It is the greatest gift that some people could spend time to teach, to interact and respond.

As someone with a bachelors, masters, phd and a postdoctoral in nuclear physics I can attest that these lectures are superb and the best thing on the internet right now covering the topic. Kudos Sean! Just bought the first book of the series and anxiously anticipating the other two.

@Prophet_Isaiah

8 ай бұрын

You need one to understand this stuff! 😵‍💫

Your generosity is almost incomprehensible

Thank you, Dr. Caroll. As a matematician, it's perhaps the best explanation of a homotopy groups to a layman I've ever seen. And in the case if you're interested: - Bolyai was Hungarian and Lobachevsky (Лобачевский) was Russian. Actually we in Russia usually refer to hyperbolic geometry as "Lobachevsky geometry". - Yep, homeomorphisms are defined as continuous bijective maps, not necessarily smooth ones. - Technically speaking, the spaces you're working with when you speak of homotopy etc doesn't even need to be manifolds. But it's probably too much of a rigor :)

This series is the best thing that happened to KZread since Leonard Susskind's "Theoretical Minimum"

That was the best description of the Riemann curvature tensor I've seen, these videos are much appreciated

This series is the absolute best thing in the world right now. Keep up the great work Dr. Carroll !!!

@Amir-vw6rk

4 жыл бұрын

Yes, u should also check "#your daily equation" with brian greene

@davidwright8432

4 жыл бұрын

'best thing in the world right now. ' - and probably for oodles of light years around - in any direction!

@Petrov3434

4 жыл бұрын

@@Amir-vw6rk Thank you for the hint. Another one perhaps are lectures on viruses by Vincent Racaniello -- kzread.info/dash/bejne/gpqnmLV8mKi-ncY.html

@mgenthbjpafa6413

4 жыл бұрын

Yes,

@mgenthbjpafa6413

4 жыл бұрын

I just assist courses sincec 2012...Dark Matter ... .

"lighten up, experts" :D ...and that is precisely why it is so hard to find a good class - it is hardly ever fun, but this is. Sean, I LOVE this! I'm not too bad at geometry - but always felt too intimidated (mostly by the 'experts' in my class) to actually pursue a scientific career. Turns out I have been using parallel transport all along in gamedevelopment for steady camera motion along a path :)

@grayaj23

4 жыл бұрын

Every class -- from math to software engineering. There's one guy in the front row who gets pedantic on every minuscule point. There's another guy in the front row who does not know what a pointer or vector is.

Officer, I was not driving. I was parallel transporting my velocity vector. I don't need a license for that.

Hello everyone 👋 welcome to the biggest ideas in the universe. Im your host sean carrol... Always glad to hear this! You are super charismatic!

Dr. Carroll - I've watched many of your videos and you have inspired me in many ways. That being said, that you referred to Gauss as a "dick" was the coolest. You are human after all. You rock, sir.

This lecture is one of the best explanations of particle physics and cosmic divergent galaxies in terms of the Theory of General Relativity. The Riemann curvature tensor is highly well defined here.

Sean! Thank goodness for you, my man! You keeping me (kinda) sane during the lock down. Thanks so much!

I thought that this series would end couple of episodes ago, but the big ideas keep coming!! Awesome!

If geometry becomes Euclidean in small scales then how does the parallel transport of the vector change it ??? please anyone answer

A new video from Dr. Sean! Stopping everything and starting to watch! =)

Sean, these videos and their Q&As are just terrific. I'm so behind because I'm watching (or listening) to each one multiple times. Thank you!!

Sean is by far my favorite intellectual and specifically, theoretical physicist. We are super fortunate to have you, thanks for your wealth of generosity for bringing us this knowledge and humanity for making it accessible and understandable! I hope someday to catch a talk of yours in person, that would really be something!

