it, I will end up in a situation where I won't know what's going on at all

I picked up some side hustles along the way, one of which is reading the extra topics from hatcher. one of the lecturers recommended a book to me, about galois theory in the context of covering spaces, I'm reading it right now, seems pretty good

tomorrow I'm seeing my advisor to discuss my progress with solving the problem for my thesis. I think I found the basis for the module, at least I proved that the set I chose generates all the other elements, remains to show that it's linearly independent. the second part of the question is the rank of the module, which is how an algebraic topology problem turned into a nasty cominatorics problem eh

today I completed the first "serious" task for my IT job, which was translating the code from java to python. I have never seen java before, but it looks a lot like c++, so I managed. I wrote 500 lines of code but I haven't tested it yet so debugging might be very painful. lol I guess that means I shouldn't say I completed the task

I am wondering if I should go to a conference, I have until the end of the month to submit a presentation. I am not sure if I can handle a trip to another city, it would be in a month, so there is no way to predict how I'll be feeling. this week I am giving a presentation about some knot theory (skein modules, bracket and jones polynomial) and it's a good pick for the conference too, which makes it a really touch choice as the hardest part will already be done. idk I guess I'll toss a coin, like I did about the IT job lmao

other than that, big thanks to everyone who interacted with my post about book recommendations! there are many great suggestions, it turned out much better than I expected tbh, I thought I would get like 2 or 3 notes. I will post a list of the books mentioned in that post, so it will be easier to find for anyone interested

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.

Hello, dear! 🌻

I saw your post wanting book recommendations. I'm sorry for your previous struggles, but I hope this list may help you find something you love!

-"The Housekeeper and the Professor" by Yōko Ogawa (The professor is a mathematician!)

-if you like Vonnegut, you may like Haruki Murakami, specifically his older titles like "Wind-Up Bird Chronicle" and "Norwegian Wood" (I feel these books do a good job of expanding on people's motivations and moods.)

-"The Elegance of the Hedgehog" by Muriel Barbery (Again, excellent at conveying emotions.)

-"Hunting and Gathering" by Anna Gavalda (This one is technically a romance - a genre which I personally would normally HATE - but it portrays such realistic characters, their struggles and their natural dialogue during fights that it actually felt more like I was reading about a collection of lives that I had the pleasure of spying on from above. I really love this book!)

-for WWI and WWII-themed titles, I'd recommend the Battlefield comics by Garth Ennis (He's SO good at writing believable characters and realistic dialogues.)

-if you don't mind high fantasy, any of the books in Terry Praychett's Discworld series about the wizards might be up your alley (You can read them independently without issue, or start from the beginning of any of the wizard titles. You can find a reading guide online! The wizards of his world are very regimented about how magic works - somewhat like mathematicians - and it's very funny.)

-the "Cemetery of Forgotten Books" series by Carlos Ruiz Zafón (I'd skip the 4th one - the main character/POV changed and I wasn't as impressed with the writing in that one - but the first 3 books are an absolute dream to read. The characters are so charming, lovable or completely horrifying, it feels like a wonderful foreign mystery series that takes place in 1940s Spain. It was really interesting to try to keep track of such a unique mystery amidst the second world war.)

I hope those help! Please enjoy your reading journey. ♡

hi, and thank you so much for the recommendations! I appreciate it a lot, those books sound really good

may I add some very red trains

red inside as well

If anyone wondered what do I mean when I'm saying our trains are "very green":

And yep inside they are also green:

(Usually not that nice looking though)

The ones in my area usually look like this ↑

(all pics are from Google images lol)

I have a possibly unusual question. since I left high school and was no longer being forced to read I completely stopped looking at fiction and other books that are not math textbooks. this is partly because I fucking hated reading as an activity – I have adhd and asd, so not only my attention regulation causes a lot of problems, I also struggle with visualizing what's happening and imagining/undestanding the intentions and mindsets of the characters. now, I am properly medicated so attention isn't as much of an issue anymore and I remember enjoying some of the books I've been forced to go through. I miss reading, overall it had a positive impact on my thinking process, I miss analyzing the human experience, so to speak

I liked Dostoyevsky, Vonnegut, Philip K. Dick, Huxley, Kafka and some other "philosophified" authors. I also enjoyed war-themed books in general such as all quiet on the western front, and everything related to the soviet union

finally, my question is, what would you recommend for someone with this taste, who also likes math¹, has a hard time with visual imagination and people but enjoys analyzing human experience, who would possibly like to see a represenation of themselves in the characters of the novel? I would love to rebuild my reading habits, but I am lost as to where to start so I don't get discouraged

¹ I included math in here because maybe there is a book about what it's like to be a mathematician (or a scientist of some kind) that give some existentialist vibes or something, if it exists I want to read it lol

13 III 2023

I remember putting it in my bio a while ago that I dream of doing actual research one day. well this is already happening, as I mentioned in some post, my advisor found an open question for me to write my thesis about

the progress for now is that I'm done with most of the reading I need to do to tackle it and I'm slowly moving forward with thinking of ideas for the solution (or at least a partial one)

this is what I want to do for the rest of my life: reading papers and trying to write my own ones

ofc I don't know if I manage to solve the problem or achieve anything at all with it but the process itself is fun

other than that I've been catching up with homeworks and assignments from work. fortunately I found an MIT lecture recordings for statistics so hopefully I might not die from boredom

watching probability and stats lectures from MIT has been my relationship's idea of netflix and chill for a while now, gotta cultivate the tradition

the algtop professor asked us to write down a full detailed solution for an exercises we did in class, because the person presenting was unable to explain it so I sent him mine

I don't know yet if it's correct but I'm pretty sure it is. I wrote this down partly because who doesn't want extra points and partly because I didn't have a chance to present it, the person who did was faster

I like how my life is right now, I want to keep it that way