[HN Gopher] How I learned to love and fear the Riemann Hypothesis
___________________________________________________________________
How I learned to love and fear the Riemann Hypothesis
Author : pseudolus
Score : 169 points
Date : 2021-01-05 12:43 UTC (10 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| credit_guy wrote:
| The video is great.
|
| But it fails on two counts. One is that it doesn't say what the
| title claims. We know now what the Rieman Hypothesis is, but how
| did the author learned to love, and more importantly to "fear"
| the RH? What's there to fear? The name of the famous movie (with
| the bomb) doesn't even have "fear" in it, so the author had
| something in mind with "fear", but after watching the video, I
| have no clue what.
|
| Second is the promise in the beginning of the video to give us a
| hint why the RH is important. What we learn is that somehow each
| zeta zero adds another harmonic to some type of series
| approximation of the (cousin of the) prime counting function.
| Which is great; if we add enough harmonics we get to approximate
| this function as precisely as we want. But why is it important
| that these zeros are on the vertical line Im z = 0.5 ? I have
| absolutely no clue.
|
| Don't get me wrong, I find that watching this video was a very
| good investment of 16 minutes of my life. But there is no need to
| overpromise, especially since the video delivers a lot as is.
| rfurmani wrote:
| > But why is it important that these zeros are on the vertical
| line Im z = 0.5
|
| Great question. The formula works whether or not the zeros have
| real part 0.5, but its implications are different. If you're
| counting primes up to x you will get the main term (roughly
| x/log x) and each zero p will give an oscillating error term ~
| x^p/p. If the real part is always 0.5 then these error terms
| are all on sqrt(x) scale, which is what you would expect from
| random variation. If on the other hand you have a single zero
| right of the line, say at real part 3/4 (and mirror image one
| at real part 1/4) then now you get a secondary error term of
| size x^(3/4) that dominates all of the other error terms,
| giving some extra structure to the primes: they're no longer
| random, you can predict where they are more or less likely.
| drbacon wrote:
| One million thanks for this comment. This is the most concise
| and clear explanation of why the critical line is important
| that I have ever read.
| nabla9 wrote:
| > title claims.
|
| Discussions based on the title are useless. Good articles
| frequently have titles that are bad or not even relevant. Title
| of an article:
|
| 1. May not be chosen by the author. Its often editor decision.
|
| 2. Article can have multiple changing titles. For clickbait
| reasons.
|
| Discussions based on title are generally worthless. In the HN
| you can ask someone to change the title to be more relevant.
| Covzire wrote:
| If one could predict with pinpoint accuracy all the prime
| numbers approaching infinity with certainty and with trivial
| resources (without having to test each one for prime-ness),
| would it have any impact on modern day cryptography?
| bawolff wrote:
| Not an expert, but the key thing is usually factoring, e.g.
| finding which prime numbers multiplied together make a
| certain composite number. The hard part is finding which are
| the right prime numbers (there are lots of prime numbers).
| Finding numbers that are prime,just generally,is a very quick
| operation already.
| gameswithgo wrote:
| Being able to find primes faster would make modern public key
| crypto better I guess, since you could find large keys faster
| and thus use larger keys.
|
| Unless the theory also leads to faster factorization methods
| simias wrote:
| On RSA-style prime-based crypto potentially but barring a
| much stronger result than merely being able to cheaply find
| primes things like ECC would probably still be safe.
| red_trumpet wrote:
| The part why he fears it is covered in the text below the
| video.
| gimboland wrote:
| > but how did the author learned to love, and more importantly
| to "fear" the RH? What's there to fear?
|
| It's made abundantly clear: they learned to love it through the
| course they were fortunate enough to take as an undergraduate,
| and they learned to fear it later, as a researcher, when they
| realised that what there was to fear was wasting their career
| attacking something that was probably just too big and too hard
| to be sensible for them to attack.
| Tade0 wrote:
| I understand that the real proof of the RH is the friendships
| we make along the way, but what would happen if we just acted
| with the assumption that it is true?
| rfurmani wrote:
| There's been a lot of research that has assumed it to be
| true and built on top of it to find deeper implications.
|
| Still, a number of people who've worked most closely with
| the Riemann zeta function, even the ones who led the
| computations and numerical verifications, have expressed
| doubts on whether the Riemann Hypothesis is actually true:
| https://arxiv.org/abs/math/0311162
| waynecochran wrote:
| This is the difference between a scientist and a
| mathematician. A scientist would consider the empirical
| evidence more than sufficient to consider RH to be true.
