I watched the live stream yesterday. It was an unbelievably clear and accessible talk!! This is one of the best scientific talks I have ever witnessed

@andrewjeromeantonio4986

4 жыл бұрын

i love these types of talks as they cater to a wider audience, not to mention it's from a fields medalist. :v

@naimulhaq9626

4 жыл бұрын

As our knowledge goes deeper and deeper (how strings compactify), the observations of eastern mystics becomes the deepest; None of the properties of any part of the web is fundamental; they all follow from the properties of the other parts,s and the overall consistency of their inter-relations determines the structure of the entire web.[The Tao of Physics].

@Zeegoner

4 жыл бұрын

@@naimulhaq9626 shut your whole mouth

It is almost 4am and I am watching this. No regrets

Mind blown. First time seeing quadratic reciprocity connecting to knots! Thank you for making this lecture available online.

This strengthens my idea that mathematicians are the nurds of science.

@simdimdim

4 жыл бұрын

I'll sit here waiting for hte comment that claims that math is the only science and everything else is applied math or worse. xD

@mikhailmikhailov8781

4 жыл бұрын

​@@simdimdim Nah, not really. The mathematical way of thinking is an alternative to the "traditional" or to the "engineering" more heuristic way of thinking. If life is a problem of walking through some possibility space and trying to find what you want there, mathematics allows you to not wonder in the dark through it and simply go where you want to be, when it is applicable and well understood or at the very least clean up the problem and reduce it to a few very difficult, but well formulated problems. The end goal is of course to get to base level reality. So far it has been a very productive endeavor and we don't know if it will continue.

@simdimdim

4 жыл бұрын

@@mikhailmikhailov8781 i was referencing an xkcd post about sciences ordered by degree of abstractness. It's kinda amusing I'd recommend checking it out

Some of these properties (like whether there is only one loop or many) are really about permutations and not about nots specifically. You can ask if a permutation has a sinlge cycle and the figure of 69% seems about right to me (I remember it from a puzzle about prisoners trying to find numbers by searching in boxes).

Wonderful! Many thinks for the presentation, and for uploading. None of this had even occurred to me. I love finding stuff like that! My background is applied math rather than pure. But I have a foot in both camps, interest-wise. The separation is artificial and I think pernicious to understanding either.

Seems like it could be applied to string theory and field lines entanglement. I also enjoy his lecture style reminiscent of feynman audience friendly lectures. The statistic part makes me appreciate laplace's philosphical essay of probabilities. I love how easily explaining difficult topics is like the bragging rights conquest in the math/science world. Thank you for the jones mention!

@GglSux

4 жыл бұрын

Edward Witten, Knots and Quantum Theory kzread.info/dash/bejne/lal-u5NtcrzHk5c.html Just in case it didn't appear in Your "feed", as it did in mine... You never can tell with Y.T. Best regards

One digit is worth 2.3 threads because 2.3 is the natural log of 10?

@CoughSyrup

4 жыл бұрын

Holy cats! I think you're right!

@ezioarno15

4 жыл бұрын

and 10 comes from being base 10?

@CoughSyrup

4 жыл бұрын

@@ezioarno15 Yessir

@bernardfinucane2061

4 жыл бұрын

@@ezioarno15 Yes, the length (number of digits) of base ten number is about its natural log divided by 2.3. ln(100)= 4.6, ln(1000)=6.9, ln(10000)=9.2 etc

@CoughSyrup

3 жыл бұрын

This leads me to wonder... is there a homomorphism between these mathematical knots and arithmetic over the natural numbers?

Fascinating!

How exactly do you generate a random tangle? It seems to me that there's no limit to the amount you can tangle the threads before you join them, so how do you sample a random tangle? Also, given a tangle, you can read off a permutation, but different tangles can generate the same permutation. If you just sample random permutations do you get the same answers to the statistical questions that can still be answered?

Would be nice to understand Morishita someday. Dense reading but the analogy seems profound.

how about trefoil knot as an analog of 2? i mean, surely someone checked that they don't match, right?

35:24 he talked about quadratic reciprpcity

aah....

I could link so many knots that you would need transfinite primes to account for them. 😂

Is this talk related to Langlands program?

