You just ooze mathematical passion, very refreshing!

@earfolds

6 жыл бұрын

Agreed; it really shows that you care about mathematics, and I can't help feeling excited about it myself.

@alexShpilkin

5 жыл бұрын

The Catsters (kzread.info/dron/5Y9H2KDRHZZTWZJtlH4VbA.html) present category theory like this (although there you do need to have some prior motivation to care about their stuff).

Don't think we don't see you trying to sneak in teaching higher category theory under our noses, +PBS Infinite Series. lol

@recklessroges

6 жыл бұрын

I am glad to hear any theory for which she has this much enthusiasm.

@anima94

6 жыл бұрын

Only now starting to read about category theory, but I didn't see any in here, what did you mean?

@AnarchoAmericium

6 жыл бұрын

If we view spaces as objects and paths between spaces as morphisms, then we have a category, a path category, or if the paths are invertible, a path groupoid. Abstracting the above idea further, you can create a path n-category/n-groupoid, where spaces are objects or 0-morphisms, paths are 1-morphisms, paths of paths are 2-morphisms, paths of paths of paths are 3-morphisms, etc all the way up to n-morphisms. John Baez gives a good exposition: math.ucr.edu/home/baez/homotopy/homotopy.pdf

This video cleared my skin and inspired me to stop procrastinating and finish my grad school application. I might show these videos to my Quantitative Literacy class next semester, I think it's a great example of how complicated things get when we look at concepts that lack many of those nice field properties numbers have, and hence why we like working with numbers whenever we can. Also, maybe some of them will get a taste of higher mathematics and decide they don't hate math as much as they thought they did!

@pbsvoices

6 жыл бұрын

I love this entire comment, but the first sentence is a prime sentence.

I was a bit worried when the host of these series changed, but you're doing wonderful! Thank you! And please don't trivialize things, I've seen other comments mentioning the complexity, but if my vote counts (and I hope it won't only be my vote), then I'm all for this level of presentation.

She's talking to me like I'm 3 years old but the topic is loop concatenation on topological surfaces (I think). I can't process this conflict. It's like Sesame Street doing an episode on the Riemann Hypothesis. Help.

@paulfoss5385

6 жыл бұрын

I more so get the impression that she genuinely has a warm bubbly personality, and that she’s excited about the topics than that she’s talking down at us.

@the1exnay

6 жыл бұрын

Paul Foss Those two things arent mutually exclusive

@timh.6872

6 жыл бұрын

I'd actually really like to see a sesame street style puppet show doing advanced mathematics, now that you bring it up...

@AaronHollander314

6 жыл бұрын

Relax Dr. Cooper

@paulfoss5385

6 жыл бұрын

I feel that they kind of are mutually exclusive, since interpreting her tone as condescending would mean that it is an affectation rather than sincere as I'm claiming.

I love the emphasis on how one topic in math can end up relating to all sorts of different things throughout the rest of mathematics! That's definitely my favorite thing about abstractions in math!

this was an absolute pleasure to watch.. thank you

I'd like to echo the other commenters requesting that you slow down a bit. A good idea is to have pause cues like in Kelsey's videos: "Here's what this result is -- pause here to think about why" and then explain it. Still, an excellent first two videos!

Since associahedra can be of any dimension, shouldn't they be called associatopes?

@whatno5090

3 жыл бұрын

-hedron is also annoyingly used for general dimensions, unfortunately

@xnopyt647

3 жыл бұрын

@@whatno5090 It sounds cooler lol, so I'm glad mathematicians stick with it

Oh boy, I can sense this is the end of cute combinatorial problems and in this series we're gonna explore ω-categories, operads, simplicial sets, higher topoi and end up by proving homotopy hypothesis. I ain't even mad.

@youtubeuser8232

6 жыл бұрын

I really hope so!

Fascinating subject to start off with! I like how the topic from the last video was extended in this one. I'm glad you've kept a moment at the end to respond to some comments and help us think further about ideas from the last video. I especially liked the small challenge you gave us at the end of this video. It would be great to have a challenge like that within the main part of the video. Kelsey used to include one often in the first half. I got a lot out of having the opportunity to pause the video, puzzle through a few of the ideas via the challenge, and then continue on to see how the ideas become more complex. Thank you for continuing this great program!

I love this! My graduate work touched on operads and A-infinity space (really, more often E-infinity spaces) a lot, I've never seen an introduction to them this accessible!

