Jean-Pierre Serre's 100th birthday editionprevious:>>16991839
>>17054862I tried to read his book Linear Representations of Finite Groups but it was very difficult because of all the errors
>>17054917The dude is a better mathematician than textbook writer
>NOOOOOOO YOU WERE ONLY SUPPOSED TO GROW SMARTER AND THEN PLATEAU AT A LEVEL JUST BELOW ME.
There was a panel about AI in math>Michael Harris>Peter Scholze>Jacob Tsimerman>Geordie WilliamsonModerated by some DeepMind guyhttps://www.youtube.com/watch?v=H7_d_sgui6o
>>17055513lol, at 10 minutes Scholze says he has never typed anything into an LLM and doesn't use a smartphont. at 11 minutes Michael Harris says LLMs are theft and should be illegal.
>>17055556I mean he's not completely wrong about LLMs. All the top AI companies have ignored every intellectual property and copyright law in existence and then left it to the lawyers to deal with the consequences, if any.
>>17055513This was good. It's charming in a way how aloof Scholze is. Tsimerman is the most clear minded of the group.
>>17055556>Scholze says he has never typed anything into an LLM he's a liar
>>17054862I'd ask this in a physics gen if we had one but alas this board is too full of opinionated schizos to sustain one so I apologize for sloppiness.Consider L^2 or a sufficiently well behaved subspace thereof. The Fourier transform can be expressed as a linear, unitary operator in this space:[eqn]\psi(x) = \hat{\mathcal{F}} \phi(x) = \int_{-\infty}^{\infty} \frac{\exp(-ikx)}{\sqrt{2\pi}}\phi(k) dk[/eqn][math]\hat{\mathcal{F}}^4 = 1 [/math] makes its powers isomorphic to the cyclic group [math]C_4[/math], and it rotates the operators x and p=-id/dx as a 90° rotation. Now the question: obviously (taking the rotation property of (x,p) as the defining feature) a group of operators isomorphic to SO(2) can be identified. What is its infinitesimal generator? Another way of phrasing the same question is, what is the phase space angular momentum operator A such that [math]\hat{\mathcal{F}} = e^{-i\frac{\pi}{2}\hat{A}}[/math]? A sloppy guess would be x id/dp - p id/dx = x^2 + d/dx^2 which would be the quantum HO for omega = hbar = m = 1 (which is a Hermitian operator that commutes with the fourier transform) but I didn't do it properly yet.
>>17055761there's sign errors in there I didn't fix but you get the general idea.
>>17055761>>17055762Nailed it, see Wikipedia, eigenfunctions of F
should I apply to university to learn maths or can I self-learn with chatbots?
>>17056462university, not even a question. You should be around people with the same basic interests and ideally a vast breath of different experiences in your most formative years. Even if a chatbot could replace curriculum (lmfao) it cannot help you grow as a person or intellectually challenge you.
>>17056462the fact that a chatbot will occasionally sperg out and get something completely incorrect with utmost confidence should be enough to tell you it's not a good idea to study material you don't know under one
>>17056462You need to build some fluency yourself or you won't be able to tell when AI is feeding you bullshit as other anons said. I do think it can be an extremely powerful tool for learning though as a supplement to traditional education.
>>17054862Ed Witten's father is 105, apparently.
>>17057041>only 5 years younger than general relativity>last published August 2025fucking hell
This guy was doing ads for OpenAI like 4 months agohttps://x.com/haider1/status/2101368221651869962
In physics one important measure of understanding is the richness of connections you can draw and the number of lenses you can see a problem through. Mathematics always seemed similar in that regard but one thing always trips me up. You guys seem to be bored when something has a lot of properties. Your eyes seem to glaze over when a space is Hilbert, a manifold is Riemannian, etc. I get that it's an old hat and chances are everything worth saying has been said about them. But the same is true for things like the harmonic oscillator or other toy problems and physicists seem to enjoy hearing new or interesting interpretations of these sorts of things. What am I missing?
>>17058496Interesting question. I think you're comparing apples to oranges. Mathematical structures like Hilbert spaces, Riemannian manifolds, etc. were developed as a way of formalizing certain widely recurring mathematical motifs: a vector space with a notion of orthogonality, a locally Euclidean space with invariant local angles/distances, etc. It's not a surprise when they appear all over the place because that's pretty much what they were designed to do. A more accurate physical analogy I think would be more general concepts like "phase space" or "entropy" rather than specific physical theories (however simple). Or conversely, a more accurate mathematical analog to something like the harmonic oscillator might be, say, I don't know, the Gaussian.
>>17058652Very interesting perspective! The Gaussian is a peculiar case in general, stupid amount of properties (central limit theorem, fourier transform, ...). If memory serves the N-dimensional analogue is the only correlation-free isotropic random distribution ([math] \rho(\mathbf{x}) = \prod_j \rho(x_j) = \rho(|\mathbf{x}|) [/math]), among other things. You're probably right I am not making as much sense as I would like to, I'm a complete outsider to mathematics as a field. At its core I would love to better understand the kind of guiding principles and mental images mathematicians draw from, I think.
>>17057722professor ching chong says "there is no reason to be this fast"lmaoprofesor bing bong there pretending not to understand power dynamics of economies at all.>gwailo... we must srow down AI
What are some hidden math? Also how do you get into philosophy of math?
Fuck my life
MATHIE ARE YOU OKAY
>Thales Pythagoras Eudoxus Plato Aristotle Euclid Archimedes Apollonius Hipparchus Ptolemy Al-Battani Al-Biruni al-Tusi Descartes Fermat Newton/Leibniz Euler Gauss Lobachevsky/Bolyai Riemann Hilbert.This progressive learning of math and logic makes sense?
>>17059313yep...
>>17059313I like going to r/math because they have one AI containment thread
>>17055556yeah just like how the mnist data set should be illegalall these workers who contributed to writing digits by handHumanity should be illegal too seeing as how we stolen from billions of years of trial and error of other organisms
>>17061100Is it *universally* applicable? Suppose you manage to do this (people have for more or less achieved exactly that for artworks)>take a copyrighted work without permission>train slop with it>prompt slop to produce copyrighted work>take output and use it without creditIs it in the spirit of the law? The only logical consequence is to abolish copyright if so.