The Cohen forces were strong in this lecture

There might be a typo in the slides. At 25:30, the chance of a long (>half) knot when tangling 100 threads in 69%, whereas on the next slide the chance of tangling 230 threads in also 69% or am I missing something?

@MrDemultiplexer

4 жыл бұрын

He starts talking about "exchange rates" roughly at 24:19.

@DeclanMBrennan

4 жыл бұрын

@@MrDemultiplexer Yes but the exchange rate is from threads to digits. These two stats are threads only.

@MrDemultiplexer

4 жыл бұрын

@@DeclanMBrennan Looks like increasing the number of threads _does not alter_ the chance of producing a long knot (more than half the length) - always 69%. However, increasing the number of threads _decreases_ the chance of getting a _single_ knot (makes sense: getting a single knot with 1000 threads is much harder than getting a single knot with 5 threads). Likewise, increasing the number of threads _decreases_ the chance of producing _exactly two_ knots (makes sense: similar to the example in previous sentence).

@DeclanMBrennan

4 жыл бұрын

@@MrDemultiplexer Thanks for the reply. That was my take too although it doesn't jibe with my intuition. That being said an intuition that was honed by my ancestors swinging through the trees may not be up to the task. :-)

You can compare knots with Gaussian primes.

How does this relate to lattices like E8?

@sajateacher

4 жыл бұрын

Taylor Moon - Did you say “lettuce”? “He ate”? (I must have skipped lunch...)

@MrM12LRV

4 жыл бұрын

It was so interesting. I feel like they are connected...

If you got lost, it will end (you are done).

I think it is not clear why he picked such big primes (other than the statistics converging). The exact answers aren't important as he says, so stick to primes less than a 1000 in your talk.

It could be better if you did provide the simulation with animation, but you’re doing a good job at explaining thanks

Very nice accessible lecture. I had a question though: it doesn't seem that the results give a convincing connection between the geometry of knots and prime numbers (probably for pedagogical choices in what is presented rather than it not existing though). The reason I say that is that it seems that one doesn't need the geometry/tangle at all to ask these questions, only the permutation they induce. If you forget the tangle, and just look at where the n'th string going into the tangle comes out, then the number of knots in the tangle is just given by the number of orbits of this permutation. Similarly, the parity of the linking number (I'm pretty sure) only depends on the permutation and not the tangle, since you're just looking at the number of transpositions between two orbits. To stress: I don't mean this to look critical at all, rather I'd just be interested in if the above were down to pedagogical reasons (i.e., that it's more visually apealing to talk about tangles than permutations), or if there just isn't that much known on the knot side of the story so the correspondences are more conjecture. The examples mentioned at the end suggested that there's more going on, but there wasn't enough detail to be convinced. Anyway, nice talk!

@steviebudden3397

4 жыл бұрын

I think (stress think, not know) you're right about the number of knots depending only on the permutation, but not about the linking number. If you take the simple case with just two strands, then if the strands twist round each other an even number of times and then link back to themselves then you get two knots, as you state, but their linking number will be that even number divided by two. So there's a bit more information needed than just the permutation. Does this make sense? If you're particularly interested you could do worse than to look up 'braid groups' and the 'closure of a braid' which is what he is sort of talking about. I think.

@jamma246

4 жыл бұрын

@@steviebudden3397 _"If you take the simple case with just two strands, then if the strands twist round each other an even number of times and then link back to themselves then you get two knots, as you state, but their linking number will be that even number divided by two. So there's a bit more information needed than just the permutation. Does this make sense?"_ I agree that the linking number needs the knot (really, we're talking "braids", see en.wikipedia.org/wiki/Braid_group [edit: doh, just seen that you mention that at the end of your comment!]). But I don't think you need the braid for the linking number _mod 2_. In your example, the linking number is even, right? Which is the same as 0 mod 2. It doesn't matter at each crossing whether one string wants to go above or below the other, because each choice just adds 1 mod 2 to the linking number. More precisely, the linking number will be 0 mod 2 if there are an even number of crossings, and 1 mod 2 if there are an odd number of crossings. But I think "number of crossings" only needs to know the permutation, mod 2.

