Applied mathematicians been real quiet since this dropped
@aleph0540
9 ай бұрын
Rightfully so.
@peterg921
9 ай бұрын
I cackled
@GARIMITO
8 ай бұрын
I have a degree in applied math from 12 years. Now relearning abstract algebra so I can tackle this concept 😩
@zhikunli475
7 ай бұрын
Analysists being real quiet too
@TheAnonymmynona
9 ай бұрын
I love the "real worldness" of creating a new proof for an allready abstract theorem. Great video
@TosiakiS
9 ай бұрын
It's a funny thing because usually people don't usually talk about category theory and functors when people talk about this fixed point theorem and the fundamental group. It may only be mentioned after like "fun fact: fundamental groups are an example of a functor." That being said, it's true that this is the first instance where people get into the idea of relating one category of things (topological spaces) with another (groups).
@HolyAvgr
9 ай бұрын
@@TosiakiS doesnt that make sense, though? Functions ARE functors over the category of Sets.
@jfb-
9 ай бұрын
however ths theorem can then be used to prove the fundemental theorem of algebra, which is pretty useful in real world applications
@benyoung8450
9 ай бұрын
@@HolyAvgr no, functions are morphisms in the category of sets you don't have functors within a category unless its a category of categories
@robertjencks3679
9 ай бұрын
We just make it applied by using this in the proof that a solution exists for some really annoying applied Dif. Eq. Problem
@thephysicistcuber175
10 ай бұрын
I loved all the "Didn't graduate texts in mathematics" memes.
@jhuyt-
10 ай бұрын
I've been trying to learn some category theory recently (as it relates to loser topics such as functional programming) and I think that my state of mind learning this topic is accurately represented by the deep fried visuals of this video
@alextrebek5237
9 ай бұрын
Oh wise one, oh depraved one! Where do i go to learn category theory for functional programming also?
@benjaminpedersen9548
9 ай бұрын
I don't think that category theory in mathematics is a waste of time, but I am not so sure about programming.
@asdfghyter
9 ай бұрын
@@benjaminpedersen9548 it was useful for finding the correct abstractions to use in FP, but there’s no need for learning it to use these abstractions
@asdfghyter
9 ай бұрын
@@alextrebek5237 there’s a free online book called Category Theory for Programmers by Bartosz Milewski. if you want a more mathematical perspective, but still not requiring a masters in mathematics, you can try The Joy of Abstraction by Eugenia Cheng
@tricanico
9 ай бұрын
@@alextrebek5237 try Bartosz Milewski's blog. (Compiled into the book "category theory for programmers".)
@yahyasonic9655
9 ай бұрын
I do believe that math shitposting stands higher than all other kinds because the real joke is that it is never trying to be one
@adsoyad2607
9 ай бұрын
this is how mathematics was meant to be presented
@hambonesmithsonian8085
10 ай бұрын
My man I’ve been chasing what are essentially biblical angels in glowstick form for a topology thesis and I’ve landed on categorical methods, (also lots of knots and braids, if you know you know.) to explain the nonsense. For you to drop this video now feels like a call from the void. Good content.
@Mobiustransformation
9 ай бұрын
What do you mean by “essentially biblical angels”? I’m an undergrad btw
@TosiakiS
9 ай бұрын
@@Mobiustransformation Probably means requiring lots of background knowledge. Though, requiring some topology and abstract algebra isn't actually a lot to a self-studier, but enough to probably never even be covered in grad school math majors in formal education.
@AexisRai
9 ай бұрын
I am very interested in how a biblical angel might assume the form of a glowstick
@Teckiels24
9 ай бұрын
are you a chatbot? nothing in your comment makes sense.@@TosiakiS
@word6344
9 ай бұрын
@@AexisRaishouldn't be too hard to make rings/wheels with glowsticks and boom budget orphanim Bonus points if it's a fiery color like orange or warm red
@thatkindcoder7510
10 ай бұрын
Words cannot express how much I love this video. Bro just explained Category theory in a way that almost makes complete sense in the first watch (better than most others)
@zdmsr
8 ай бұрын
The kingdom hearts memes helped
@darkshoxx
10 ай бұрын
4:50 😆Cohen O. Mology 😆 Fun fact, Brouwer was a constructivist, meaning avoiding contraditction proofs at all cost. And he's most well known for his FPT 😅 Stan Grothendieck!