|
| Of course there is hope that a proof of RH would reveal
| some deeper understanding -- not merely a conformation.
| hansvm wrote:
| I'm not so sure a scientist would consider RH true at
| this point. Math has a rich history of conjectures
| appearing true for the first n dimensions, the first k
| inputs, the first z partial/tangential proofs, etc. The
| evidence for RH isn't any stronger than that for a
| plethora of eventually-proven-false conjectures.
|
| Do you know if anyone has tabulated stats on conjecture
| proofs? That could be a fun dataset to examine.
| jjgreen wrote:
| As Richard K. Guy's strong law of small numbers states:
| There aren't enough small numbers to meet the many
| demands made of them.
| mensetmanusman wrote:
| The fear was alluded to.
|
| If so much technology is built around a hypothesis, it is
| essentially building a large house of cards if the hypothesis
| turns out to be incorrect.
|
| Imagine if the world set up a monetary system and what would
| happen after 100 years if suddenly everyone could print money
| when a fatal flaw in a system was discovered.
| wodenokoto wrote:
| The title relates to the article and the video is a supporting
| piece of media.
|
| The article talks about the personal story of how the author
| first met the Riemann Hypothesis.
| legel wrote:
| Fantastic video! For me seeing this math for the first time was a
| very beautiful surprise.
| st1x7 wrote:
| Vaguely related question - what is the best textbook that you can
| recommend for studying complex analysis?
| rfurmani wrote:
| I've heard very good things about Visual Complex Analysis, as
| well.
| lldbg wrote:
| "Complex Analysis: An Introduction to the Theory of Analytic
| Functions of One Complex Variable" By Lars Ahlfors was very
| enjoyable to me. However, I read it after I already knew
| Complex Analysis.
| ellis-bell wrote:
| i second tristan needham's visual complex analysis.
|
| some other good ones:
|
| * the one we used in my undergrad course was fisher's complex
| variables which is great if you're learning for the purposes of
| applications. it's a cheap dover book.
|
| * rudin's real and complex analysis (if theory is your thing.
| note that rudin's books, while great, do require a good
| background in math).
|
| * as the article mentions, eli stein has a series of books on
| the four main branches of analysis. i believe the second book
| is on complex analysis.
| Moeg wrote:
| Am I supposed to Google what the hypothesis is? Is it really too
| much to ask of writers to provide context first?
| Santosh83 wrote:
| Distantly related question. It was claimed that Alpha Zero taught
| itself to play incredibly high level chess from the very basic
| point of playing with itself millions of times after having been
| fed the basic building block rules of chess and nothing else.
|
| Now I'm wondering if something like this can be made to work for
| discovering new mathematical relationships or solving existing
| hypothesis by training a neural network with what we know of
| maths so far and then letting it "play" with the equations?
|
| Am I being too naive here?
| TheCraiggers wrote:
| >Am I being too naive here?
|
| I don't think so, but I think the trick will be how to score a
| new mathematical relationship as "interesting". Basically, a
| fitness metric. With Chess, you have the obvious win/loss
| metric, and also number of turns and possibly time. With math,
| how do you quantify the level of interestingness between 2+2=4
| and p=np?
|
| This isn't a snarky question, by the way. I'm genuinely
| interested in how this might be done.
| whatshisface wrote:
| It would be sufficiently amazing for AlphaProof to simply
| answer the questions posed to it. The training data would be
| pairs of theorems and proofs as already existing in the CoQ
| and other proof checker literature.
| groby_b wrote:
| Unless you have a prover that can answer "is this actually
| a _correct_ proof ", your fitness function is essentially
| "does this look like a proof"
|
| And if you have a prover that can answer that question, you
| don't need to train a neural network to give you the
| answer.
|
| Proving maths demands 100% precision and recall, at which
| point employing ML doesn't make sense - the point of ML is
| (horribly simplified) stochastic reasoning, not finding
| truths.
| PartiallyTyped wrote:
| But ML could in theory accelerate the process. For
| example, AlphaZero uses ML to highlight the correct paths
| in search algorithms. The same search algorithms exist
| for stockfish and both can reach the same conclusions,
| but, AlphaZero ends up looking up much shorter distance
| than stockfish, yet it wins. If anything, it is
| significantly more human-like than stockfish.