@steviebudden3397

4 жыл бұрын

@@jamma246 No, I don't think so. I think that adding one more twist to the braid before closing up will change whether you end up with one or two knots. If you have an odd number of crossings then you will only end up with one knot and so no linking number at all. You need to have an even number of crossings to get two knots and so you need to add two crossings to change the linking number between odd and even. So I think what you have is: Crossings mod 4 = 0 gives two knots, even linking number, identity permutation. Crossings mod 4 = 1 gives one knot, undefined linking number, switch permutation. Crossings mod 4 = 2 gives two knots, odd linking number, identity permutation. Crossings mod 4 = 3 gives one knot, undefined linking number, switch permutation. So while you do throw away most of your knot information, I agree with you on that, I think we need a little bit more than just the permutation. I feel confident in saying that things will get a lot harder with more strands as well. :D

@jamma246

4 жыл бұрын

​@@steviebudden3397 _"If you have an odd number of crossings then you will only end up with one knot and so no linking number at all."_ That's a good point. I oversimplified when I was talking about just counting crossings though: For the linking number, you take two components (by definition) and count the number of crossings between them (you ignore crossings of the knot with itself). Again, it's irrelevant whether a string goes "over" or "under"; mod 2 it's the same. But, as you've excellently brought up, you should only consider crossings between two different components! Whilst you don't need the full power of the braid for the linking number mod 2, a priori you need to know which (adjacent) transpositions happen and in which order. For example: suppose that I have 5 strands. I could consider crossing strings (a random example here) like: (3,4) then (2,3) then (1,2) then (2,3) then (4,5) then (3,4) An (adjacent) transposition (n,n+1) means that you pass strand n under strand n+1, or over... I don't say which! Here's a picture of it: 12 3 4 5 | | | | | | | X | | X | | X | | | | X | | | | | X | | X | (it's kind of hard to draw as text and it's probably better to draw horiztonally, but hopefully it makes sense). We can work out how many components there are: A) 1 -> 4 -> 1 B) 2 -> 2 C) 3 -> 5 So there are 3 components. For example, to see that strand 1 leaves at 4 in A: 1 stays at 1 after (3,4) and (2,3), moves to 2 at (1,2), moves to 3 at (2,3), stays at 3 at (4,5), moves to 4 and (3,4). Picking any pair of A, B, C, you can work out their linking number mod 2 just from this sequence of transpositions. Again, you don't need to know which strand goes over and which under. There's a complication though: whilst it's true that every permutation can be expressed in this way as a composition of transpositions of neighbouring pairs, this expression isn't unique. What if we reordered these tranpositions, and thus maybe get a different braid? Well, there's the Coexeter representation of the symmetric groups, which allows you to: a) swap the order of consecutive adj. transpositions s and t if they involve different strands i.e., if they swap order of (n,n+1) then (m,m+1) so long as n and m differ by at least 2. b) remove (n,n+1).(n,n+1) [which means swap a pair, swap back again]. c) swap something like (n,n+1).(n+1,n+2).(n,n+1) with (n+1,n+2).(n,n+1).(n+1,n+2). (try drawing a picture! Looks like something called the 'Braid relation' from the braid group.) If you change by rule a), you don't change linking number mod 2. Indeed, these just correspond to moving one crossing forwards a bit and another back a bit, not changing the braid in a meaningful way and thus preserving linking numbers. Changing by rule b) doesn't change linking numbers mod 2. If strand n and n+1 were different components, then you get 2 new crossings, so no difference mod 2 (whether each is over/under). Finally, for c). You can just draw a picture and see that each strand crosses any other exactly once in this rule (involving 3 strands), for both composition of 3 transpositions, so again changing by this rule doesn't effect linking number mod 2. So I think that's the proof! Sorry it's rather long. The point is: - give me a permutation... - I can express it as a sequence of adjacent transpositions - this gives a braid diagram except I don't say which crossings are over, which are under - but that's clearly enough to define linking number mod 2 for that representation of the permutation as a sequence of transpositions - but changing a transposition by one of the above "moves" also doesn't change linking numbers, and all ways of expressing the permutation in such a way are related by such moves. So which original representation of the permutation we chose didn't matter and we're done.