>>17056462If you only want to learn mathematics (no intention to make a carrier out of it) then buy textbooks and work through them. Only use AI after you've done the exercises and want a feedback on it.
Stupid question perhaps but: doesn't it already require the axiom of choice to pick a representative from an infinite quotient set? I would suppose you don't always have to (see for example the standard construction of the rational numbers from Z^2, since you can just pick the element with the smallest nonnegative denominator) but I wouldn't be able to give a general answer.
>>17062577a representative for each element in an infinite quotient set I mean.
>>17062577Depends, what do you need to pick a representative for? If you just want to construct the quotient, as long as you are able to describe the "classes", each object of the resulting quotient set can be taken to be the entire "class" iirc, no?
>>17062579I am not a logician myself, but the answer is yes. The statement "Every quotient map [math]X \to X/\sim[/math] has a section" is indeed equivalent to the axiom of choice.However, sometimes the equivalence relation has a "canonical" representative which does not need the choice principle. Very often this is well-orderedness. Your example of [math] \mathbb{Z}^2 [/math] has a clear canonical one namely the reduced fraction.
>>17055556lmfao and those people are supposed to be smart?
>>17055761ask chatgpt
>>17055556>Scholze says he has never typed anything into an LLM and doesn't use a smartphontbased, I envy him
I dont understand what a vector bundle is
>>17063997Locally every point of the bundle looks like the space **times** a vector space. Then there are some other topological formal conditions.The easiest way is to just look at a single point (even though the formal definition uses neigborhourhoods), then a vector bundle assigns to each point on the topological space a vector space. If you are familiar with manifolds you already know the most important example, namely the tangent space. A vector bundle is simply a ""generalization"" of this idea, where instead of the tangent space we attach some other vector space to the base space at each point.
>have to try my half-baked conjectures in le meme counterexample generator before I put them in a paper just to make sure I don't get "scooped" by some biostatisticianI can't wait until normies stop pretending to care about math again.
>>17063974based
>>17054862
I'm beginning to I won't know anything substantial about Topology from this undergraduate topology course.
Any recs to quickly learn the basic concepts of logic? First order of predicate logic mostly but my professor is already on second order and I didn't learn shit in bachelor so I'm behind. Any lectures, textbooks, etc. would be massively appreciated.
>>17066865What have they taught thus far?
>>17066865Intro topology is designed for analysists just like intro calculus is designed for engineers. If you were expecting to learn about coffee cups and donuts then you'll probably have to wait for the next level course.
that thing you're working on?it's already solved
>>17068390Based. I never get why people feel bad that AI solved it instead of people. If anything, it's a sign that AI can do maths, thus accelerate scientific progress and make our lives better (or worse, but it's worth the risk). All the best to OpenAI.
>>17068370Man I wish, I would have loved to learn about uniform structures in undergrad. On a more serious note, intro topology is just intro topology the only "analysis" part is the connection with metric convergence. Manifolds for example are as important in topology as in analysis.Of course you can tailor it more towards analysis but that just depends on the lecturer.For example, I was taught the subspace topology using commutative diagrams, because my professor was a higher cohomology theory tard.
>>17068370>>17068710These days, would be a damn miracle to find an undergrad topology course not oriented towards analysis-prep.
>>17055435All this and for what?
It's overhttps://raw.githubusercontent.com/openai/math/main/overview.pdf
>>17055556>Michael Harris says LLMs are theftbased
>>17068828Yes, the theft is based lol
I left grad school 9 years ago, why haven't you given up on maths and pursued something that will actually bring you success and money instead? read the room, you are no longer required
>Quasi-Riemann hypothesis:Excludes zeta and Dirichlet L-function zeros beyond 7/8. Major progress, although it does not prove full RH.>Hodge conjecture for CM abelian varieties:A major special case, with claimed consequences including Tate for abelian varieties over finite fields.>Hilbert’s tenth problem over \mathbb Q:Settles whether an algorithm can decide if arbitrary polynomial equations have rational solutions.>Unique Games Conjecture:Resolves a central question about how accurately hard optimization problems can be approximated.>Nonsofic groups and Kaplansky counterexamples:Produces groups contradicting several foundational expectations in algebra.>Free group factor isomorphism problem:Claims these fundamental operator algebras are all isomorphic.>Both Mahler conjectures:Settles central convex-geometry questions in every dimension.>Global smoothness of relativistic Vlasov–Maxwell Resolves a major mathematical problem governing plasma dynamics.ummmm math bros?
>AI spergs elected to flood a slow general, begging for attention because all their frog threads have a dozen posts or less>p-pls entertain me /sci/ I am so lonelyI don't think I will. Post math.
can anyone with access reup this shit https://link.springer.com/article/10.1134/S0081543824020111
Not sure where else to ask this, as I'm afraid that bringing it up candidly to my professors may (possibly subconsciously) cause them to write me a less-emphatic recommendation. Lately it seems like all published math will be divided between pre-2026 and post-2026, with post-2026 work essentially having an asterisk by the authors' names to indicate that they should be given significantly less respect/prestige. I'm a math major who's currently applying to graduate school, and am coming to grips with the fact that this would really bother me. Everyone has their own motivations, but personally I am mostly driven by competitiveness and the desire to do difficult things that impress peers (including myself). This, when typed out, certainly comes off as the "wrong reason" to do something, but from what I understand they are similar to the reasons that many mathematicians were drawn to the field.Am I overreacting? Should I bail while I can? Just do a phd since that's what I'm set up for, and spend that time planning my pivot?I know those are impossible questions for a stranger to answer, so I guess I'm mostly just looking to see if anyone feels similarly (or would granted the same facts, but disagrees with my premise).
>>17069311Just bail now, the glory of having citation metrics and stuff is all over, which may be a good thing for maths but not for your motivation. No idea where you should go though, maybe become a tech bro.
>>17069311You forget a crucial point about academia: people talk. I know of professors who have a radiant H-index by the standards of their community who, outside of the very well known clique of people citing each other in a circle, are not as highly regarded as their numbers would suggest.
>>17069311You should be ashamed that these are your motivations.
>>17069320NTA but why exactly? It's natural for a human to feel that way. Do you think scientists and mathematicians should somehow be superhuman? Why do you expect anyone to meet such a standard? Do you simply hate others?
