The Multiplication Multiverse | Infinite Series
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- Опубликовано: 28 июн 2024
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What happens if you multiply things that aren’t numbers? And what happens if that multiplication is not associative?
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Multiplication of numbers is an associative property and we can make sense of “multiplication” between things that aren’t numbers but that’s not considered as associativity. And since we’re talking about associativity, you might wonder about that other property of real numbers: You know: when multiplying two numbers, swapping their order doesn’t change the answer. This property is called commutativity. But keep in mind: it’s a very special property to have! Not everything in life is commutative. For example, getting dressed in the morning... putting on your socks and then your shoes is NOT the same as first putting on your shoes and then your socks.
Link to Resources:
The Fundamental Group: ‘Loop concatenation’
www.math.uchicago.edu/~may/VIG...
Written and Hosted by Tai-Danae Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!
And thanks to Nicholas Rose and Mauricio Pacheco who are supporting us at the Lemma level!
Hi everyone, Tai-Danae here. It's nice to e-meet you all! Feel free to drop me your questions below! You can also find me on Twitter at @math3ma.
in my opinion you are going too fast, I can imagine how difficult it will be for someone who didnt ever see how a basic group is formed or how parametrization works to follow the main idea.
Hi Tai-Danae, nice to meet you!
zamalin2 you can always change the speed!
I think pausing just a little longer after sentences such as at 1:20 might help, as well as slowing down a little. Other than that, I'm really looking forward to more episodes with you!
Random Q for you: What kinds of infinities are there in music? I know that one could make musical fractals which(doesn't have to but) leads to at least one infinity are there others? o.O
Hey y’all! Many of you have asked a great question: Why are we calling loop concatenation a “multiplication”?? Technically, loop concatenation is a “binary operation,” i.e. a way to combine two inputs and get one output. Addition and multiplication of numbers are probably the most familiar examples.
In practice, mathematicians often use the word “addition” to describe binary operations that are commutative. And although I don’t mention it in the video, loop concatenation is not commutative! (Do you see why?) So we just call it “multiplication” instead.
And thanks for all the feedback! I’ve got another idea to help us get used to this notion of multiplying things that *aren’t* numbers, and it won't involve fancy algebra language. I’ll share it on my blog (www.math3ma.com) hopefully next week. Stay tuned!
- Tai-Danae
Hey Tai-Danae, great first video! I have a question, is there a specific reason topologists define loops in terms of their parameterisation instead of saying that loop functions that have the same image are equivalent? it looks to me like they are instead defining a loop in the space M as a path in the space IxM such that the ends of the paths are the same in a projection of the curve onto M
Thank you! Great video! Very interesting.
In furtherance:
Often there is often one 'natural' binary operation which is simply (sometimes trivially) defined, and a diversity of 'natural' binary operations which are less trivial and which distribute over the trivial one. The term addition is often used to distinguish the simple operation from the others.
E.g. operations on polynomials where addition is termwise (simple) but where multiplication is a convolution.
en.wikipedia.org/wiki/Convolution
Amazing episode, can you guys do more videos on Topology.
My intuition tells me that loop addition, whatever that may be, would not be constrained within the [0,1] interval (though part of me not knowing about this field says it'll never not be [0,1]).
my favorite commutative non-associative operation is winning in rock-paper-scizzors. B)
(✊ vs ✋) vs ✌ ≠ ✊ vs (✋ vs ✌)
sofias. orange Awesome! The operation might be better called "winner"
That's awesome!!!
"Now, to do this each car must travel at twice their original speed... but that's fine." I did not expect to crack up like that, quality delivery.
Shout out to the green screen guy for somehow getting her hair keyed correctly. I know that wasn't easy! You can see the insane amount of motion blur they used every time her head even slightly moves!
Infinite series is still is good hands!
2:55, 5th line, I assume that's a (-1) rather than a (=1)?
yep
Also, 4th line: "Anti-communatitivity".
Ha ha nailed it
"We'll discover that all of the different ways of multiplying 100 different loops in a topological space can be encoded in a 98-dimensional polyhedron called an associahedron." That sounds absolutely mad! I love it. I wonder what you'd consider the identity in loop multiplication.
First, we need to determine if there is an identity at all. And based on how it is defined, I doubt there is one, but I didn't do the calculations, so I couldn't know.
There indeed is no identity for this particular way of defining the operation, but that's okay, it doesn't mean we can't encode it.