@asdfpoiuyify
9 ай бұрын
the juxtaposition of “esoteric academic topic” and extreme shit-post meme humor is exquisite. Like an abstract math version of @explosionsandfire. Plus also NL meme. Well done, subbed, pls make more videos.
@jesusvasquez4734
10 ай бұрын
Dude just one video? I was going to binge watch your channel, great job!
@__christopher__
Ай бұрын
Well, that makes binge watching the channel much easier, doesn't it?
@ducksies
10 ай бұрын
Category theory is useful when you're studying type systems
@jacobpaniagua8785
9 ай бұрын
I really vibe with this video and its sense of humor. Top notch content. Best new math video style I’ve seen in ages.
@filipnagy3535
10 ай бұрын
so the sole application of category theory is a slightly more annoying proof for brouwers fixed point theorem?
@TensorLiquidExp
5 ай бұрын
According to the undergrad who made the video yes. And he didn't even manage to do it right... the one application of CT he knew of
@ramongallardocampos5241
9 ай бұрын
Maaaan this is such a good video and your way of explaining things makes it real clear
@tylerduncan5908
10 ай бұрын
Did not expect the NL reference lmao
@KurtGodel-po3zl
10 ай бұрын
amazing video. This is high quality stuff. Thank you very much.
@ConnorMcCormick
9 ай бұрын
I was working on a problem that involved morphing objects into non objects and it was almost impossible to figure out how to come up with a constructivist example of a non object that was not an object and then boom this video drops. Thank you!
@nicolasoyarce9734
9 ай бұрын
This video should win the contest. Continue doing more
@Neubulae
10 ай бұрын
oh boi what kind of humor do you have i totally lost my shit, excellent job m8
@andrewfischer-garbutt2867
10 ай бұрын
Awesome video I feel like I understand something that always confused me! Also love the jokes so funny!
@TheoremsAndDreams
9 ай бұрын
You don’t have to stop at the disk (n=2). For n>2, the n-disk (or closed n-ball) and the (n-1)-sphere (which is the boundary of the n-disk) have the same fundamental group, i.e. the trivial group. But, that’s only the first homotopy group. Since the n-disk is contractible, all of its homotopy groups are trivial. For the (n-1)-sphere, the (n-1)st homotopy group is infinite cyclic (isomorphic to the integers). Homotopy-equivalent topological spaces have isomorphic homotopy groups. So, the deformation retract, which is a homotopy equivalence, tells us that the n-disk and the (n-1)-sphere have the same homotopy groups. But, they don’t, and that’s the contradiction.
@DiegoTuzzolo
10 ай бұрын
this is ur 1st video and I loved it. subscribed to see more 🤟🏻
@steliostoulis1875
10 ай бұрын
now this is a real math video
@rhodesmusicofficial
10 ай бұрын
MAN i haven't even finished the video, but your sense of humor is something i will cherish, this is the kind of math content im trying to watch 🔥🔥🔥 i already know abt the proof for banach fixed-point thm using fundamental groups so im interested to see how it connects to funny arrows theory edit: fundamental group = functor kek ngl this video does kinda feel like algebraic topology with category theory spray painted on, and i did kinda know abt pi1's functiness already, but still it's so kool to see a nerdy ass video abt "higher" mathematics with such a humorous and lighthearted tone, great stuff ima bouta watch and like the rest of ur videos rfn edit: just checked the description kek
@peterchindove7146
2 ай бұрын
Absolutely fantastic concise video. Thank you.
@2394098234509
Ай бұрын
Oh boy this will come in handy next time I'm holding a 2-dimensional disc in one hand and another 2-dimensional disc in my other hand and I need to know whether there's a fixed point on the two discs wrt a map.
@raph2550
9 ай бұрын
That's it! That's what math vulgarisation has been missing: memelords
@rupertsmixtapes812
10 ай бұрын
you like category theory and went exactly where my mind went with homotopy theory 10/10
@ablobofgarbage
9 ай бұрын
This is one of the most convoluted proofs by convolution i have ever seen
@trymbruset3868
2 ай бұрын
I never realised that "figuring out the fundamental group" is a functor. Colour me impressed I learned something!