My favorite series. I appreciate the detail you get into, compared to most. Excellent vids 👌🏻

Incredible as always and great timing! I’ve been teaching myself differential geometry in an attempt to ready myself for Riemann Geo and GR : )

Thanks for making more advanced videos! I was just listening to Eric Weinstein talk about how we need more advanced physics information out there for the general population vs the usual pop-sci physics stuff, and this series is definitely setting the bar high on advanced educational content!

This was great, love the longer vids, and the new pop-ups edits above your head is appreciated.

I'm ready to buy the book(s) from this series of lect- ups, of videos. Absolutely stunning material, thanks prof. Carroll

For some reason when he said "So, there's good news and bad news when it comes to topology" around 52:45 that struck me as hilarious for some reason - I love the off-the-cuff style of these lectures.

This is one that I've been waiting for.. looking forward to watching this later!

Thanks for making something that is waaay over my head a lot easier. Teachers and professor like you should get paid like professional athletes.

These videos mean so much to me! Thank you, Sean!

This series is astoundingly good. Thank you very much for your time,Dr. Could you show a bit of the math about parallel transport in the Q&A? For example, do parallel transported vectors change their length when changing direction? Maybe a radial velocity becomes tangential velocity in a curved spacetime?

Thank you so much for these. As a hobbyist and someone who never retained any of my math education, attempting to find a clear definition of a Riemann Curvature Tensor or any similarly complex concept has proved very difficult. I'd be very interested if you made these lec..videos into a book. Kind of like a 'Road to Reality' except for people with smaller hat sizes.

I don't want the lockdown to ever end if means Sean will go back to his daytime job. Please keep it up, Dr. Carroll. This is super helpful!

Thank you for going into the “Mathyness” in this pop physics video. So grateful. Thanks!

Thank you so much for making these!

Thank you so much Prof.Carroll for great series.

Great! I've heard of this tensor - it's so nice to see it! Thank you!

Great explanation of a tensor - thank you so much for these lectures - videos.

🤯 Thanks Doc!😷 Phenomenal gift to us all, and a delightfully casual presentation that keeps me coming back for more.

Really enjoying Sean Carroll”s videos!!!

The reason why hyperbolic geometry was the first non-Euclidean geometry discovered, is that it is easy to show that no parallel lines is inconsistent with the other axioms as they were then currently formulated of Euclidean geometry. For example there are an infinite number of different lines between the north and south pole of a sphere which contradicts the first postulate of Eucliden geometry - two distinct points determine a unique line. So this is why the focus was on many (infinite) number of parallel lines through a point not on the line and parallel to the given line.

Leibnitz tried to prove the parallel postulate by a proof of contradiction - by using a different postulate and looking for contradictions. But when he discovered that the resulting geometry was perfectly free from contradictions he was certain he had made a mistake and never published it - which says something about the respect people had for Euclid. We've found it in his personal papers.

@SrValeriolete

3 жыл бұрын

I don't think it's just respect, geometry on non-flat surfaces seems wrong because we don't experience it most of the times.

@markkrueger6751

3 жыл бұрын

@@SrValeriolete I think it's definitely a combination of the two. Leibnitz came from a period that would have would have placed the "classical genuises" on a very high pedestal.

Superb lecture. Thank you.

All these videos are profoundly informative. You aren't going to get this level of knowledge from most other videos on KZread with maybe the exception of Science Asylum, Veritasium and Ask a Space Man. Great work Sean!

@Littleprinceleon

2 жыл бұрын

Marvin Ash has some good ones on QM, too. IMHO

Very informative and very well done, it shows that you sir are a teacher! :) There were a lot of things that I finally understood and others that I heard of for the first time. It is an art to put together basic and high knowledge and the mix to be understandable by any listener... I really wish that you sir never stop doing this series but I know that sometime in the future you really have to move on...:)

@naimulhaq9626

4 жыл бұрын

Oct onions by Sean will be a real treat.

Thank you for putting out this content, this is very useful!

I love learning about the history of these ideas and the people that brought them to us. I love your explanation of the Reimann theory. I love it and I appreciate it. I want to know this. ❤️

Great explanation of Riemann curvature tensor

"These are hard concepts": such an understatement after having just summarized an entire semester of differential geometry in 45 minutes.