@steviebudden3397

4 жыл бұрын

​@@jamma246 Hello again. Well done with the braid diagrams, they work really well! I was thinking of trying to draw some myself, but I wasn't brave enough. Emboldened by your example I'll have a go. However, I'm not sure that I've understood you properly. It's the: " - this gives a braid diagram except I don't say which crossings are over, which are under - but that's clearly enough to define linking number mod 2 for that representation of the permutation as a sequence of transpositions " part that I'm not sure about. Consider a braid like the following. 1 2 | | X | | X | | (Actually that drawing works quite well) If the first crossing is left to right and the second right to left then we'll have two unlinked loops, crossing number 0. If instead we have both crossings as left to right then we'll have two linked loops with crossing number 1. That's a different crossing number mod 2. If you look at his picture of linked knots in the video (29:00 onwards) particularly at the left hand knot (with linking number 1) then you can see two crossings. Indeed in any picture of this link (the Hopf Link) then you must have an even number of crossings. - every time the left hand loop goes "into" the right hand circle it must come back out again.This why, when computing the crossing number of a pair of knots, you count all the crossing +/- according to sign, then divide by two. Without that last part then all linking numbers would be even.

I hate to be that guy, but Ed Witten did not find the Jones polynomial in quantum theory. He found a different knot invariant that applies to higher dimensional analogs of knots.

Around 19:30: haha, he doesn't explain why the estimates get better as the numbers get larger, but I bet it's because whatever the formula/theorem, etc, it's a limit...so when it approaches infinity, the estimates approach some formula, etc...so it makes sense for the estimate to only get better as the numbers get larger, because they get closer to being infinity...

11:21 "Every number is a product of a prime number." Is that true? (Edit; could be...? I think I get it. Obviously terrible at math.)

@zubrz

4 жыл бұрын

if it's not a product of 2 or more numbers or prime numbers - then it's a prime number itself

@Zeegoner

4 жыл бұрын

@@zubrz Wrong. 1 is a number (read: Natural *number*), so the first part of your sentence ("If it's not a product of 2 or more *numbers*") is incorrect.

@zubrz

4 жыл бұрын

@@Zeegoner : )

@Zeegoner

4 жыл бұрын

@@zubrz Very important not to mislead

@zubrz

4 жыл бұрын

@@Zeegoner i agree with you, nice that you mentioned this point about 1

So the universe counts in base 10/2.3.... woah... (Keanu Reeves meme...)

@ProductionPresents

4 жыл бұрын

It might be because 2.3 is approx ln 10

Skip to 4:30 for the start of the talk.

@Mauromoustakos

4 жыл бұрын

What for? What is the reason? In that way, you will miss the wonderfull color of the hair of the one who presents the speaker.

But no math... :(

very disappointed... i read the title as Primates and knots and thought this was going to be about chimpanzees

@garyraab9132

4 жыл бұрын

jkickass View the Cognitive Tradeoff Hypotheses video, that is once you descend from swing in the branches. ;)

I have a question: Did Mankind INVENT Mathematics or did it DISCOVER Mathematics? Same with Logic.

@jawaji7435

4 жыл бұрын

YES

@ElSachinoo

4 жыл бұрын

google.com

@steviebudden3397

4 жыл бұрын

That's a BIG question, too big to do justice to here. But I'll just say that in my opinion we invent it. Maths only potentially exists before somebody writes it down (or however they express it). The same could be said about the steam engine or anything else that we traditionally call invention. That is my very brief take on the question.

@NoActuallyGo-KCUF-Yourself

4 жыл бұрын

It's only a confusing question, because the question itself is confused. It uses "mathematics" in two different senses. In the sense that mathematics is a language with its own symbols, notational conventions, and symbolic ways of expressing logical relationships, it is man-made. The logical relationships themselves are discovered and then translated into the language of mathematics. In short, people discover mathematical relationships, but invent mathematical notation to express those relationships.