| gjulianm wrote:
| The search space for proofs is far bigger and less
| structured than a Go game. A lot of proofs also require
| creativity, in the sense that they aren't just "find the
| set of logical steps from 'initial condition' to
| 'proof'", but they require auxiliary theorems,
| definitions and constructions that help make the problem
| manageable.
|
| Another problem is that a lot of proofs starts by
| exploration. For example, certain bounds for functions or
| convergence rates are proved without knowing what will be
| the end result.
|
| Of all the problems that ML could be applied to, I find
| mathematical proofs one of the least promising. One, I
| don't see enough similarity between proofs that would
| allow any algorithm to learn useful patterns; and two, I
| don't think most mathematicians would trust a proof that
| cannot be understood (even if the output is a set of
| steps for a formal system, I imagine translating that to
| human language could be quite difficult).
| touisteur wrote:
| I wish we'd start with general loop variant/invariant
| generation...
| whatshisface wrote:
| Proof checkers are an existing technology. That's what
| CoQ is.
| lacker wrote:
| Indeed, existing automated theorem provers do spend a lot of
| time discarding true statements as "insufficiently
| interesting". In general, the longer a statement the less
| interesting it is. The easier it is to prove with simpler
| statements, the less interesting it is. And if it's
| superceded by a more general rule it is less interesting.
|
| For example, an automated theorem prover can easily prove
| many statements of the form "A or not-A or (any long
| statement)". Those are all uninteresting tautologies that
| should be discarded.
|
| It's interesting to theorize about improving these heuristics
| with AI methods, there is some work along these lines like
| ENIGMA-NG.
| mensetmanusman wrote:
| From a finite set of axioms you can generate an infinite number
| of proofs because complexity is unbounded.
|
| 'Proofs' in human parlance are those which we believe to be
| interesting or aesthetic or useful.
| rfurmani wrote:
| Adding an extra point, one thing that makes this difficult is
| that a lot of this kind of research is not about applying
| existing ideas but coming up with something that has never been
| done before, and neutral networks are a lot worse with
| creativity than with pattern matching. It's also hard to get
| enough training examples, especially since you'd either want
| these to be in strict logical language or train a system to
| recognize whether a hundred page proof in natural language is
| rigorous or not. So it's like a very long maze where noody
| knows if you're getting closer to the exit or not, and in most
| rooms you have to come up with something original or discover a
| new dimension. And there's the aspect whereby a lot of
| significant research is: create a new abstraction, realize and
| convince others that this abstraction is useful and gives you
| new intuition that lets you solve new problems.
| ellis-bell wrote:
| you might be able to do something like "predict the next prime
| number" or "predict the next zero of the Riemann zeta
| function".
|
| you could try something like this for statements in a formal
| axiomatic system, but know that you're running up against
| things like the halting problem / entscheidungsproblem / godel
| incompleteness. so it may be possible to train a neural net to
| decide the veracity of a statement and do so more quickly than
| a human might, but you would inevitably be running up against
| things that are truly undecidable in nature. which is not like
| go or chess where although they are difficult, they are
| decidable.
| sanxiyn wrote:
| I am not sure how math is unlike chess. Chess has three
| outcomes: win, loss, draw. That seems exactly like math,
| having true, false, undecidable.
| gjulianm wrote:
| Yes, but it isn't useful. A conversation also has three
| outcomes (neutral, you like the person more, or less) and
| yet it doesn't make conversations similar to chess.
| whatshisface wrote:
| Chess and math are both searches over countably infinite
| trees.
| gjulianm wrote:
| And a conversation is a search over a countably infinite
| tree too.
|
| My point is that even though some things have technical
| similarities, in practice they are very different
| challenges.
| whatshisface wrote:
| Math and chess are both very easy to check. Proof
| checkers and chess rule checkers both exist.
| "Conversation checkers" do not.
| whatshisface wrote:
| There is a sense in which that is a great idea, and a sense in
| which it is a not so great idea. If AlphaProof could prove
| theorems as superhumanly as AlphaGo wins at Go, that would
| really be something, they might be able to claim all of the
| millennium prizes. However, the longterm usefulness to
| mathematics may be somewhat limited. Mathematicians are
| interested in proofs not so much because of the statements, but
| because of what those proofs add to our knowledge of the way
| math fits together. It isn't just about "true" or "false," it's
| about why. If AlphaProof generated a long and unreadable proof
| with no discernible structure, it would not be such a large
| contribution to mathematics as if a human figured out "why" and
| wrote it in a way other humans could understand.