How is Arnold's ODE for a first into to DE. I am a math grad but I have never studied applied math or physics except for CS.
>>17069352Well I certainly hate you. How about you tell me where you are right now and we can "compete" for who's going to be the least raped. I think I already know the answer.
Some guys did a panel after the 400 openai proofs droppedhttps://www.youtube.com/watch?v=voDtZqijetw
>>17069511tl;dr? or tl:dl?
>Math before>You are poor but do it for the love of the game>Math now>You are poor but AI sucks the fun out of everything I’m tired boss. Should have done quant or shit years ago.
>>17069539Literally how does AI suck the fun out of anything?
>>17069565Some of us actually enjoy solving problems ourselves. It's a triple digit IQ thing, you wouldn't get it.
>>17069572And how exactly does AI stop you from doing that?
>>17069587Becuase a big part of the appeal is solving something no one else has done before.Even if it's a trivial problem that's been over looked.Now you can just say Computer solve for X.
>>17069311You've picked about the worst possible place to ask a question like this, the schizoid AI doomerism you see here really is not representative of people actually in the field.>Lately it seems like all published math will be divided between pre-2026 and post-2026, with post-2026 work essentially having an asterisk by the authors' names to indicate that they should be given significantly less respect/prestige.You were right to not bring this up to your letter-writers, because it's completely retarded, with all due respect. You're projecting a degree of your own prestige-hunting onto the field that it simply doesn't have.>Everyone has their own motivations, but personally I am mostly driven by competitiveness and the desire to do difficult things that impress peers (including myself).At least you're honest with it, I suppose. Even before AI there's no glory in math. Other mathematicians aren't going to bask in your radiance because you published in Annals once. If your ultimate goal is to just be a BMOC then I'd suggest becoming the resident mathfag at some tech startup or something.
>>17069593 You could always work on something no one has ever solved or will ever care about. But you're a narcissist, so you'd rather solve something that gets you status and recognition than actually engage with a problem out of curiosity or for the sake of science, technology and philosophy.
>>17069646>You could always work on something no one has ever solved or will ever care aboutliterally the vast bulk of "vibemathed" lmfao
>OpenAI dumps a bunch of papers, with about 1 out of every 7 including a lean proof>Lean proof is a mechanical translation of natural language proof the model arrived atSo, how does anyone verify that the translation is accurate, and that the assumptions the proof was derived from are the correct ones for framing the problem? I'm not a mathematician, but even as a code monkey I find myself needing to fix AI hallucinations at work.
>>17069698In principle, the whole idea with Lean is that you only need to check that the initial statement is correct, which is orders of magnitude easier than constructing a proof or even checking the proof yourself, formalizing each step with a library of existing theorems is what it is designed to do. It is not remotely foolproof and bugs have been found that allow you to "prove" any false statement. People say they've been fixed but of course an LLM is exactly the software you would dream up to find and exploit new ones when incentivized to produce a Lean cert.
Saw the OpenAI announcement today. I'm not smart enough to know if this is a big deal or not. It seems like a big deal. Can someone fill me in?
>>17069717It's a big deal but no one will know exactly how big for months while experts check their claims. We also have no reason to believe their claims about how much compute it took or how many problems they failed to solve, which has huge implications for how impressive the results are.
>>17069511>mfw being filmed without consent
>>17069515>math isn't about results, its about math frameworks reee
>>17069515>this is fantastic! congrats to openai
>>17069316>>17069318>>17069320>>17069599This is literally just a copied post from /r/math. What happened to the quality of this general. Go to proofsandprompts if youre this fucking retarded.
>>17070020> What happened to the quality of this general. People foolishly assume good faith posts while the board is being raided.
>>17055556>Michael Harris says LLMs are theft and should be illegal.how is he wrong? at least in how LLMs are being used now and not some theoretical fantasy world where these companies actually respected copyrights, they are theft
>>17070072nta but you're fucking retarded. Training is not plagiarism and this has been found to be the case in even the very first frivolous lawsuits from years ago. It's only in some smaller cases in European countries that rulings against AI corporations occurred, and even in those it's exceptions like song lyrics where even memorization could be considered reproduction.Calling it plagiarism let alone "theft" as if legal fact is braindead. Plagiarism concerns misrepresenting another person's work or ideas as one's own. The AI cases concern statutory rights of reproduction, distribution, and derivative use; you're not seeing those models claiming to have proved unramified GLC, it's just all-around fucking retarded and legally illiterate
>>17069572It's a double-digit IQ thing to doomerpost about AI on 4chan instead of doing math.
>>17070100Yeah, I don't really care whatever extremely stupid precedent they set says. I don't think many would argue Dodge v. Ford was a good ruling, even though it is the basis of corporate America. In hindsight it fucking sucks and we all know it. Enjoy your world where no one wants to write books anymore because they can just be trained on or fed into a prompt and reworded to the point they can be considered transformative and then resold.
>>17069717Yeah it's absolutely a big deal, nobody except schizo undergrads on 4chan are denying this. What is legitimately a problem is that the repo just contains the results and very terse proof papers, and none of the research artifacts the agents created. And just from my personal experience with GPT ever since getting access to a free Pro subscription through my institution, it really does feel very obtuse at times. Like thinking logs aren't showing up (probably to prevent distillation) and some replies read like what you'd see in the Encyclopedia of Mathematical Sciences. Just really terse and difficult to follow; you can kind of prompt yourself to better explanations but more human-readable output generally seems to be an issue right now. They're undeniably capable though.