I thought about it since the last video and on this one. There are many comments talking about the speed. As is evident in the comments, there are many different ways how people perceive "speed" in this case. Many people process speech differently. Some will say she is too fast, some will say she is too slow. One has to consider non-native speakers (like myself). This "speed" is just about listening to the words and understanding and recognizing sentence structure. Pauses (along with intonation, volume and speed) are a form to emphasize and make it easier to process. I found this was improved in this video. The other thing is the pacing of the content -- how the subject matter is presented and how quickly new concepts are introduced that the viewer needs to understand and process. Here it matters how familiar the viewer is with the subject to determine whether it's perceived as "fast" or "slow". For example, I felt the recap at the beginning to be very slow, because I was already familiar with these ideas. It's difficult to find the right balance and make a video like this accessible to most people. This video already felt better than the previous video. I still prefer Kelsey's presentation, though. And I think it has nothing to do with speed or pacing … Keylsey (to me) always felt like she just walked in, saw the viewers and just thought of a cool thing to explain to them, just because she can, then somebody took the video and animated to it. It felt very spontaneous and off the cuff. Tai-Danae (again, to me, very subjectively) feels like she is working off a script. Reading off the tele-prompt. This can lead to rushing, especially if they have to do multiple takes. At the end of the video, when she answers the comments she seems to talk freely, which naturally introduces some pauses as the speaker thinks about what to say next. This is a very natural way of speaking and feels more like a conversation. So I don't think it's about "speed" at all. Maybe it's just different and unfamiliar. Tai-Danae is doing great so far and I'm sure she will improve her presentation skills as much as I will get used to her style.

That was incredible ! Thank you very much, your enthusiasm is so contagious !

This is fantastic. I love the pace! Killing it.

@TheLaughterAsylum

6 жыл бұрын

Landon Proctor hi ease see this also kzread.info/dash/bejne/iZ-Js86Sd8LdlKw.html

I'm *really* liking the direction this is going. I'm sensing some n-cat lab goodness.

Great job. I can tell you enjoy the topics. That makes it so much better than those channels where someone just reads what their researchers prepared while proving they know nothing about the subject because they don't even know how to pronounce some major words.

Awesome vid! Love that this is sort of meant for someone who has worked in the area a bit, while remaining entirely non-technical.

@1:47 there is a mistake in the "two ways to multiply"? There are two instances of (ab)c (ab)c instead of what I assume should be (ab)c a(bc)

I love your presentation style; A cauldron of bubbling mathematical enthusiasm! Perfect for teaching. Thank you Tai-Danæ.

Damm, I like it - and your passion is infective.

To be honest, i think it was the very first time that i was actually waiting for a new episode. Interesting topic

You are wonderful. This video was wonderful. My mind is absolutely blown by the depth and beauty of the mathematical landscape. Keep up the fantastic work. Sincerely, your fan!

Glad to see Infinite Series is still in good hands.

Loved this episode a lot. Definitely got me interested in homotopy theory. Will be checking out these references in the video description. Please keep it going!

I really enjoyed your fast pace... That video was interesting as I didn't get very deeply into Algebraic Topology in my first course in Topology.

The homotopies between points to make paths, between paths to make faces and between faces to make solids reminds me of the progression of Categories, to Functors, to Natural Transformations etc.

9:00 for everyone who is wondering: Top-right is quantum mechanics Top-left is sin and cos for half angles

The conceptualization of transforming these different multiplication orders to one another, begs a couple of questions. 1) Do points along these paths correspond to points on the associahedra of slightly different binary operations? If so, how would the two binary operations be related, and would this allow the construction of a sort of fractional algebra as to express conveniently one in terms of the other? 2) Why, for 5 loop multiplied with one another, would the associahedron be 3 dimensional, would not a 2-d 14-gon suffice? My intuition suggests that the faces coincide with some sort of ordering of the parenthesis, but admittedly I have not done the requisite math to confirm or deny that suspicion. The dimensionality of the respective associahedron seems arbitrary; is it?

@FrogsOfTheSea

6 жыл бұрын

If multiplying two loops gives you a point, and multiplying three loops gives you a line, and multiplying four loops gives you a 2D shape then it’s definitely intuitive that five loops would give a 3D shape. It makes sense that each extra loop that you throw into the mix would increase dimensionality by one, since you’re gaining an extra degree of freedom in terms of how you set up your multiplication.

@stevethecatcouch6532

6 жыл бұрын

A 2-D 14-gon would suffice if you were interested only in the homotopies between the product loops. To look at homotopies between those homotopies, the 2-D 14-gon needs to be embedded in something other than a plane or one face will overlap the others. If you want to look at homotopies between homotopies between homotopies, you need a 3rd dimension.