|
| Of course, it is possible that clever users could tease useful
| information out of the black box truth oracle. Perhaps by
| modifying the statement of the conjecture and checking each
| modification, they could discover some parts of the structure
| of the problem, which would help inspire a useful proof.
| Furthermore, it would be helpful to never again waste time
| trying to prove a false conjecture.
| idolaspecus wrote:
| Your comment inspired an interesting (to me at least)
| thought. I'm imagining a world where mathematics becomes too
| advanced for even the brightest human being, but progress is
| preserved by proof-generating programs running in data
| centers. The data centers churn out exabytes of mathematical
| proofs, and humanity employs an army of what are effectively
| philologists/semioticians to dissect, mine, and excavate
| these proofs for meaning.
| brzozowski wrote:
| And when your surpassing creations find the answers you
| asked for, you can't understand their analysis and you
| can't verify their answers. You have to take their word on
| faith--Or you use information theory to flatten it for you,
| to squash the tesseract into two dimensions and the Klein
| bottle into three, to simplify reality and pray to whatever
| Gods survived the millennium that your honorable twisting
| of the truth hasn't ruptured any of its load-bearing
| pylons. You hire people like me; the crossbred progeny of
| profilers and proof assistants and information theorists...
|
| In formal settings you'd call me Synthesist.
|
| --Peter Watts, Blindsight (2006)
|
| [1]: https://www.rifters.com/real/Blindsight.htm
| dwohnitmok wrote:
| Ted Chiang has a (very) short story exploring this idea,
| with metahumans replacing data centers here:
| https://www.nature.com/articles/35014679
| agar wrote:
| Thank you for that link. A very interesting story, with
| surprising philosophical depth given its brevity.
| karmakaze wrote:
| We already have machine-assisted proofs which are structured
| by people but executed by machines. AlphaProof would be that
| but structured by machine. As with the machine-assisted
| proofs, there's value in knowing and building upon the result
| even if the methods themselves can't be adapted for other
| use.
| lacker wrote:
| _If AlphaProof generated a long and unreadable proof with no
| discernible structure, it would not be such a large
| contribution to mathematics as if a human figured out "why"
| and wrote it in a way other humans could understand._
|
| Maybe not directly. But if AlphaProof managed to prove the
| Riemann hypothesis, I bet it wouldn't be long before human
| mathematicians had managed to decipher the proof and explain
| it in a way that other mathematicians found useful. It's just
| a lot easier to re-explain something that other people have
| already proved, rather than proving it yourself.
| Jabbles wrote:
| Godel's incompleteness theorem doesn't fill me with the
| same confidence as you that "a true theorem should have a
| useful explanation".
|
| In fact, looking at chess engines tells us that sometimes
| (though not always) brute-force calculation of many
| possibilities gives us better moves. In the same way some
| true statements might just be an exhaustive calculation.
|
| Perhaps the four color theorem is the most famous example
| of a proof that has lacked (still lacks) a lot of useful
| lemmas in relation to its length:
| https://en.wikipedia.org/wiki/Four_color_theorem
| virgil_disgr4ce wrote:
| True, and important. However, I suppose it could be argued
| that if the only results AlphaProof could obtain were long
| and unreadable proof(s) with no discernible structure, it
| might suggest something about the epistemic assumptions we
| tend to make about Occam's Razor and the (so-called) beauty
| and parsimony.
|
| On that angle, it could be further argued that humans are
| biased towards 'readable' proofs and that there are many
| undiscovered 'unreadable' proofs that we simply have no way
| of obtaining without automation of some kind.
|
| Anyway I assume there are volumes already written on this
| subject...
| whatshisface wrote:
| > _the epistemic assumptions we tend to make about Occam 's
| Razor and the (so-called) beauty and parsimony._
|
| The vast majority of ways to write a program are unreadable
| messes. The shortest way to write a program is sometimes an
| unreadable mess, mainly when it's been code golfed. The
| longest ways to write a program are _always_ unreadable.
| Proofs work the same way - and if Alpha Go selects randomly
| from all of the ways to write a reasonable-length proof, it
| is virtually guaranteed to pick an unreadable one.
| postalrat wrote:
| A long unreadable proof would not mean a small proof
| doesn't exist.
|
| I feel like a unreadable proof would be useful to simply
| know something is true while a small proof would be more
| useful to perhaps gain some sort of insight.