>>17070140Well then just admit to being a lowly shitposting activist and don't obscure it behind faux legal reasoning to pretend your immature seethe has any valid basis. It's also not just about one precedent or one legal system, the training argument holds up in most courts around the world for a good reason. >fed into a prompt and reworded to the point they can be considered transformative and then resoldYou know that there are thousands of trash, derivative textbooks out there, hell with the dawn of the internet it was common practice of textbook authors to copy sections from Wikipedia and edit the Wiki article before publishing of the book. Spend any time at university and you'll go through several trash courses where the prof is shilling his shitty botched-together textbook.If anything, chatlogs with these AI services are subpoenable material. AI doesn't create a copyright loophole. You just do not understand what training means and what copyright infringement is. You don't have copyright on a specific explanation or a proof. People will create slop textbooks at a greater scale thanks to AI, so what? Just don't read it. Stop being such an alarmist depressed bitch lmfao
>>17070183>You know that there are thousands of trash, derivative textbooks out thereCool. Scams exist therefore ...let's make it super trivial for scammers to make more? What a compelling argument. I've yet to see a single good use out of this technology that wasn't either some throw away novelty, or automating away deep intellectual pursuits or just a tool for straight up fraud like indian scam ads and impersonations. But it's all okay because 60 year old boomers who can't navigate a smartphone said so?AI creates all kinds of chaos and violates trust and base core assumptions we have held for decades. Everyone who made software since nearly its inception operated under the assumption that a compiled binary was a safe way to protect their source code because the amount of work required to reverse engineer it would be many years worth of tedious labor. That assumption no longer holds and has extreme ramifications. Carefully constructed legal agreements over decades between the corporate world and FOSS developers are being violated in very serious ways. For example, a programmer who pours years of work into his software and licenses it, even with a cuck license, usually expects attribution as per the license. The attribution being at least part of the reason they are willing to be so permissive with it. But the AI slopper can hit the copy button, shit out a rust rewrite of the same fucking program in an afternoon and pretend they made it, bypassing attribution entirely while hiding behind the plausible deniability of the soup of the dataset.Anyone who thinks being able to build training data on works they do not own is either complicit in the fraud or a retard that has not even thought long about any of the consequences.
>>17068454but can the AI prove the result?
>>17070210Let me name a few more.Everyone who is losing their job to AI is having an existential crisis. No one stands to gain anything at all except the ultrawealthy. It completely undoes the benefits of the industrial revolution by devaluing the ingenuity of man. Many are losing desire to do the things they once loved. Mathematicians look like they're about to start jumping out of buildings. They dedicated their entire lives to a pursuit of obscure and arcane knowledge, regardless of whether they did it for prestige or whatever, they deliberately chose the harder path when they could have gone into finance and made fuckloads, which speaks to how devastating this is for them and their purpose.FOSS developers are closing themselves off from the world to combat the mountains of AI generated pull requests. It is killing the last way juniors were getting trained; through the mentorship of these developers and the communal pursuit of building open source software. FOSS itself is on its deathbed. The open internet philosophy following after universities (its birthplace) is on its deathbed.Oh, and do you like competitive video games? Well, you're not going to for long. Cheaters will now be able to reverse engineer anti-cheats and figure out all of the ways cheats are detected from the perspective of the client, meaning the only way anti-cheats will be able to operate will be from the server side looking at the data the client is sending. This used to always be a mixed approach but a lot more power is now going to shift in favor of the cheater.Developers are not going to tolerate their sourcecode being decompiled so they're going to shift towards cloud gaming.There are just nearly unlimited ways this fucks up literally everything if you just sit down and think. How exactly does this help the world become a better place? Ignoring the grandiose claims about curing cancer with puppy dogs and rainbows, everything is just getting worse around us.
No of course I don't like "competitive" video games fuck off
>>17070224>pick the weakest claim of the group and discard it outright without even trying to refute it while ignoring every other argumentI forget which fallacy this is, care to remind me?
>>17070227I don't care about Reddit nonsense like logical fallacies I just hate competition go do one.
Listen I'm sorry, I actually agree with the thrust of your argument and shouldn't have said that. I've been having 30 years of bad days in a row and decided to take it out on an innocent person just because they used a word I dislike.
>>17059588Missing Tarski.
Some people may be retarded some of tbe time, but all peoplevwon't be retarded all the time. We're entering a new era, but it will be fine.
This might be a longshot, but is there a reason why there is not much discussion (online) around the Baum-Connes conjecture disproof? I am not in C^* algebras but a related field and I thought it was a pretty important conjecture.
>>17070262A new era of machine assisted retardation.
Three, six nine.Twelve, fifteen, eighteen.Twenty-one, twenty-four, twenty-seven.Thirty.
>>17070321It just hasn't been a very hot field for a while. There were some poeple working on it in my department, but they were usually looked at us dying breed.
>>17054917Math is for midwits who specialize in only one field.
>>17055513AI > Maths
Maths is the most useless thing there is.
If you haven't been cited by an AGI, you're basically a non-mathematician.https://citedbyagi.com/
>>17070862It's funny because it's mostly true. Math is both the most important thing for any civilization to progress beyond sticks and stones and it's also one of the least important things currently because there's more maths than ever in history of mankind yet very little genuine, major progress, if we exclude AI that is, humanity's only hope.
I wonder what all the tourists will do when the hype ends and they need to find a new personality
I wonder what all the buckbroken fake mathtards will do when AI conquers the galaxy and they need to find a new personality
>>17070969Math is not the most important thing to the development of civilization at all. Unless you consider engineering maths, but you really shouldn't because engineering doesn't required any proofs to work, only precise calculations. Pure mathematics is completely useless, a logical game, perhaps the most beautiful game ever invented by men, but still just a game.
>>17071344What a load of nonsene. Science and engineering both rely heavily on mathematics, including the proofs present within. Mathematics was vital element for progress ever since Mesopotamia and Ancient Egypt, it was used for taxation, trade, calendars, construction etc. Even proof-heavy maths and abatract algebra have found their uses. Without boolean algebra, computability theory and number theory there would be no digital computers. Without quantum mechanics and linear algebra (Hillbert spaces, operator theory) also Schrodinger wave equation, there would be no lasers and semiconductor chips. Even category theory has its practical uses lol. I'm currently studying Sobolev spaces, it's pure math, yet every piece of modern engineering simulation software relies on it and its proofs. Not to mention quantum mechanics, aerodynamics and more.
>>17070946It's called SI.
>>17071344What the fuck are you even doing on this board, genuinely? You have no idea how science or engineering developed. Seriously, dude, grab a book.
>>17070946>no one from my university>only one paper even close to what I'm working onshite
>>17054862THERE'S A SOUND AT THE WINDOW, THAT HE STRUCK YOU, A CRESCENDO, MATHIE
>>17054917You're expected to correct those yourself.