Enjoying these lectures very much! Actually feel I can begin to understand these topics better, and the Topology part seems as if it foretells the development of string theory(?)

Awesome lecture, thanks so much!

Really like the communication in this one

fab. it never occurred to me that you couldn't compare two vectors at different points in space without bringing them together . lol. i find that genuinely very very thought provoking :) all wrapped up with locality and what something being space really means

These are amazing.

Thanks for helping me out with continuing my education.

@ToriKo_

Жыл бұрын

+

really appreciate this professor！ you are doing something grate！

Fantastic Lecture!

39:49 Hi Dr. Carroll. Thank you for the great lectures, it would be very helpful if you can make a separate lecture on tensor calculus.

You really explain things amazingly well........................

Are the Gaussian circles rings or disks? I'm talking about the section where Gauss speaks of seeing geometry from the point of view of someone living on the circle.

Thanks, Sean! If you want a 'one word' for the videos - call each a 'presentation' on the Relevant Topic! Avoids 'lecture', which for some (not me!) has ominous memories of exams etc.

Still the best videos anywhere on the internet

The 5th postulate is a little more general: It says that if the angles in the interior add up to exactly 180, then they are parallel. But it should be noted that both angles don't have to be 90 degrees. For example look at this figure ( =/= ). If the diagonal line that cuts through the equal sign, forms two angles that add up to 180 (same side), then they are parallel. Just a fun fact! =D This episode was one of my favorites!

Every episode that I watch, I get a brain ache, but its always a good brain ache! This video has deformed my brain into both a coffee cup and a donut, ...hmmmm donut!

One hurdle I had with understanding non-Euclidean geometry was a notion of line. I was so used to “straight line” type of thinking that it was really hard to imagine that whole geometry would not blow up if lines were not straight. To my surprise I found that Euclid didn’t give definition of a line. He defined it algebraically by describing properties any line has to have such as “To draw a straight line from any point to any point.” (thanks Wiki). This made me appreciate how advanced ancient Greeks were because it looks very much like modern math. Once I realized that “line” is anything with requested properties it became easier to understand other geometries. It seems we can even have y=x^3-lines and Euclidean theorems would still hold. Amazing level of generality!

Of course they each are huge subjects which deserve videos of their own-they're not gonna get them; I tried to squeeze both of them into a single video. -Sean Carroll, Physicist

Now we hitting the good stuff

Thank You!

Could you give a short motivation on (co)homology groups as well in the Q&A please? I struggle to get an intuitive approach there. Thank you for this series!

@zapazap

4 жыл бұрын

I found this good. It is not what you might think of as motivation, but it worked for me. Good presentation of *calculating* the groups. kzread.info/dash/bejne/i4J2y5aCqJbJiZM.html

Thank you so much for all that you do. And it ain't just coming out my black hole. I really mean it. Thank you!

Excellent!

Love the blackboard!

also dont know if he talked about this but when Reinmann died his house keeper threw out a whole bunch of papers that he was working on. apparently Reinmann didnt publish unfinished work so we most likely lost some incredible discoveries 😞

21.23 just lighten up guys. Awesome!

These are great videos. Thank you so much for explaining things so well. You have chosen the right level. Btw - what is the app you use to present?

Major bummer! I already thought about tea, chocolate cookies and the video!

@pikkutonttu2697

4 жыл бұрын

It is working after all!

20:58 Metric: infinitesimal length. 24:25 38:31 43:00 46:21 1:00:32 1:08:28

This one was a brain melter. Good stuff.

It's nice to know there a positively curved universe where the circumference of a circle is exactly 2r

Sean Carroll should write a textbook about everything :)

@joelcurtis7447

4 жыл бұрын

Agreed! He's working on a QM textbook. Can't wait for that. Would also like a QFT book from him. He should do what Susskind did and work these videos up into a book series. He's so good at explaining without leaving the important stuff out.