@TheOneMaddin
9 ай бұрын
2:15, I think the ray should emanate from f(x) through x and then to the boundary. Otherwise the map is not continuous at the boundary.
@leandro8897
9 ай бұрын
Wat
@MikelMath
9 ай бұрын
My concern exactly! I was going to add a comment, but happy to see someone beat me to it. From f(x) to x then the the boundary.
@categorygrp
9 ай бұрын
I checked my copy of Hatcher and it's done from f(x) to x to the boundary, as you say, in there.
@categorygrp
9 ай бұрын
they actually have it corrected in the description
@OrenkoVivo
9 ай бұрын
underrated comment
@gravysnake78
9 ай бұрын
The first graduate level math video that didn’t make me fall asleep 🔥🔥🔥
@chenardpierre8270
9 ай бұрын
Great job. This is a very good introduction to Category theory which I have always found rebuttal.
@xovi4902
9 ай бұрын
I think it's dubious to claim that this is an application of category theory... maybe this is an uncommon way to teach it, but I recall that when I was in college, I saw this exact proof of Brouwer's f.p. thm. (in an algebraic topology course), which did not mention category theory whatsoever. Granted the fundamental group is a functor, but there isn't really any need to introduce it as such for this proof to work - you can understand the proof just fine by understanding that there's this concept of a fundamental group that lets you associate a group to any based (...or path-connected) topological space in a way that has some nice properties. This is the only part of the proof that can be categorized (or categorified? I forget the usual word for this) - category theory doesn't inform the construction of the g-map at all, and, correct me if I'm wrong, my assumption is that this proof (as well as algebraic topology in general) existed before category theory, and the only novel thing here is that the proof can be restated in a slightly different language. That being said, I love the style of the video.
@GNeulaender
7 ай бұрын
I think category theory was first introduced specifically to clarify some of the constructions on Algebraic Topology, such as the fundamental group and homology/cohomology. Yes, using it is most likely unnecessary for the proof and showing that the fundamental group is a functor, omitted here, is basically showing it to be an invariant under homotopy. Still, the proof becomes much simpler after the introduction of category theory, or just more clear, which is not to be discarded! When one starts working with more advanced topics in algebraic topology, algebraic geometry and others, this clarity makes a huge difference. In some sense, I think of it in a similar way as set theory: it isn't necessary to do Math, we have done so much before it was created, but it's madness to tackle more complicated subjects without it today.
@marufmamun6592
7 ай бұрын
Yeah. that pretty much sums up the usefulness of category theory for most mathematicians Lol The proof just relies on the fundamental group being homotopy invariant but the video is still hilarious 😄
@TensorLiquidExp
5 ай бұрын
A different perspective to look at an old thing definitely counts as application. Historically it's perhaps the greatest kind of applications that led to many conceptual revolutions. This is perhaps the difference between competent and incompetent thinkers. e.g. What can be done with Lagrangian formalism that can't be done by Newtonian mechanics? asked the incompetent thinkers in 18th century (nobody would ask such question now after it turned out this change of perspective was crucial to both GR and QM and particle physics!). Or remember how people had been calculating derivatives and integrals since before Newton introduced the new perspective of Calculus, or how Newton could use his version of Calculus to solve the variational problems, why bother with variational calculus? etc. Reminds me of Ronald Brown's essay groupoids.org.uk/pdffiles/eureka-meth1.pdf >“What can you prove with exterior algebra that you cannot prove without it?” >Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. >In my time, I have heard it repeated for random variables, Laurent Schwartz’ theory of distributions, ideles and Grothendieck’s schemes, to mention only a few. Personally I am just happy that the incompetent people didn't win (even though they probably did slow down progress by quite a bit)
@xovi4902
5 ай бұрын
@@TensorLiquidExp You're fighting against nobody here - it's certainly not mainstream opinion yet, but most (esp. younger) working mathematicians agree that Grothendieck was right about the importance of categories. This doesn't change the fact that it is bordering on dishonest to put the snake lemma on the thumbnail of a video and claim something is an "application of category theory" (when the word "application" in this context usually refers to a cutting-edge contribution to another field), when it turns out that the video just amounts to "we can take this well-known proof and restate a single part of it slightly". This isn't even the only significant bit of undergrad mathematics that can be restated in the language of categories.