>>17055762>there's sign errors in there I didn't fix but you get the general idea.t. Jean-Pierre Serre
Why wouldn't this work?
>>17070946I'd be interested in the distribution of dates of publication of these citations, maybe compared to the citations of the average human-authored paper. In my experience AI will readily cite recent papers of dubious relevance just for having come out in the past 3-5 years, will only sometimes remember to cite foundational works (granted, that's also often true of human authors), and has to be explicitly wrangled to find important references for anything in between.>>17071455Imagine a drunk ant trying to follow a straight line for a mile. It's drunk, so it wobbles around the line, moves a little off the path and has to correct, and it ends up walking kind of a sinusoid along the line, so that it actually walks 2cm for every 1cm down the line it moves. The effect that is although the ant ultimately stayed very close to the line relative to its length, it still had to walk 2 miles. The point is that length isn't "continuous" as a function of paths; you can have two paths very close together with one much longer than the other, and a "limit of paths" doesn't necessarily correspond to a limit of lengths.
>>17071455But if you cut a box infinity times wouldn't it stop existing?
>>17071470You have to be very careful
>>17071455The other guy already explained it well so I will just supplement technical stuff. Basically because under the hood it is an example of interchanging two limits that don't commute. The circumference of a curve you can write as a radial plot of a differentiable function is[eqn]C = \int_0^{2\pi}\sqrt{1 + \left(\frac{d\gamma(\phi)}{d\phi}\right)^2} d\phi [/eqn]And while you built a sequence of curves that approach gamma, their derivatives don't approach its derivatives. So just because [math]\gamma_n\to\gamma[/math] does not mean [math]\gamma_n'\to\gamma'[/math]
>>17071455
>>17071455You have a bunch of points spread evenly over a circle's circumference. You use them to approximate its length by computing the distances between adjacent points and summing them up. How do you compute the distance between two adjacent points?A) Use the Pythagorean theoremB) Imagine a stair step connecting the points and add up its side lengthsIf you choose A, you're disagreeing with this meme, which chooses B.
The line is fuzzy no?
>>17071374You're only mentioning Applied Mathematics, my criticism is towards Pure Mathematics. Applied Mathematics doesn't require the use of any results from Pure Mathematics to work.>B-but what about this very abstract concept that has actual real world applications?That's wrong, the real world application of that concept would exist regardless of the mathematical formalism that was required to prove it. Pure Mathematics is a just a game, a mental masturbation, a way for intelligent people to feel good about themselves and to pretend they're useful to society when all they're doing is really playing with words while all the actual real world progress is being done regardless of their proofs and formalisms. And there's nothing wrong with that really, some people just can't live in society, most of them are highly intelligent, academia is like an asylum for them, if it wasn't for that they'd be burdens on their families and probably very unhappy. I support the uselessness of math and academia, I just think you should acknowledge it as such instead of pretending you're actually doing something important.
>>17071987NTA you're replying to.I know you're baiting, but this either tautologically false or purely semantic. >Applied Mathematics doesn't require the use of any results from Pure Mathematics to work.What exactly do you think "pure mathematics" is? Just logic, category, set and modal theory? Let me give you an easy example. The finite element method for numerical PDEs (the thing almost every engineer uses on a weekly basis) was derived and *proven* using the weak formulation of differential equations which heavily relies on ""pure"" math.The line between "applied" and "pure" math is so vague in certain fields that the whole distinction is retarded.>That's wrong, the real world application of that concept would exist regardless of the mathematical formalism that was required to prove it.This is irrelevant. No one could have thought of certain techniques to solve real world problems (especially numerical ones) if not for the underlying theory.>real world progress is being done regardless of their proofs and formalisms.In certain serious industries (like mine), it is normal for industry to work together with mathematicians to solve problems which does involve proofs and theorems.Let me state this clearly since your mind can only deal with absolutes, is all ""pure"" math "useful" probably not. Is the claim you're making even half truthful, absolutely not.
>>17069311The field of mathematics to an extent has been fueled by this kind of ego, and ai is a big ego check (you can no longer ever be the smartest thing in a room like when you grew up). To do math in the future is going to require a degree of humility that a lot of present day mathematicians will struggle with. This kind of ego check won't stop at mathematics and will eventually spread to all of human society. If that kind of competitiveness is what drives you and you won't to keep it, you're only real option is to go into some sort of competitive gaming where competition is explicitly between humans and bots/ai is banned. Things like professional sports/esports, chess, poker, other games, all of that type of shit since even though an ai is much better at those games than any human could ever possibly hope to achieve, it doesn't matter since the goal is to be the best human, not just the best.
Please recommend me a Lattice Theory textbook, friends.
>generalizes derivative Show your gratitude, /mg/. Can you prove its uniqueness?
>/mg/ is dead nowadaysThis is sad, 5~6 years ago these threads would hit the bump limit in like 4 hours or so.
>>17074087Imagine the amount of spam and shitposting. But I guess this isn't much better.
Any good reason why fixed point notation for groups uses an exponent like K^G or V^G for fixed point vector spaces? Or is it a coincidence/lack of new symbols that it's similar to a function space like Y^X
>>17074102Not really. There used to be a lot of high quality posts even during the COVID /pol/spam era. I remember there being one anon who would post interesting questions with that Stein's;Gate girl's picture.
>>17074336There was the one Turkish problemposter (might be the same one, it's been years) whom I think stopped shortly after the 2023 earthquake. He definitively survived it, since he made a post clarifying as such, but I don't remember him posting much after that. I wonder how he's doing.
>>17074357Yeah, we are talking of the same person.
>>17074336>>17074357I've actually got a backlog of simple but interesting problems that I could post periodically, sort of olympiad-style. I'd considered doing this before but decided against it because of the danger of you-know-what shitposting, but if there's some genuine interest in it I could drop a Kurisu picture every now and then.
>>17074596>Kurisu picture every now and then.Based??? Show us some mathy math problems anon, I might give em a try.
>>17074596Can you post Ritsuko instead?
>>17074641Okay, here's one. I might post like one a week or so if they don't engender too much shitposting. Difficulty may vary.>>17074660Sorry, I don't post hags.