Hey Tai-Danae, great first episode! Im a swedish 27 yo that has got a masters in engineering and your presentation pushed the boundry of what I know - thanks! Since I've taken quite a lot of math classes in uni, following your presentation wasnt that hard - but for somebody that still is in uni or in high school, the speed of which you introduce and jump between concepts and terms might be hard to understand. I've followed the PBS series on youtube since the first episodes, and all presenters did this in the beginning - so im confident you'll do a great job!
7:14
My mind just imploded
Literally Brainfucked
Before Associahedron - ahh, this is some interesting stuff about the associative property, I don't really get vectors but overall it's understandable and somewhat intuitive.
After Associahedron - ahh, this is why I won't study maths in university : brain hemorraging.
"The cool thing is that this forms what we call a Lie algebra...... ok, moving on."
Do not slow down.
This is a video. We people can go back and listen, pause. Cant wait for the next one!
I miss this series. And I think Tai-Danae Bradley was one of my favorite hosts for it.
I love where you're taking the channel. And I like how you're showing the notation while you're saying the statement to low-key introduce people to the language of mathematics.
Also, dropping the definition of a Lie algebra in for funsies - dig it. I'd love a short series on the importance of Lie algebras in modern math and physics, with some explanation of their properties (it might be tough to hit the right level for this channel, granted).
I wouldn't mind Tai-Danae speaking a bit slower.
Congrats on the first video as one of the new hosts :) Yay for more Infinite Series :D
yaaay, Gerstenhaber algebras mentioned in popsci video :)
Face and hands. Very creative idea for a host!
Amazing video! I really like the fast pace.
I really love this series. Thanks for all the great vídeos 🎉
Also, welcome to the channel! I like the pacing and the depth of abstraction, this seemed to go deeper faster than Kelsey tended to.
This video is amazing!! I'm like already super interested in topology, but I've never really thought about it much before. This is a PERFECT introduction! I can't wait for the next video :) :) :)
very happy with this channel. How it has started, and where it is going keep it up! (im a math major)
Amazing! You are fitting really well and preserving the vibe, the complexity and the speed of the content!, Thank you very much. Was worried for the show, not a anymore!
I don't know why people are complaining about a fast pace. I had no trouble keeping up, even though the idea of multiplying loops is completely new to me. You're doing great, and welcome to Infinite Series! I loved your first episode and I'm hyped for the next one :D
at 0:38 the face shifting is really jarring.
This is so cool! I don't have much mathematical insight to add, but this is really cool! Great first video, Tai-Danae!
Thank you for doing more math!! This is so great!! I'm sharing this to all of my physics friends!
a worthy successor indeed.
Loved this ep! Good work can’t wait to see more from the new hosts
Very cool! I very much appreciate hearing of so high-end stuff in such a easy going way (well it is fast, Mr. Alexander is right, but the beauty of those clips is that you can stop and rewind them as your please).
Thank you Tai-Danae. Nice Episode! Yes, you spoke quite fast but it was also rather clear and articulated, I liked it.
The pace is fine. People who complain are not used to Gabe's original pace for sure. That being said, you should know that if something isn't familiar for you already, you probably want to give some pauses and review some points. It's fine really.
This video helped me understand somethings that I was just not quite understanding with Homotopy. Thanks
Great first vid! I like the speed - the great thing about YT is you can always pause & rewind.
Hmm so you say this leads to rich mathematics? What my Professor did in his lecture was just define two curves x:[a,b] -> T an y:[c,d] -> T to be equal when you have a continuous function f:[a,b] -> [c,d] with x = y°f, so speed does not matter. But we also did it in the context of integrals over curves in R^N and not in a general topological space. The integral over such two loops are then the same.
The driver of that car is mathematical drunk and dizzy driving in those loops
Great Job ! , love the topic and all the stuff you said
Happy to see we’re going faster! Go go go!
Really good video guys! It seems a little fast though it would be easier to follow at about 80% to 90% the speed
Wasn't sure if the new folks (you) were going to be as good as Kelsey. I found this video and your delivery great. I'm really looking forward to seeing more of your videos! I thought the pace was perhaps better then before, even though I usually like slower delivery videos...or some reason, not with this video. Great job
I find it hilarious that there is Lie Algebra and Poisson Algebra, Together they form the Algebra of Poisonous Lies
Putting some loops together sounds intuitively like addition. On the other hand concatination is often represented as multiplication (at least they share operator symbols)
Hey! You have a very nice voice to listen to, and it makes the information very easy to listen to and comprehend! I'm sure you're going to do great, hope you enjoy the hosting!