@Mr4thdimention

6 жыл бұрын

I too am curious why we get the particular shape we get. In particular, what is the relationship between the points that make up a single face on the 3D polyhedron?

Wonderful set of videos! Great job!

I feel like im watching one of the early episodes of PBS spacetime where Gabe blew my mind with the intro to general relativity videos. Love the new pacing the show has and love this new mini series! :)

OMG THAT WAS BEAUTIFUL

How can we structure abstract spaces? This questions seems to be at the heart of much of mathematics and science. I feel, as ever, that my mind has expanded its reach just a little. What an elating feeling! Cheers!

it's not the pace so much as the lack of pause, but great none the less, i really enjoyed it, this channel is awesome, always making me think!

I love the new speed of the channel!

THANK YOU. This was fantastic. Time to start supporting you on Patreon

this is so deep, fantastic!

Loop concatenation as defined on the video isn't commutative because you have a sense of order, i.e. if you concatenate a and b, you run a first and then b. In other words, if a(s) and b(s) are loops and the multiplication is defined as: (a • b)(s) = a(2s) if 0

It's not immediately obvious to me why those "homotopy graphs" aren't just complete graphs. isn't there a homotopy between any two vertices? Just smush the different colors of intervals around a bit and voilà.

@rmsgrey

6 жыл бұрын

What confuses me with the homotopy graphs is why the range of available states is bounded by homotopys between loops generated by the binary loop concatenation process rather than allowing, say, a loop that spends 97% of its time going round a, then 1% each on b, c and d. Though, yeah, the reason for the geometry of the graphs isn't obvious - why a flat pentagon rather than a 3D shape? Particularly considering that there would appear to be 3 degrees of freedom involved in partitioning the second into 4 chunks.

@Tupster

6 жыл бұрын

They aren't graphs, they are convex hulls.

@sofia.eris.bauhaus

6 жыл бұрын

Jason Wilkins okay then. why arent the associahedra not n-simplexes? :P and what is the difference between their insides from their outsides? maybe i just turned really stupid, but i feel like i didn't get anything from this video

@Tupster

6 жыл бұрын

They are convex hulls because they are linear combinations of terms. Their structure is determined by how you can arrange parenthesis. I guess that doesn't correstpond to n-simplexes? Outside wouldn't be a linear combination because the weights of all the terms would add up to more (or less) than one.

@Tupster

6 жыл бұрын

rmsgrey, I think you are confused because it isn't a graph. .97a + .01b + 0.01c + .01d is perfectly allowable. This sum of products of weights (where the weights add to one) is called a linear combination, and the space of all the possible combinations occupy is called a convex hull. It is easy to think of a,b,c,d as, say, 3D points, but less obvious when they are some abstract algebraic object like how terms associate, but the concept is the same.

To be honest, this episode made waay more sense to me that the last one. Also, i really like where the questions got explained. That really cleared some things up I think?

Doing a great job. Really enjoying your videos.

I really enjoyed this talk, Thank you.

This was super cool!

I cried. Tears of happiness.

Thank you for teaching general audience higher category theory

This was cool. I feel like I finally understand what HOTT was talking about with it's "homotopies have computational content" (or w/e the exact wording was).

Whoa, whoa, whoa. Some of us have never heard of this before. This is a whole lot of information in ten minutes. I felt somewhat okay with the episode before this, but I can't keep up.

Don't get stuck up on the multiplication. In my university class we sometimes even put indices on the multiplication Symbols (cause there are so many different ones). It really is just a Symbol and depends in the definition you give it. You could also use little Smileys but that would get tideous, dots are just simpler and mathematicians are lazy.

This was super cool! I learned a lot. I would like to see a video on Bernoulli numbers plz

This is amazing!

Talk about Model Theory!

Fantastic.. loves the way they deal with mathematics..👍👌👌

Wow! This.... This is just awesome😂!!

I may have yelled out loud when I recognized the Catalan numbers before they were named in the video. _> Also, I met Richard Stanley once. He was an interesting person. Smart as hell, but still very interesting. I wish I had gotten to talk to him longer.

Hey, welcome to the channel. I'm not usually comenting but decided to do because you're kinda newguy here and I loved the topic a lot. If you're also working on the content of episodes, you really are a good addition to Infinite Series :)

this is so cool !!!

I'd like to see a videos about Homotopy Type Theory! And keep it up, I love these videos!

There is also some recent applications of homotopy in logic, involving proof assistants and Voevodsky ; see Homotopy Type Theory

Hmmm I have a question. What would it look like for there NOT to be a path between two paths (i.e. the polygon can't be filled in)? Could you show us an example?