@mikhailmikhailov8781

4 жыл бұрын

The particular way we think about mathematical problems and formulate them is artificial, but much of it is sourced from nature. If we are going to encounter an alien civilization, they are going to be thinking about number theory and geometry and will likely ask many of the same questions and find many of the same things. If there is something arbitrary about math, its as arbitrary as God deciding to use Geometry to create this world.

I don't understand whether the speaker is seeking approval to teach a course in this subject, he admits the talk is a summary of work from others. You can't reduce geometry to a physics problem or some kind of simulation.

@sarthakgirdhar2833

3 жыл бұрын

The guy is a field's medalist for god's sake. He knows what he is trying to do

@willjennings7191

3 жыл бұрын

@@sarthakgirdhar2833 Blind acceptance defeats the point of that award.

happy to have seen kzread.info/dash/bejne/impnr7mTebGZl7A.html before this one

You explicitly stated that ‘every number’ is a product of prime numbers. 1 is a number; so kindly show us the prime factorization of 1 (in a way that doesn’t resort to goofy assumptions). The statement should have read: Every composite, non-prime number has a prime factorization. For the love of mathematics, careful how you state the fundamental theorem of arithmetic.

@AndreRhineDavis

4 жыл бұрын

Actually, the fundamental theorem of arithmetic states that every positive integer is equal to a product of prime numbers. 1 is the empty product, i.e. the product of no numbers, and hence a product of prime numbers.

what use is all this nonsense

the introduction to these videos is not really interesting to people on KZread, is it? Why put it there?

@ipudisciple

4 жыл бұрын

4:30

@TheDavidlloydjones

4 жыл бұрын

Quite right, LOL. I think people need to learn that recording a speech given to a live audience gives you some, but only a part, of the material that might make a KZread.

@TheDavidlloydjones

4 жыл бұрын

@@qed100 So?

@TheDavidlloydjones

4 жыл бұрын

@@qed100 It's not my complaint, but if you want to know more about the speaker because you've found their paper interesting, Googe is your friend. E.g. en.wikipedia.org/wiki/Akshay_Venkatesh. Interestingly, the introduction doesn't hit one of the more interesting things about this guy: he's a major league physicist at the same time as being a ver-ree edgy mathematician, and all with fine style.

@nebularwinter

4 жыл бұрын

@@qed100 thanks for your insight and for taking the time to lesson me

An example of the answer to a question I didn't know how to ask, properly, as a Mathematician would know. The Observable Universe properties of mathematical reciprocal positioning formulae is e-Pi-i resonance imaging projection drawing principle-> inclusion-probabilities = prime-ary numberness, and the sum-of-all-histories, excluded remainders, that are potential possibilities of frequency density and relative rate of change intensity, ..Modular Arithmetic = Superspin/circular logic, because existence substantiation is continuous creation connection of temporal modulated standing wave positioning pulses, much as it is described by analogy in the content of the video.., thank you. Moving into square roots and roots to any functional relationships is to discover dimensionality in numberness, Polar-Cartesian type coordination reference to frequency-resonances/co-factor modulation. Logarithmic mathematics forced on us by e-Pi-i connection/existence is definitely "an Art".., it's the 3D+T fields of Geometrical Orthographic Projection Drawing and Perspective derived from the coordination of QM-TIMESPACE Principle In-form-ation formulae in String Theory, Mathematical Abstractions.

Too speculative. No hard proof there is a relation.

@satyu131089

4 жыл бұрын

But it's important to build analogies like this. We understand most things, not just in mathematics by analogy.

@Ballosopheraptor

4 жыл бұрын

SO, nothing should ever be talked about as a conjecture until it is proven... right.. that's how the history of math and science works. Stop being a useless naysayer and actually say something useful, or just shut up.

@randairp

4 жыл бұрын

@@Ballosopheraptor Think about how many papers *assume* Riemann's Hypothesis, by your argument we shouldn't have to care. Think of all the outstanding results founded upon the Church Turing thesis or the Curry Howard Correspondence. Think of, for instance, how much we know about complexity theory isn't whether P = NP, but rather IF "some statement is proven" THEN P = NP. Mathematicians need these kinds of implications and relationships, otherwise, it isn't really clear how to prove a conjecture in the first place.