Tai-Danae does a great job here. I think the writing has too little detail though, and assumes that people are familiar with linear algebra. You could use division and subtraction as examples of non associative/non commutative multiplications as warm up mulltiplications and then go into the linear algebra for the more advanced viewer.
The background and your shirt made me though for a while that your head was floating in the video
This is what I've been working on for the past 2 months! I'm working on 4 loops with 3 vectors.
Nice video, Tai-Danae!
Great topic! Love it. Also glad to see you so enthusiastic. For those of us at home, would you mind lowering the tempo a little bit? The video was only 8 minutes - with such rich material (and the same script) you could have paced it out to 10 minutes. I realize you're excited. We're all looking forward to see what part 2 of associativity looks like.
Man, Tai-Danae goes light speed compared to Kelsey. Love this channel, though.
I like it, though... and, if you remember Gabe's speed from his SpaceTime episodes, you know Tai-Danae is just the tip of the iceberg haha
I love this show an though I'm not an mathematician in any way, Kelsey always had a talent to make me understand and appreciate those often difficult topics.
I think Infinitite series is still in good hands, you made a great first impression. Still i would really appreciate it, if you speak more slowly and keep in mind that not all have a mathematical background. An 10 or 11 min video at a slower pace and with maybe with a few more examples or explanations (eg why or how vector multiplication is different) would help me a great deal :-)
I'm looking forward to next weeks episode.
Thanks for sharing!!!
Great video! Speaking of multiplication, please consider doing an episode on the Clifford product
An algebraic topologist! I'm excited about the direction of this series.
I like the "socks * shoes != shoes * socks" comparison :)
And suddenly I'm remember my Abstract Algebra class. The nightmares are coming back.
this was really interesting! i have no idea about this level of maths but the way you explained it made sense! :) also, i googled associohedron as soon as you said it and that looks interesting too!
subbed! :)
I like the new hosts (still miss Kelsey). Good luck to the new hosts.
Thank you. This is close to what I'm working on now. Combining simplex, torus, spheres, in vector space, not sure if this leads to Hilbert space, or a phase space.
Hope this makes sence to you.
The experiments involves B fields and their interactions.
me before watching the video
multiplying things that aren't numbers? Illuminati confirmed
Oh, god, please ! x)
Lol! :-)
She's talking about homotopy groups (which are associative) - almost. Usually we define equivalence classes where 2 loops are the same if they can be continuously deformed into each other. In her example, it's a simple matter of having the car speeding up/slowing down appropriately; the most important part is the path it travels, and that is the same either way.
Very interesting topic and I'm looking forward to the sequel! I recall that (under multiplication) the quaternions are non-commutative and the octonions are non-associative.
Gj Tai, relieved that you turned out to be a good host :) looking forwards to that 98d polyhedron !
Great Video ! I learnt a lot.
In my opinion, you are a very good host and you helped me stay interested.
Oh i cant wait for the next episode! !!😍
Everyone is saying is you're going too fast. You're going "too fast" for me too, but the truth is, in order for me to actually understand this, you'd need to go so slow it would take you several hours. At least. I dont know why I like watching these video's, when I know I won't really grasp it. But you're not too blame for that, you're an outstanding presenter.
Nice vídeo, excited about the nexo
This is way over my head. Like I understand but I don’t comprehend. Welp, back to Eons. I like how they talk about dinosaurs.
HEEEY!!! Nice to meet you too Tai-Danae! I'm excited to see what you will bring to us!
The sock shoe, shoe sock thing is a bad example because if you are looking at it from a "are they both on my feet" standpoint, then it works just fine. It's like saying four times two is eight. Yes, but if you are looking at which number is specifically multiplied first, the two has to come before the four otherwise it's four times two. You aren't looking at the functionality of the numbers in terms of how other numbers might look at them funny throughout the day, it's the fact that it's cumulative, just like socks times shoes equal sock-shoes.
Hi! great video in general. Mathematician here, so this video was just a refreshener, i would like to point out 2 things:
As many people said: you could probably slow down a bit. It will be less info but maybe you will deliver even better. (This applies to Gabe as well, it was my general complaint with his performance in Space Time).
I think this was an step up on the bare minimun level needed to understand the subject. I think you are delivering complex concepts in a more abstrack way than the former host. You might use more concrete examples (not only to known basic math, but examples from real life; reviewing what you just said and stuff like that)
I don't mind, i enjoying this videos, you are doing great in general, keep the good work!