>>17074740Wait, isn't it sufficient for b) that since [math] L_u: \mathbb{R}[x] \to \mathbb{R}[x] [/math] where [math] L_u v = u \otimes v [/math] it follows that [math] L_u \in \mathrm{GL}(\mathbb{R}[x]) [/math], so all you gotta show is [math] L_a L_b = L_{a \otimes b} [/math]?
>>17074785Not incorrect but not really meaningful progress either; [math] L_aL_b=L_{a\otimes b} [/math] is equivalent to associativity, just notationally repackaged.
>>17074835ah damn it, I was hoping the fact that it is linear would immediately imply a natural homomorphism of that sort but you are right digging up a theorem like that would be cheating as it is akin to looking up the answer.
>>170747401 is a nothingburger. Just swap the vertices around and you swap the lengths.
>>17074978That would imply the set of possible lengths is the same for AC and BD, which isn't true. For example, if [math] AB=3, BC=4, CD=5, DA=6, [/math] then the set of possible lengths for AC is [math] \{1, 2, 3, 4, 5, 6, 7\}, [/math] while the set of possible lengths for BD is [math] \{3, 4, 5, 6, 7, 8, 9\}. [/math]
>>17075041>>17074740He's got a point though. The geometry doesn't have to be symmetric to exploit symmetries. I didn't fully solve it yet but the reasoning is arguably clear at its core.0) Any quadilateral ABCD is fully parametrized by 8 Cartesian coordinates. 1) Without loss of generality you can assume the center of mass of ABCD to be at the origin (the problem is translation invariant), leaving 6 coordinates.2) The problem is rotation invariant, leaving 5. Fixing 4 distances leaves degrees of freedom, and ultimately all you have to show for every integer choice of AC there exists an in some sense equivalent choice of BD.3) You can choose the unit square ABCD as the origin of your 5D cartesian parametrization. That geometry has a point group symmetry (D4 for mathematicians, C4v for chemists because we are confined to 2D. I will use the chemist's notation. See: https://gernot-katzers-spice-pages.com/character_tables/C4v.html).3.5) Note that all point group operations are isometries and correspond to switching the labels of the quadrilateral around.4) Taking all 8 Cartesian displacements of the 4 vertices you get the following characters for the reducible representation: (8,0,0,0,0). That decomposes into A1+A2+B1+B2+2E5) A2+E are rotation + translation, you can discard those to obtain your 5D subspace: A1+B1+B2+E. Any choice of AB,BC,CD,DA can be realized in that space up to rotation and translation, so it doesn't affect the combinatorics.6) Changing the labels [math]A\leftrightarrow B, C\leftrightarrow D[/math] in this parametrization corresponds to applying [math] \sigma_v [/math].7) A1+B1 are [math] \sigma_v [/math]-invariant. E is a 2D representation but can be chosen such that it is diagonal in [math] \sigma_v [/math], so any quadrilateral in that 3D subspace satisfies AC=BD and is therefor out, so I broke the problem down from 8 to effectively 2D.
>>17055556Unbelievably based.>>17055661And the LLM users predictably have to deny reality.>>17056462I don't know of a single person in real life that learned from chatbots.
>>17075350Oh, we have an unhinged luddite over here.
>>17075585How am I luddite?
>>17075677>Unbelievably based. And the LLM users predictably have to deny reality.You sound like a deranged anti-AI mobster.
>>17075690not him but it's so funny to see a mainstream headline regurgitating npc scream "we are the resistance"
>>17075712They don't have anything else to live for.
>>17075690>deranged anti-AI mobster.That's not a thing but it should be. I wouldn't mind seeing a couple of "AIgoon found dead" headlines among all the "AI generated 600,000 lines of Lean" ones.
>>17076077Oh, it IS a thing. Bold of you to assume they can write anything interesting though.
>>17075690>Unbelievably based.This is a personal opinion on someone who never used LLM chatbots. How is celebrating someone not using your product a positive action forcing AI people to do anything?>And the LLM users predictably have to deny reality.This is once again a statement of fact. We aren't forcing LLM users to do anything.>You sound likeWhich is another way of saying "you aren't, but you might be". Get your LLM chatbot to argue for you next time. This is embarrassing.
>>17076089>it IS a thingThere's a moblike organization killing AIgoons? Holy based.
>>17076161>This is a personal opinion on someone who never used LLM chatbots.Yeah, an opinion no one asked for. >This is once again a statement of fact. It is a statement of delusion that directly proves my point.>We aren't forcing LLM users to do anything.I didn't say you were. >Which is another way of saying "you aren't, but you might be". In this case it's crystal clear actually. Now that we've figured out your dishonest question was actually rhetorical and you are indeed a luddite, why are you wasting everyone's time?
>>17076439>I didn't say you were. You said I was a luddite. Luddites smashed machinery. Luddites killed people. You're implying I was doing some positive action against AI users, when I'm not. You're painting not liking a technology as being luddites. Post math or I'll preemptively accept your concession.
>>17076619>nuh uh I am le peaceful ludditeEmbarrassing.
>>17076634Concession: accepted. The whole point was the contradiction of a "peaceful luddite" you moron.
>>17069247Why do you care what gatekeepers think?
>>17054862I'n getting back into math with some ezpz junior maths olympiad questionsvery comfy and better than 4chan or solitaire
>>17069247I checked sci-hub but it's not there, you could make a sci net account and request it
>>17054862For those who continue to study math on their own after graduating from uni, what is your motivation behind this? I know it's just a curiosity but how do you keep yourself encouraged to do this?
>>17077796At a small scale, I like having mental puzzles to think of whenever I'm bored. However, math is made of math, and I find that learning things occasionally has a resounding effect. I gain deeper insight into what I've studied in the past and intuition that aids in the study of more advanced stuff. This all feeds into my "big picture" understanding math as a universe of interconnected structures, which is the main reward for me. It probably helps that it is far removed from what I actually do.
>>17077796Why would I go to university for something I don't want to do?
>>17078376That makes sense. You just keep going because you enjoy it. I was overthinking it a bit.
>>17078407I know a guy who graduated in mathematics and told me he wasn't terribly interested in it outside of industry. The guy is now unemployed and existential. You don't go into something like pure math because of reasons like his. If you want industry competence involving numbers, then you become an accountant. If you go into mathematics because you enjoy it, then you either go into academia, or become an accountant and do math recreationally.