So I clearly went off on a tangent, learning this via quantum mechanics* rather than my usual field (3D / physically-modelled computer graphics)... but honestly, this is the first explanation of non-Euclidian geometry I've ever understood. I've been using vectors in similar ways for so long now that - seeing parallel transport demonstrated like this - I can't believe this didn't dawn on me long ago (I was never good at 'math theory', but if I can visualise it in my head, I get it just fine). The topology stuff I could imagine including at some point in the near future as well; for example, mapping textures to arbitrary geometry, possibly using curvature tensors to project texels in 3-space based on surface normals. * Thanks Sean; I got here via some of your quantum mechanics talks, and I think I may be hooked.

Yes. People, even a major at maths, should have to recognize pedagogic excellence, especially about vectors and high level maths, I saw young people fight and fail, fight...because math is not easy, even to those that understand the concepts but cannot do the calculations, nor those who don't see in three, four or more dimensions and suffer for that. In metric fields ...What is keeping a parallel postulate, Riemann Curvature tensor parallel transport.....the connection, the curvature...smoothly deformed spaces, topological invariants.

If this goes on, Sean will end up with hair like the much-used photo of Albert Einstein :) Apart from that, I think Seans videos in general is amazing because I actually understand stuff, that I didn't expect myself to understand... (or rather.. I understand what leads to the theories, even when some of the theories are hard to wrap your mind around because they are counter-intuitive...)

Any other non scientists here who just enjoy listening to Sean talk about cool shit? Half of the fun is just trying to keep up lol

Thank you for this wonderful series, Professor Carroll. There is one thing in this video that would like to comment on: at multiple places, 1:04:46 for instance, you referred to a member of the homotopy groups as "topologically equivalent maps", which I found a bit misleading. The members of the homotopy groups are "equivalent classes" of maps, instead of maps themselve. Any two "topologically equivalent" maps in fact represent the same member in the homotopy group. I think this should be pointed out as it is somewhat important for what follows, and it is not too hard for the non-professionals.

BTW: If anyone was annoyed he left us hanging what the fundamental group of the plane minus two points is: it is the "free product" of two copies of the integers, which is indeed not abelian.

Questions for the Q&A: Is the first homotopy group of a 1-Sphere mapped to Euclidean 3-space (a circle in Newtonian space) trivial? It seems like the winding number of a circle around a missing point is irrelevant, as it can just go 'above' or 'below' the missing point to avoid it as it smoothly transforms. (This would generalize to an n-Sphere mapped to an n+2 space, I assume?) Some versions of the story Physics tells of reality depict black holes as *actual* holes... is this equivalent to "missing points" in spacetime in any way? In other words, does the formation of a black hole fundamentally change the topology of the universe? Alternatively, is that what lead to the idea of black holes leading to other universes, analogously to the way a 1-Sphere can map to two different 1-Spheres? With respect to curvature... we often see mass depicted as a depression in a rubber sheet or 2D wireframe plane. In this analogy, black holes are depicted as depressions that go so deep that a hole is torn in the rubber sheet and/or fabric of the universe. But we also often hear the verbal description that the singularity is the point at which "curvature becomes infinite". But in that depiction, the curvature at the bottom isn't infinite; in fact it's very nearly zero, with all of the real curvature happening at the event horizon. What would it *really* mean for curvature to become infinite? Is there any way you can think of to visualize this more accurately? Could this have any interesting implications for the true nature of the singularity? This is, of course, assuming GR, not quantum gravity (in which, I presume, the singularity is not expected to persist and will turn out to have been an artifact of the math of GR being pushed past its domain). The disc with opposite points defined, and the way the even-numbered windings can contract to zero while odd-numbered windings can only contract to 1 reminds me of the way some of the curled up dimensions are depicted spontaneously unraveling into macroscopic dimensions in String Theory... is that a coincidence? Is that where that particular topology example is heading later on, or am I off base on that similarity?