@freyc1
28 күн бұрын
@@TensorLiquidExp Nobody sane really asked that question about analytical mechanic because you could actually do many new things with it, contrary to what the incompetent historian of physics will say.
@petrospaulos7736
9 ай бұрын
I love the fact that facts are loved!
@aaron3157
10 ай бұрын
2:20 i had to pause, that absolutely killed me
@Sara-cy3wo
Ай бұрын
Wow I am surprised this made so much more sense than a course I took on algebraic topology 😊
@zaidtayeh5899
10 ай бұрын
Great job homie
@TheRiverNyle
10 ай бұрын
This is one of those goosebump-giving videos bro, fascinating af
@eduardodecastro8858
9 ай бұрын
Loved the content keep it up!
@VishnuNittoor
10 ай бұрын
Awesome job!!!
@CreativeMathProblems
10 ай бұрын
I am wanted in 47 countries for various war crimes
@samocali
10 ай бұрын
@@CreativeMathProblems rookie numbers
@Alceste_
9 ай бұрын
From afar this looked like an educative video and it turned out to be some of the funniest stuff I've seen in a wall (tho for a targeted audience, I'll admit -- people that at least reached high school I'd say).
@profesorjan7614
10 ай бұрын
Love the humor !!
@PhantomKING113
9 ай бұрын
I like your funny words, magic man.
@valkharitonov9346
9 ай бұрын
Speaking of fixed point theorems, Lawvere's fixed point theorem is another neat application to mathematical logic and computability theory
@p3xo
9 ай бұрын
i have no idea what any of this means but i’ll keep watching bc it makes me feel smart
@zaitzerzazza2830
10 ай бұрын
Amazing video!
@doraemon402
10 ай бұрын
It's 5 AM, haven't slept all night and this made, not my day, my life
@MrCreeper20k
10 ай бұрын
I thought the "homo / topped" joke was funny though it feels a little too spicy to include in a math video submitted to a competition. I can feel the stress of the judges deciding if its even okay to consider this video lol. The math was real interesting though! Maybe a little jargon-heavy but I think thats more of a personal skill issue since explaining all terms to satisfaction would probably take a while.
@jkid1134
10 ай бұрын
A very jocular tone throughout, and at a breakneck speed, and in tandem they sort of transform it into entertainment instead of education (I for one am not sure I could reproduce even a sketch of the proof I just watched). Well, that was my experience with it, anyway, but then again, even in the video he pokes fun at the fact he's breezing over things you've never heard of. So yeah, an odd submission to be sure.
@MrCreeper20k
10 ай бұрын
@@jkid1134 Ooh jocular is the perfect word. Haven't heard that in a while. I am going to steal it back into my vocabulary now lol.
@OutbackCatgirl
10 ай бұрын
ehh, it's not at all spicy, as a homo top
@sheilbanerjee997
10 ай бұрын
Damn this insightful and intelligent af! I love maths!
@lwmarti
7 күн бұрын
Q: What do you call someone reading a category theory paper? A: A co-author.
@ranggakd
9 ай бұрын
I have no idea what you are talking but im listening while eating my lunch
@eternaldoorman5228
9 ай бұрын
This is hilarious. The punchline especially, ... Brouwer would be laughing his head off, ... Which reminds me of a dream I had last night about someone being who was run over by something much bigger than a taxi, ...
@donovanholm
10 ай бұрын
Thank you for this.
@phthalo7401
9 ай бұрын
Sometimes I think about switching majors, but now I gotta persist for Homotopy theory.
@TheRiverNyle
10 ай бұрын
Amazing job bro! I suppose a new and growing field, Topological Data Science (TDS), must have a lot of applications from Category Theory as well Edit: for those who didn’t catch on, it doesn’t
@aleph0540
9 ай бұрын
Data Science ticks me off. They straight up just steal concepts and nomenclature from other fields and rebrand.