@danielrogalski905

6 жыл бұрын

I'm presuming you could do this by creating a discontinuity in the function, for example by creating a point that could not exist because of division by zero. Of course, if that happened the entire loop would disintegrate, so I'm thinking you would have to create a family of loops, where loops could exist within loops and you would try to look for a path between two or more "superloops". That way one or more of the subloops could be frayed, but the larger structure of the loop would remain intact. Then one of those subloops inside a superloop would be broken up, so certain paths would not be filled in and there would thus be a hole in the resulting associahedron. Of course, this is way beyond my ken but that's what I'm thinking would be likely

I think it's fantastic that Mathematics is being explained in this lucid way. It's still difficult because mathematical concepts are built on a series of mindbogglingly abstract ( but demonstratably irrefutable) structures. It will be very helpful to have some breaks in the video that can illustrate the application of these theories in the real world that we can relate to somewhat. That will help make my neurons "associate" more with the topic. Another way would be if we could inject some comedy - but that might be asking for too much :p

Wonderful!

You should explain (at about 3:50) why, for K4, the pentagon is not instead a 4-simplex (and why for K5, the enneahedron is not instead a 13-simplex, etc). For the benefit of other readers who might not already know... this is because each edge of an associahedron must represent a single application of the associative property (i.e. must reposition only a single pair of parenthesis). Thus, there cannot be an edge from ((ab)c,d) to (a(bc))d, as this would require altering two pairs of parenthesis. We can construct a homotopy to link these two vertices, but it will not be a proper edge of the resulting polytope, so we get a pentagon instead of a 4-simplex.

Great video, great resource to learn Thanks a lot providing free content and knowledge

nice video different from traditionally going to the fundamental group..more honing in on loop spaces at a little lower level then the fundamental group (i.e breaking down the property of composition without equivalence or homotopy to get some really cool results)

Cant wait to the physics hiding behind this!!!! Im loving this! 💕💗❤

I like this new host ~

You are a great host, keep it up!

Play at 0.75 speed.

@ObjectsInMotion

6 жыл бұрын

I play at 2x speed usually. I recently downloaded an extension that raises the cap and I watched this at 2.5x because I felt she was slower than the last host.

Just superb! Now around five loops, the homotopies go on for forever. Clearly. Your assignment is to find out first, what happens for for forever, and then what do we have to do to get an infinite number of fors into this game?

I had found Catalan numbers while counting total number of binary trees possible with n-nodes. It has a good recurrence relation.

Wow! Awesome :). I thought your explanations were super clear but I do agree with some of the comments that pausing longer on certain tricky points would be helpful for non mathematicians. Doing Kelsey proud though :)

To me, it makes a lot more sense to think about multiplying loops as a function of material used rather than time taken. I find it easier to visualize the difference between different multiplications when I think of the bar as representing how much of a fixed length string would be used to complete that part of the loop c:

This is VERY cool

I started wondering if operads were going to get mentioned, and then they did! I hope there are a few episodes to come on them.

That was amazing. Thank you so much. A question that I'm left with is how the A infinity spaces depend on the underlying topological space. Are there clear ways of relating the base space's fundamental group (for example) to the structure of the resulting associahedron?

Jim Stasheff was my external PhD supervisor :-)

@JohnSmith-cl3ez

6 жыл бұрын

whoa! that's pretty neat, small world mon librepenseur; your paper at the Fenner Conference, AAS etc was intriguing. I look forward to hearing your lecture at USC innovation center on plasmonics at some point. et, your reflection on last years' M Giustina et al of IQOQI's Bell Theorem discussions. (which has implications for QDotler Memceivers). not to mention, your discourse about H deGaris, memristors, spintronics, optronics etc... To see so many from Australie's space program speaking is awesome.

@DavidRoberts

6 жыл бұрын

John Smith different David Roberts, I think. It's a very common name. I'm a pure mathematician :-)

At least in these first two episodes, compared to Kelseys', what I'm missing is the "pause here to try to figure this out yourself" moments. In this episode, for example, you could have first reminded us why loop multiplication as originally defined is not associative, then challenged the audience to devise an equivalence relation on loops that makes it associative, while still preserving some other important distinction among loops. Seeing the solution is more meaningful after having struggled with the problem.

i more or lessed accidentally clicked this video but i am entranced by the info wowow

Please keep doing videos on topology! I hope the series covers homology and cohomology...

It’s be great to see an episode on homotopy type theory (HoTT), something that’s a pretty hot topic in programming language research, and how it relates to the homotopy theory discussed here. Also some category theory would be great too.