| touisteur wrote:
| Maybe that'll be time to bring in AlphaProofReductor or
| AlphaProofCanonizer. I'm finding this worry a bit strange.
| I'm guessing tools evolve together depending on need, so as
| long as humans want to know 'why and how' there'll be some
| effort in reduction?
| Bjartr wrote:
| The trick is coming up with a reward function that drives
| exploration when "played" against itself. Chess has an easy
| one, win the game. General mathematics doesn't have anything so
| easily specified.
| whatshisface wrote:
| It does, math has "prove the conjecture."
| rfurmani wrote:
| We can't even solve Sonic the Hedgehog without getting
| stuck in dead-ends in the game :-)
| https://openai.com/blog/retro-contest/
| 317070 wrote:
| Well, math is two parts.
|
| There is the "prove this conjecture I have" part, which I
| think will be taken over by machines rather soon (say,
| within my lifetime). I cannot imagine there is something
| extra-ordinarily hard that mathematicians do, that is
| somehow not captured by how a Go player approaches the
| game.
|
| There is also the "come up with an interesting conjecture"
| part, which I think is a lot harder. It is extra-ordinarily
| hard to specify what exactly makes some conjectures more
| interesting than others. So I reckon this part of math will
| be much harder to automate in the long run. It will
| probably require a lot of cross-contamination from AI that
| manage to write interesting books for example.
| gjulianm wrote:
| > I cannot imagine there is something extra-ordinarily
| hard that mathematicians do, that is somehow not captured
| by how a Go player approaches the game.
|
| It's not "extraordinarily hard" but the issue is that
| math does not have a well-defined search space. A lot of
| times, proving statements requires quite a lot of
| intermediate results, additional tools and definitions...
| There is more creativity in math proofs that people
| think, and I don't think ML algorithms will be able to
| reproduce that. At most I see specific algorithms for
| limited purposes.
| mpettitt wrote:
| Matt Haig's "The Humans" talks about humanity finding the answer
| to the Riemann Hypothesis being feared universally - it's
| simultaneously very important to the plot, and a convenient
| MacGuffin.
| wodenokoto wrote:
| The accompanying video popped up on my YouTube recommendation
| list last night and it is really good. Like, you married a
| numberphile video with 3blue1brown, good.
| st1x7 wrote:
| I don't think it comes close to 3blue1brown's video on the same
| topic - https://www.youtube.com/watch?v=sD0NjbwqlYw
|
| Still nice of the video in the original post to give a bit more
| of a historical context, instead of focusing entirely on the
| mathematics. It could have done without the building analogy
| around the beginning though, that didn't come up later and was
| just distracting.
| pseudolus wrote:
| +1 for the 3blue1brown recommendation. Between the two, a
| non-mathematician can acquire a layperson's comprehension of
| the Riemann Hypothesis. Additionally, John Derbyshire (of the
| unfortunate views regarding race) wrote a very good book
| "Prime Obsession: Bernhard Riemann and the Greatest Unsolved
| Problem in Mathematics" that covered both historical and
| mathematical aspects of the Riemann Hypothesis [0].
|
| [0] https://www.amazon.com/Prime-Obsession-Bernhard-Greatest-
| Mat...
| EricMausler wrote:
| That video made complex analysis click for me
| jbj wrote:
| The vertical scaling of the counting function with log(p) went a
| little quick, does someone know an easy to digest resource to
| this? also what would be the logic for now including the p^n of
| primes, p ??
| war1025 wrote:
| I felt that way about most of the video. I never took advanced
| math theory classes, but I have enough background that those
| classes would probably have been the next step for me if I had
| continued on.
|
| Seemed like every time I felt like he was about to explain
| something in a way that would click for me, then all the sudden
| he left out the last two sentences and moved on.
|
| Still a neat video though.
| jbj wrote:
| Some of my lecturers throughout Uni, always glanced across
| the entire auditorium to see if anyone needed clarity of what
| was said before switching slide/erasing the blackboard.
|
| Having those extra seconds after something has been said,
| while being able to watch graphics/text/figures helped the
| information sink in for me.
|
| This moment to digest what was said is not really available
| in this video, but I guess youtube is different from a
| lecture in the sense that the whole thing can be set on
| pause.
|
| I think I have the impression that the visual explanation
| here appeared and sometimes even disappeared before him
| mentioning the "take-home-message".
|
| I am still very impressed with the explainer and the visual
| assistance though.