I always thought you could put your shoes on before your socks...
Thanks for the video and welcome to the show Tai-Danae! The Jacobi Identity got me thinking how do mathematicians "perceive" a Lie Algebra? For instance I perceive a group as a structure in which one can solve ax=b (implying the algebraic properties); I might perceive a vector space as a structure in which objects can be decomposed (skipping the lengthy properties and focusing on the idea of basis). I'd like to know how mathematicians perceive the other algebras you mentioned for the the sake of understanding them better myself, maybe that could be a topic for infinite series. Thanks for your time.
I know this might be late but anyways. There are things called lie groups, which are basically higher dimensional surfaces( manifolds) with a group operation. You can think of Lie groups as symmetries of some other surface (or manifold) where the symmetries are continuous e.g. think of symmetries of a circle. You can rotate the the circle which is a symmetry of the circle. Moreover, you rotate the circle continuously by any angle. So symmetries of circle form a lie group. Since Lie group itself is a manifold, we can see what happens at the tangent space of the identity element. The group structure gives the tangent space a lie algebra structure. TL;DR, You can think of lie algebra as an differentiation ( or differential/. Infinitesimal) version of group multiplication ( technically conjugation).
wow amazing beginning!!Congrats
Not too slow at all! Keep up the pace and go deeper! People can rewind if they need to, it's a relief to not have something watered down and spoon fed.
Hi, Tai-Danae, I like your articulation. That was a nice introductory video... ^_^
Similar to matrices (or second order tensors if you like) quaternions also do not commute, but are associative. Quaternions can be used to represent 3D rotations. I have read that 4D rotations can also be represented by octonions, which also don't commute, but are also not associative.
Although I think you could slow down your delivery a little, I love the topic of choice and your presentation of the material! Abstract Algebra is my favorite area of mathematics, and I'm excited to see how associative structure ties into Geometry in the next episode!
That's beautiful!
You rocked this!!! Good job!!!! Excellent presentation!!!! You are a worthy successor to Kelsey!!!!!
Love Kelsey, but Tai-Danae is having a stellar debut on the channel. She went deep, but not too much and has a pleasant way of talking and explaining stuff.
Would love to see an episode on lie algebras!
Well done :D
Nice video.... I associate it with the greatness of thus channel and the mind-blowingness of math
Also ties into complex numbers (loss or orderness), quaternions (loss of commutavity), octonions (loss of associativity), and sedonions (you now have zero divisers.
Nice topic, well presented. I was all ready for the video to dwell overly long in the familiar territory of Matrix multiplication but then it took an interesting turn to loop concatenation. I didn't have any issues with the pacing except for the massive cliff-hanger it ended on... wait we can encode instructions in shapes? Tell me more.... *video ends*
Good video, well done, very clear. Thank you. Unfortunately I could not quite grasp multiplication in the form of loop concatenation (am I saying that correctly?) -- that is, not on the first viewing.
But for this series, given the choice between a presentation that is too slow and one that is too fast, I'd put up with the too-fast one rather than plodding along, which can be more frustrating.
You're an awesome host Tai-Danae! Cool topic also!
Nice! I like the style of the new host, and this is a cool topic on top of that!
Nice to meet you. Never got to greet a new host before :D
Associahedron: Coolest word I'll learn today.
Learn another: amplituhedron
At 0:56, there's also the third way of multiplying (2*5)*3
Sidhant Rastogi associativity only means that 'moving the parenthesis' doesn't change the output, not changing the order of the the numbers themself.
I'm surprised, if i remember correctly, sci show gets a lot more push back when it comes to new hosts than this. Well, keep up the good work-- I'm excited to see the rest of this (infinite) series!
Poisson algebras are like Commutators and Poisson Brackets right?
kudos! luv ur style
Right angle clusters in number groups and their distribution patterns give excellent geometry. How do we rearrange all these numbers on a number line.
Something like 345 689i .
YAY a channel that changed hands without gaining an impenetrable accent
Btw You lightly touched on non-commutativity without mentioning (Hamiltonian) Quaternions. HOW?
Great video! Just one comment, why link to notes on the fundamental group? Up to homotopy (defined on the 2nd page of the notes), concatenation is associative, which may confuse people.
Sad to see Kelsey go but the new hosts look great. Tai-Danae is awesome and it's wonderful to have the old PBS Space/Time guy back!