4:21 about the orange and purple edges, here's my understanding. Orange edge #1: (ab)c ~ a(bc), multiplying d from the right => [(ab)c]d ~ [a(bc)]d; Orange edge #2: Let e = (bc), then [ae]d ~ a[ed] implies [a(bc)]d ~ a[(bc)d]; Orange edge #3: (bc)d ~ b(cd), multiplying a from the left => a[(bc)d] ~ a[b(cd)]; Purple edge #1: Let f = (ab), then [fc]d ~ f(cd) implies [(ab)c]d ~ (ab)(cd); Purple edge #2: Let g = (cd), then (ab)g ~ a[bg] implies (ab)(cd) ~ a[b(cd)]. Hope this helps.

This looks like an interesting complimentary companion to the enclosure of a constant by a quantized field(?), eg in the "One Electron" theory, the wave-particle location is the inverse or reciprocal state of the available possibilities, for an electron to exist in relation to the universal probability environment. (I'm trying to interpret loop quantum gravity by a rational connection. Temporal superposition connection is equivalent to "All things are connected".., by loop-cycles of difference => general Homotopy because it's Modulation Mechanism/Function cause-effect of Quantum Fields) Another excellent presentation thank you. This explanation leads to the Four Colour Mapping story?

These videos are awesome

excellent, so excited for combinatorialist! awesome stuff! so, it seem some have done non-standard analysis, et seen S inaytullah or K Wilber lectures then. I jest, you're a post-doc, so it's all good hehe. thus, the knot theory have some practical applications then hehe, as Perelman reflected. I look forward to your vid on Bayes Theorem, futures modelling (WFSF) and on the E8 lie structure, or nth dimensional topology, 4Glomes etc. can you also cover the mathematics of plasmonics?

yes. i think it stands to reason. if you ask what happens when you multiply things that are not numbers, then you have to ask what numbers really are

Is there a way to define a concatenation of curves that begin and end in the same point that is commutative or associative or both?

Commutivity means that a*b=b*a. But if you go through loop a and then loop b, that’s in a different order than going through loop b and then loop a.

Great show. I'm looking forward to some good maths. You'll be brilliant I'm sure. :)

Is there an f(n) which generates the Catalan Numbers?

@ZonkoKongo

6 жыл бұрын

Daniel Shapiro maybe check out OEIS

@viktort9490

6 жыл бұрын

The catalan function ?

@MrTeknotronic

6 жыл бұрын

The Wikipedia page for Catalan numbers has it

@dcs_0

6 жыл бұрын

ah yes , f(n) = (1/n+1)*(2n choose n) Thanks guys

very interesting!!!!!!!

10:12, what's that weird blue bar on the left?

@dcs_0

6 жыл бұрын

wow good spot!

@MagicGonads

6 жыл бұрын

The proof of the reimann hypothesis

@dcs_0

6 жыл бұрын

Magic Gonads HAHAHA!

@googolplexbyte

6 жыл бұрын

10:15 last frame before 10:16

@chaoaretasty

6 жыл бұрын

Glad it's not just me, I was worried my monitor might have had a problem

Consider rings R_0 and R_1, using the same underlying set R, with the same addition. However, every product in R_0 results in 0, whereas R_1 uses a more-usual multiplication. Up to isomorphism, there is only one group of order 2, namely S2, the symmetric group on two elements. Because this group is abelian (in fact cyclic), we can permit multiple ring structures on it. However, only these two are permissible on S2. Let 0 and 1 be its elements, where 0 is its additive identity (the zero of our proposed ring). Obviously in a ring, all products involving a 0 return a 0, so we need only consider the product 1 x 1. If we return a 0, we would obtain the "all products return 0" ring on S2 -- this is our R_0. If we return a 1, we would obtain the field of integers modulo 2 -- this is our R_1. Obviously things would be more complicated with more ring structures permissible if we replaced 2 with an integer like 12 or 315. But for any integer N > 1, it is clear that both of these ring structures can hold on the cyclic group of order N.

So how are the orange and purple paths different at 4:10?

Man, I am strugling with this video... so much math!

All always enjoy watching your videos guys. I have a question about “Convolution” I study electrical engineering and one of my professors asked us to fined a physical explanation of convolution rather than the mathematical definition and shifting and all that.

Great video. I just didn't understand how we went from a 2D Pentagon to a 3D Polyhedron with 14 vertices. Why not a 14-sided 2D polygon?

Sooo THIS is how we'll traverse worm holes! It's all so clear now... I think?

Hey everyone Sure, she talks a little fast, but you can always pause the video and rewatch it.

soo good

I think the speed of your speech is fine. Yes, it's a tad faster than what other people do, but I have to say I find that rather pleasant.