@TensorLiquidExp
5 ай бұрын
A growing field centered around homology? CT probably does appear a lot for competent mathematicians. Of course if you keep looking at the gymnastics category isn't designed for you wont find it. You can probably avoid categorical view completely for every concept and make your homology computations 200 pages longer, that is very possible too! probably get paid more as a data scientist as your boss appreciates how much work you put in :D
@mathemelo
8 ай бұрын
I love the vibe of this video. I'm more into analysis myself but this is pretty much how I view maths as well, through a constant prism of spicy memes
@keithpeterson4005
9 ай бұрын
Thanks for the free advertisement on categories.
@avi123
9 ай бұрын
a monad is a monoid in the category of endofunctors.
@DavidConnerCodeaholic
9 ай бұрын
Is Zorn’s lemma apropos to finding (or not finding) a specific fixpoint? I donno. I hear “equivalence classes” and “can’t find it specifically”, and Zorn comes to mind … though I’m not sure if you implied the second part.
@lakshaymd
9 ай бұрын
This is an amazing shitpost. 8:48 would be the perfect thumbnail for this. Subscribed.
@landy4497
10 ай бұрын
loved the humour
@Skyb0rg
9 ай бұрын
Great video! I’m expecting the follow-up video that expands on the “non-constructive” aspect, because explaining how to type this proof in HoTT (“there exists” vs “there merely exists”) is pretty related to this proof, since that requires you to define the topological space of proofs up to homotopy.
@jjtt
10 ай бұрын
nice video bro, one small problem tho, a well known theorem i pulled from between my asscheeks states: no diagram chasing = not real category theory (everything flew over my head as i'm a mere functional programmer loser)
@peterchindove7146
2 ай бұрын
You could have actually mentioned that the vector gauge field can be understood as an element of bhTop and that this element explains phase changes in charged in particles in the absence of a magnetic field...but hey, what does an applied mathematician know?
@df-163
7 күн бұрын
Why is the function g surjective on the unit circle? I didn't quite understand the reasoning for this
@bowiebrewster6266
9 ай бұрын
I have never heard so aggresively “call it F for function” 👺
@jacobleider4591
10 ай бұрын
Oy, this is the new wave, isn’t it
@micayahritchie7158
9 ай бұрын
Wait I've seen this one. It's a classic. (This exact arguement was in my topology class for the fixed point theorem)
@andleepfarooqui7874
10 ай бұрын
My favorite proof of Brouwer’s fixed point theorem is the Sperner’s Lemma one~
@masonskiekonto590
9 ай бұрын
99% of the proofs in abstract maths seem to shove away complexity in calculations taking us from start to finish to sheer mental gymnastics needed to string together correct observations about properties of the system at any given time. It feels like you need whole courses to understand simple derivations because each step seems like pure nonsense that came from nowhere.
@Rahat2056
9 ай бұрын
Genuine question, what would be the sequence of maths one needa to be verses in to understand this video? Im a stem major and I've only needed Calculus, Diff eq, linear alg, and very briefly glossed over some basic topics in discrete math.
@DarGViD
9 ай бұрын
Depending on how deep you want to understand the video of course. In principle, General Topology + Group Theory should be enough. If you're familiar with them, you can study some basic Algebraic Topology like the homotopy stuff in the video. Category Theory is a little different. Theoretically, you don't need to know any prior maths to understand it. So you can just try to read/watch an introduction to Category Theory and be done. The problem is, Category Theory usually needs examples to be well understood. So if you haven't taken any Sets Theory/Topology/Group Theory/Ring Theory/Diff. Geometry, then Category Theory would look very bizarre and dry. Imagine studying integrals and derivatives, and proving theorems about them without being familiar with polynomials, sines and exponentials.
@christopherellis2663
9 ай бұрын
Which verification of the Nonsense Factor in Applied Gibberish relieves me of all further care.😅 Instinct says, "Don't go down this path 🤔
@adrianhindes3537
9 ай бұрын
This was the most metal fucking proof of a fixed point theorem I've ever seen.
@OniSMBZ
9 ай бұрын
I was just about to say ''okay now again but constructive this time.'' Category theory mind trick?
@alanknguyen
9 ай бұрын
A true classic!