| spicymaki wrote:
| This is a great video. I think finally I understand the basics of
| Riemann hypothesis and the zeta function z(s). Thanks for posting
| this.
| odyssey7 wrote:
| If someone wanted to learn more about this type of mathematics,
| without matriculating at a university, what would be some good
| starting points?
| rfurmani wrote:
| This Numberphile video on the Riemann Hypothesis is also very
| good: https://www.youtube.com/watch?v=d6c6uIyieoo
| SkyBelow wrote:
| 3Blue1Brown on YouTube has a number of great videos into math
| that help give a visual understanding without needing to sit
| down and do the pure number crunching. I'm not sure he has one
| on the Riemann Hypothesis in particular, but he does cover a
| number of items that would be somewhat related and help build a
| level of comfort with being exposed to the math, even if you
| can't do the pure number crunching.
|
| Mathologer would be my second recommendation. Once again not
| sure he has touched the Riemann Hypothesis, but he does a
| number of great videos where you can start expanding being
| comfortable with often untouched fields of math (like modular
| arithematic) while introducing a how seemingly unrelated fields
| end up having connections.
|
| Videos can go between 20 minutes to an hour and I often enjoy
| watching one every few days.
| ryanianian wrote:
| Depends on your level of {experience, intended engagement,
| patience for pedantics, intended outcomes}.
|
| I've found that 3Blue1Brown (youtube) gives a great starting
| point for many topics. Provided you actively participate in
| them, the videos will build up some intuition so you know what
| questions to ask (and why they're worth asking).
|
| The author usually links to followup resources so you know
| where to go next. I've found his resources a bit hit-or-miss,
| but they give me enough to know if I want to continue
| exploring. If I do, I usually check out a good textbook (I
| refer to https://news.ycombinator.com/item?id=17617825 often).
| frankfrankfrank wrote:
| From Georg Friedrich Bernhard Riemann's tomb:
|
| "Denen die Got dienen mussen
|
| Alle dinge zum besten dienen."
|
| Translated:
|
| "All those that serve God
|
| All things best serve them."
| jhncls wrote:
| Wikipedia seems to disagree (and includes a photo): "Denen die
| Got lieben mussen, alle dinge zum besten dienen."
|
| Which it suggests translating as "For those who love God, all
| things must work together for the best".
|
| https://en.wikipedia.org/wiki/Bernhard_Riemann
| simonh wrote:
| It's also one of the verses that inspired Voltaire's
| character Dr. Pangloss to declare that "All is for the best
| in this, the best of all possible worlds."
| wussboy wrote:
| Which is a direct quote from a Bible verse, and would make
| more sense (but be less mysterious)
| earthboundkid wrote:
| Specifically, Romans 8:28.
| rfurmani wrote:
| I love this quote:
|
| > A meeting on how to extend the GPY method was immediately
| organized at the American Institute of Mathematics in San Jose,
| California. As a bright-eyed and bushy-tailed grad student, I
| felt extraordinarily lucky to be there among the world's top
| experts. By the end of the week, the experts agreed that it was
| basically impossible to improve the GPY method to get bounded
| prime gaps. Fortunately, Yitang Zhang did not attend this
| meeting. Almost a decade later, after years of incredibly hard
| work in relative isolation, he found a way around the impasse and
| proved the experts wrong. I guess the moral of my story is that
| when people organize meetings on how not to solve the Riemann
| hypothesis (as they do from time to time), don't go!
| rfurmani wrote:
| That said, I do wish I were at AIM's meeting on the Riemann
| Hypothesis :-) https://aimath.org/rh2018/
|
| The lecture videos are online and are a good way to get both
| historical context and connection to how people are thinking
| about it currently.
|
| This seminar talk on how to fail to prove RH was also great
| https://www.math.rutgers.edu/news-events/list-all-events/ica...
| Jabbles wrote:
| Is that a link to a VoD or a historical event with no
| recording?
| mdoms wrote:
| Off topic, but does anyone know where one gets started on making
| motion graphics like in that video? This combination of cool
| maths and animated visuals is so interesting to me, but I
| wouldn't even know where to begin. What kind of software is used
| to generate this kind of visualisation? Are there free or
| affordable online courses? Is there a job market for someone with
| this ability and also with software development chops?
| xevrem wrote:
| I believe the maker of 3blue1brown has a open source kit for
| just that: https://github.com/3b1b/manim
| mdoms wrote:
| Thanks for the link.
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