The absolute state of this general. Would it maybe help if I made these a little easier?
https://www.youtube.com/watch?v=zONGVaONKe8It'll all be okay.
>>17074740From the triangle inequality the number of possible values for AC is the smaller of 0 andmin(AB+BC, CD+DA) - max(|AB-BC|, |CD-DA|) + 1.With a bit of case analysis, I was able to rewrite this as the smallest of 0,2min(AB,BC,CD,DA) + 1, andAB+BC+CD+DA - 2max(AB,BC,CD,DA) + 1.Since this is symmetric under interchange of A -> B -> C -> D -> A, the number of possibilities for BD is the same.You should crosspost some of these on lambdaplusjs.
>>17078967I should probably add some more explanation.We can construct points A,B,C so that AC takes any value in the interval [math][|AB-BC|, AB+BC][/math].Similarly, we can construct C,D,A so that AC takes any value in the interval [math][|CD-DA|, CD+DA][/math].Joining the possibly degenerate triangles together at points A and C, we see that AC can be any value in the interval[math][\max(|AB-BC|, |CD-DA|), \min(AB+BC, CD+DA)][/math].Since AB, BC, CD, and DA, are all integers, the number of possible integer values for AC is either 0 (for the case the interval is empty) or[math]\min(AB+BC, CD+DA) - \max(|AB-BC|, |CD-DA|) + 1[/math].Now to get from the first expression to the second, first consider the case where[math]2\min(AB, BC, CD, DA) \le AB+BC+CD+DA - 2\max(AB, BC, CD, DA)[/math].Assume without loss of generality that [math]\min(AB, BC, CD, DA) = AB[/math].From[math]AB \le BC+CD+DA - 2\max(BC, CD, DA)[/math]we have[math]AB + BC \le CD + DA[/math][math]BC - AB \ge CD - DA[/math][math]BC - AB \ge DA - CD[/math]so[math]\min(AB+BC, CD+DA) - \max(|AB-BC|, |CD-DA|) + 1 = 2AB + 1[/math].Then consider the case where[math]AB+BC+CD+DA - 2\max(AB, BC, CD, DA) \le 2\min(AB, BC, CD, DA)[/math].Assume without loss of generality that [math]\max(AB, BC, CD, DA) = AB[/math].From[math]BC + CD + DA - AB \le 2\min(BC, CD, DA)[/math]we have[math]CD + DA \le AB + BC[/math][math]AB - BC \ge DA - CD[/math][math]AB - BC \ge CD - DA[/math]so[math]\min(AB+BC, CD+DA) - \max(|AB-BC|, |CD-DA|) + 1 = BC + CD + DA - AB + 1[/math].
>>17078967>>17078989Hey, winrar, very nice.Part b remains open, and at the risk of spoiling a little, there's a reason I grouped these two together.
>>17074740The second one feels related to spin, which I noticed when I realized that this formula works for like only half integers including negatives prob, which suggests some sort of recursive decomposition or wtv. So like, [math] \begin{array}{r c l}x^0 = (x^{1/2})^0 = \left[ \tfrac{1}{2} \right]^0 := & [0] = 1[0] & = (1, 0,0,...) \\x^{1/2} := \left[ \tfrac{1}{2} \right]^1 := & \tfrac{1}{2}& = (0, 1, 0...) \\x^{1/2} \otimes x^{1/2} := & \tfrac{1}{2} \otimes \tfrac{1}{2} = 0 \oplus 1 = 1[0] \oplus 1[1] & = (1, 0, 1, 0 ...)\\\left[ \tfrac{1}{2} \right]^3 = & \tfrac{1}{2} \otimes \left[1 \oplus 0 \right] = \left[1 \oplus 0 \right] \otimes \tfrac{1}{2} = 2 \left[ \tfrac{1}{2} \right] \oplus \tfrac{3}{2} & = (0, 2, 0, 1, 0\\\left[ \tfrac{1}{2} \right]^4 = & \left[ 0 \oplus 1 \right]^2 = 2 [0] \oplus 3 [1] \oplus [2] & = (2,0,3,0,1) \\\end{array}[/math]So with [math] \left[ \tfrac{1}{2} \right]^0 [/math] and [math] \left[ \tfrac{1}{2} \right]^2 [/math], you got a basis for any degree 1 polynomial since they're equal to basis vectors (1,0) and (1,1), and if you have [math] \left[ \tfrac{1}{2} \right]^4 [/math] you get (2,3,1) so now you can do any quadratic, etc. and for any [math] \left[ \tfrac{1}{2} \right]^n [/math] the highest term is gonna have coefficient 1, so you def have a basis. So ideally any polynomial with half integer exponents can be decomposed into some linear sum of half integer exponents, and all you have to show is that multiplying three half integers are associative. Negative exponents had some weird thing but I didn't think too hard on that.So, no real proof given here, but there's def some easy path with decomposition if you write out the exact pattern. If anyone has any spin related comments to give, that's cool.
>>17079135>multiplying three half integers are associative.I mean multiplying three 1/2's sorry
>>17079135oh it's a pacal triangle thing, took like 5 secs to see that
>>17079064>Part b remains open, and at the risk of spoiling a little, there's a reason I grouped these two together.I see, it follows fairly easily from (a).We can restate the definition of [math]\otimes[/math] as[math](a \otimes b)_p = \sum_{(i,j,p) \in T} a_i b_j[/math]where [math]T[/math] is the set of integer triplets [math](i,j,p)[/math] which may be the side lengths of a possibly degenerate triangle. Then[math][(a \otimes b) \otimes c]_l = \sum_{(i,j,p),(p,k,l) \in T} a_i b_j c_k = \sum_{i,j,k} M_{ijkl} a_i b_j c_k[/math]where [math]M_{ijkl}[/math] is the number of possible integer values of [math]AC[/math] for [math]AB=i, BC=j, CD=k, DA=l[/math]. Similarly[math][a \otimes (b \otimes c)]_l = \sum_{(l,i,q),(q,j,k) \in T} a_i b_j c_k = \sum_{i,j,k} N_{ijkl} a_i b_j c_k[/math]where [math]N_{ijkl}[/math] is the number of possible integer values of [math]BD[/math] for [math]AB=i, BC=j, CD=k, DA=l[/math].Since [math]M_{ijkl} = N_{ijkl}[/math], [math]\otimes[/math] is associative.