@physira7551
10 ай бұрын
The homotopy part was great
@megaing1322
9 ай бұрын
Is it just my headphones messing up or are the audio levels in the video all over the place? Otherwise, amazing video.
@MegaGuiaGamer
10 ай бұрын
I laugh so fking hard, you're awesome
@IronicHavoc
9 ай бұрын
In my experience, "How to convince yourself category theory is useful" was starting to learn Functional Programming
@nameq
7 ай бұрын
cool! how did you convince yourself fp was useful ?
@nirinarabeson
10 ай бұрын
this is some based maths for #SoME3 💯
@canaDavid1
9 ай бұрын
Important clarification: the disc must be closed. Else it admits a translation such as "move every point to the midpoint of it and a specific point on the perimeter"
@newwaveinfantry8362
2 ай бұрын
Brower's fixed point theorem doesn't necessitate the concept of a category and predates it.
@teemuaho4807
10 ай бұрын
3:08 listen you cant just drop a ryab on us like that you gotta give a content warning at least
@Red-Brick-Dream
9 ай бұрын
The Springer text meme is just 🤌
@kaloka521
10 ай бұрын
Assuming the Disc is in R^2 im fairly certain this can be shown using only analysis and some very basic topology (R^2 cuz that way path-connected iff connected, which makes it easier)
@zestyorangez
10 ай бұрын
I remember seeing a proof of the non-existence of a map of the disk to s1 whose restriction to s1 is homotopic to the id map on s1 in one of my textbooks and i think it was done essentially how it was done in the video just without any references to category theory.
@nicolasoyarce9734
9 ай бұрын
There are a lot of proofs of this theorem from different perspectives, i think that in the book of Evans PDE, in chapter 8 theres a more analytic proof
@georgH
9 ай бұрын
I thought it would be a video about Haskell :D
@DontWatchWhileHigh
8 ай бұрын
Top G approved
@coreylapinas1000
9 ай бұрын
Me when the algorithm recommends a higher maths video: "Huh, so its like that. I understand everything now" [doesn't get it at all]
@backhdlp
10 ай бұрын
3:16 Nooooooooooooooooo (literally crying right now)
@pedrohenriquesacramentodeo3458
10 ай бұрын
I believe you might have commited a small mistake when describing function g. Namely, for the disk to map to itself and g to be continuous, we must have the ray go from f(x) to x, not the other way around as it appears in your imagine at kzitem.info/news/bejne/r6CumJmacYR4nKg.
@GeometricSquirtle
10 ай бұрын
math.stackexchange.com/questions/803499/continuous-function-on-closed-unit-ball convince yourself you can replace x with f(x)
@pedrohenriquesacramentodeo3458
10 ай бұрын
@@GeometricSquirtle The link uses the function I described. Maybe there is some confusion between us. The function created by considering the ray from x to f(x) is a continuous map from the circle to the circumference, it just isn't necessarily the identity at the circumference. Say f(x)=-x for example, then r(x)=-x at the circumference.
@GeometricSquirtle
10 ай бұрын
@@pedrohenriquesacramentodeo3458 upon further review, you are correct. You get a gold star after class
@MattHudsonAtx
9 ай бұрын
Programmer got scammed! Great video. I might still learn me a Haskell one day.
@philkaw
10 ай бұрын
Basedest math video of the year
@alextrebek5237
9 ай бұрын
How do i apply this to programming tho mother- 🤬
@Rahat2056
9 ай бұрын
I enjoyed watching this video. It made my brain hurt. I wish I understood it better because it seems cool. Mathematics is something else 😭
@rogergalindo7318
9 ай бұрын
haskell boi sad because no monad is a monoid in category of endofunctors (after this i understand less what a functor is)
@pomtubes1205
9 ай бұрын
A monad is a monoid in the category of endofunctors.
@pillmuncher67
9 ай бұрын
What's the problem?
@georgebabus2030
9 ай бұрын
🔥🔥🔥🔥
@BramCohen
9 ай бұрын
Not actually understanding category theory I *think* this proof extends one dimension higher because a ball also only has one element of its fundamental group but a sphere has two. Not sure about higher dimensions. It hints at a constructive proof because you can take two different elements of the fundamental group and attempt to map backwards from all the deformations from one to the other.
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