There must be something to how you say what you say, cause I had an idea about non Euclidean geometry imagining a version more like a sine wave than a diverging or converging parallel line. Then after thinking about it, that's basically GR which you almost immediately mentioned in the video as I had the thought.

Is the surface of a taurus an example of hyperbolic geometry, at least on the "inner" surface (the face that you can see the center from).

You said you could classify topological defects in cosmology by the homotopy groups. Could you elaborate on this? Is it a theoretical approach to analyse spacetime topologically in order to predict strings or monopoles, or can you somehow measure the homotopy group of our spacetime and then know that there must be strings/monopoles? Thanks!

I understand QM, entanglement, special relativity, QFT, geometry etc, but I found the topology stuff really hard to follow.

@47:05 I'm "incredibly complicated abstract stone(d)" by these lectures.

So a Tensor is basicly a equation field (for some quantity) that you apply to every (or some) degree of freedom in a certain space and get an answer, wether its air pressure in the atmosphere, even a frequence in a song or curvature in some dimensional space, at a certain point (or any point in this certain space) ?

For a novice -- why exactly is the formula for infinitesimal length = Aa2 + Bab +Cb2 ?? --- if c2=a2 + b2

@Sean Carroll ...First of all, thanks for your amazing educational series! Now, I have a question regarding 37:36, i.e. parallel transportation of vectors in curved space. I cannot get it to add up: You take a vector, "parallel-transport" it in a loop, and come up with a different vector. If you here imply that the vector changes with regard to the curvature it passes through, it should in my mind result in the same vector when it arrives back at the origin?! What am I missing? To clarify my question, I reason that the net sum of change of the vector ought to be zero in a closed path. Would your closed path transportation be equivalent to just transport it along v1 and v2, and then construct the difference of the original and changed vector? However, I don't see these two ways to be equivalent. I'm sure I'm missing something obvious here. Still I would be very grateful for a short explanation, if you (or any one else more elightened than me) happen to read this!

I don't understand the parallel transport example on the sphere. The orange vector starts out pointing straight up, but as we progress up the geodesic he has it lean over more and more, which isn't keeping it parallel. If you keep it parallel then it will still be pointing straight up when you get to the north pole. From this I conclude that parallel transport doesn't keep the vector parallel with its original value. So what does it keep it parallel with?

Do graph theory I hate math but love when I can shortcut the work and just see the concepts.

And if you're still accepting questions for Q&A: is there any use of let's say "non-standard" topologies in physics? E.g. non-smooth manifolds, manifolds with holes, maybe even non-Hausdorf spaces? I've heard once about 'topological quantum field theory', but frankly speaking have no idea what it is, is it somehow connected to using some non-trivial spacetime topologies?

@tetraedri_1834

4 жыл бұрын

There is a notion of quantum metric, which mesures in a sense how different two wave functions are. As the name suggests, this defines a metric to the space of wave functions, and if I remember correctly, topological quantum theory is studying the effects of the topology defined by the quantum metric to the physics. I think that for example topological insulators and topologically stable quasiparticles are studied using this machinery, but don't quote me for that. It has been long time since I took a course on quantum physics, and we only scratched the surface of this stuff.

I love the note tool - is that GoodNotes?

Could you expound a bit more on the concept of "embedding" in spaces in the Q&A episode?

You may have mentioned it, but what would the fundamental group be of something like (S2)+(S1xS1)? It seems like it depends on whether your "fixed point" is on the sphere or the torus.

Mathematicians don't go to holiday parties because they can't tell the difference between DEC 25 and OCT 31. 😃

the definition of a circle that you give is a collection of points but it doesn't obviously have a direction around its circumference which is necessary for the fundamental group of S¹->X to be Z. If we were mapping a 2d projection of a corkscrew so you go round and round touching the same points without ever being able to say you could have got there by going the other way from the start point would the group for X-point be the naturals because no matter which way you wind it's always (+1)? Would this question be related to the arrow of time, or gravity (Naturals) vs the EM field (Integers)?