>>17079135There's definitely a link here and Clebsch-Gordan supplies the recursive decomposition you're thinking ofTo use only whole integer spins, we can use SO(3) instead of SU(2)More precisely, working over K_0(Rep(SO(3))): ([Vi][Vj])[Vk] = [(Vi ⊗ Vj) ⊗ Vk] = [ Vi ⊗ (Vj ⊗ Vk)] = [Vi]([Vj][Vk]) so ⋅ is associative let A be the basis of a[x] and B be the basis of K0(Rep(SO(3))) let φ:A->B, φ:x^n->[Vn]φ extends uniquely to an isomorphism φ* as it's a bijection between bases of free a-modules From the Clebsch-Gordan decomposition formula, we have [Vi][Vj]=sum k=|i-j| to i+j [Vk]So proving φ*:(a[x],⊗)->(K0(Rep(SO(3)),⋅) is a homomorphism and thus a magma isomorphism is trivial This means φ preserves associativity, so ⊗ is associativeExtending this to R[x] by R bilinearity, we preserve associativityBecause associativity on basis elements implies associativity everywhere
>be me>math undergrad>has to take linear algebra this semester>studies for 7 hours from Liesen>only advanced 20 page-ishhow do i fix this? is 3-4 pages per hour normal or am i just retarded?
>>17079409>looks up bookJust post questions with pictures here or on the simple questions thread. Or hell, even AI could help with this type of book, and guaranteed there are a butt ton of youtube videos on this stuff. Or go to office hours. There are lots of ways to digest this stuff faster if you're finding difficulty.Also if you're doing the exercise problems then yeah it'll take longer too.
Has there ever been a more overhyped piece of crap/slop than hatcher's algebraic topology? Holy shit is that book disgusting
>>17080375See my response to the larp thread you made.
>>17080375Only one larping is you kek if you knew about friedl's notes you would piss on hatcher
>>17080399Meant for >>17080390
>>17080399See my response in the dedicated thread instead of butting in on our conversation.
applied math or actuarial mathi want to have a job but i dont want to fucking
>>17080516...fucking work with excel spreadsheets all day
>>17079227nice
>>17080516>>17080519Good news: you don't have to anymore, but you there's no any other paid work that makes use of your math degree
>>17081921genuinely what the fuck do i do? i'm too stupid for research and even if i werent it doesnt pay shit here
>>17068851A Certified Hyperbolicity of the Jensen PolynomialsDoes this AI startup's proposed certification help the Reiman Hypothesis study?https://www.linkedin.com/posts/reevetech-contextplus_certified-strict-hyperbolicity-of-the-jensen-activity-7521055348435255296-Ab2a
I am an enginigger who wants to learn applied ML in my field. Any good statistics books for plebeians like me?
>>17082095>MLclaude.ai
You guys solved the hard one (nice) but not the easy one from last week >>17078498
>>17082112ask ai
>>17082112AI says the possible triples [math](x,y,z)[/math] are the points in the convex polytope defined by the inequalities[eqn] x + y \leq 1 \\ x + z \leq 1 \\ y + z \leq 1 \\ x \geq 0 \\ y \geq 0 \\ z \geq 0[/eqn]334 people choose [math](a,b,1-a-b)[/math], 334 people choose [math](a,1-a-c,c)[/math] and 333 choose [math](1-b-c,b,c)[/math] to make [math](m_x,m_y,m_z) = (a,b,c)[/math].
>>17078498Generalize to the n=0 and k=0 cases with the convention that an empty list has sum 0. Denote the set of sums that meet the criterion as S(n,k). For k>0, a sum in S(n,k) either starts with a number less than k and is a member of S(n, k-1), or it is k minus a member of S(k-n, k-1). We obtain a recursive formula for |S(n,k)||S(n,k)| = |S(n, k-1)| + |S(k-n, k-1)| for k>0|S(n,0)| = if n=0, 1; otherwise, 0which is satisfied by |S(n,k)| = n choose k.You should consider cross-posting to some other imageboard; there are many to choose from. Raise the quality of a site that actually wants its quality raised.
>>17082144Nice, I had in mind myself a recursive casework on the exact value of the first term followed by the hockey stick identity, but your way's a little cleaner (though obviously ultimately equivalent). The form of the answer does also beg the question of a bijective proof.>You should consider cross-posting to some other imageboard; there are many to choose from. Raise the quality of a site that actually wants its quality raised.No, I'm too stubborn and don't even use other imageboards, as long as at least one person like you responds in good faith I'm dragging this general down with me kicking and screaming
>>17082112Whatever x is, at least 501 people must have chosen [math]x \geq m_x[/math]. These people must choose [math]y \leq 1 - m_x[/math], so [math]m_y \leq 1 - m_x[/math] making [math]m_x + m_y \leq 1[/math]. Similarly [math]m_y + m_z \leq 1[/math] and [math]m_z + m_x \leq 1[/math].Any triple within [math]\{(m_x, m_y, m_z) \in [0,1]^3 | m_x + m_y \leq 1, m_y + m_z \leq 1, m_z + m_x \leq 1\}[/math] is possible. Let 333 people choose [math](m_x, m_y, 1 - m_x - m_y)[/math], 333 people choose [math](1 - m_y - m_z, m_y, m_z)[/math], and 333 people choose [math](m_x, 1 - m_z - m_x, m_z)[/math]; the choice of the remaining 2 people does not matter.
I graduated and I'm studying for master's. But how the fuck am I supposed to learn 6 heavily difficult courses in a single semester? Is this normal? There's so much stuff already, Functional analysis is by far the most mindbreaking, I have to memorize over 150 theorems, proofs and definitions for the final exams. Lmao. I barely understand differential topology and I'm stuck trying to grasp Gödel's completeness. I already feel like dropping out.
>>17083957Just cheat nigga. All students before you did this and everyone after will. You not gonna be recalling proofs by heart at work (if you find one kek).
>>17083966I would if I could, but I can't really cheat without getting caught and failing.>if you find one kekI'm pure math student, so no.
