The Biggest Ideas in the Universe | 13. Geometry and Topology 

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

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 :)

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

Your generosity is almost incomprehensible

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

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!

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!

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.

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

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?

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

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.

"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 :)

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.

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

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

DrCar I’m already a huge fan and watched hundreds of your talks. I ran into this randomly searching for answers “geometry of the universe “

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.

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

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

Really enjoying Sean Carroll”s videos!!!

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.

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.

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

Great explanation of Riemann curvature tensor

Superb lecture. Thank you.

Really like the communication in this one

Thank you so much Prof.Carroll for great series.

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!

really appreciate this professor！ you are doing something grate！

Sean Carroll should write a textbook about everything :)

Now we hitting the good stuff

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!

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(?)

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.

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

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. ❤️

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

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

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

Thanks for helping me out with continuing my education.

I am starting to get it. Thanks Professor

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

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 😞

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?

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.

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

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

Still the best videos anywhere on the internet

Love the blackboard!

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

This one was a brain melter. Good stuff.

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.

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?

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) ?

These are amazing.

I found the way you and Riemann think about space utterly complicated. I usually think of space as a infinitesimal graph (points and links considered equal and random at the smallest scale).. so the shortest path is the shortest sequence of links (and the length is the number of links).. the straight line is the links that lead the furthest (using the shortest path definition) from some other point that defines your direction (here, a point alone doesn't carry direction).. etc.. there's can't be "unparallel transport of vector" without curvature.. there can't be "rotation" of a point (since a point has no dimension).. there can't be a vector or angle definition without multiple points (for the same reason)

Using small differentaion neighbourhoods seems to run into trouble when expressed as fields. If you lived on donut and figured out a metric field could one figure out that for ex -100 x and +100 x refer to the same point? If a worm hole formed the jump from flat to coffee mug topology seems hard to represent. With the special relativity sphere, de-sitter space I have wondered whether it is two disconnected pieces or one connected one. Define the shape as events 1 second from the central event. If you draw it it seems to be a future bowl and a past bowl. One could try to draw a great circle on it by picking a tangent and parallel transporting in the direction pointed. Because the bowl has a lightlike asymptote far away from the center it seems it could be possible that the proper lenght over all of the coordinate space could stay finite. If that compares to 2pi would it be fair to characterise it as flat, positively or negatively curved? It also seems that as one goes into far west future and far east past the distance between them approaches zero. Would this be sufficient to conclude/guess that the branches actually connect that way? If you have an asteroids screen the argument that the flat rendering places some locations far away is poor argument that the points are not closeby on a torus. Is there a way to make such connectivity judgements for arbitrary potentially weird spaces?

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?

The Scottish theologian Thomas Reid was the first to discover non-Euclidean geometry in 1764. See his Geometry of Visibles in An Inquiry into the Human Mind on the Principles of Common Sense.

If there is no 'natural' or 'canonical' way of defining parallelism, i.e. of saying which vector at one point is 'in the same direction' as another vector at a nearby point, then what constrains our definitions of parallelism and therefore of curvature? Given any curve, can't I just define the velocity vectors to the curve at each point to all be parallel to each other, and thus the curve is trivially straight? But then I can make any curve at all 'straight' and the concept seems to lose meaning. If the metric determines the connection, which defines parallelism, which determines curvature, then what determines the metric? Doesn't the metric represent a coordinate system, and are we not free to choose any coordinates we like? But perhaps it is whether or not you can find a coordinate system in which the metric is the Minkowski metric that tells us about curvature or flatness.

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

Thank You!

What's purple and commutes? An Abelian grape.

I don't think you can have a winding number greater than 1 on an *invertible* map of the kinds you're showing. For a winding number of 2 on the plane-minus-a-point, the line must cross itself somewhere. Same for a map to a circle, no?

Shouldn’t the fundamental group of the torus be Z x Z (instead of Z + Z)? I would think that you need to specify a pair of independent winding numbers for each direction, not only a single one for one of the directions.

shouldnt the middle latitude line have a circumference equal to 2 pi r? if you slice the sphere at the middle, it will look like a circle you drew on a flat surface

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

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

Excellent!

if you situate yourself inside a sphere is it still a sphere? does it still have the same curvature?

I just searched for lecture #15 in this series but couldn't find it. Is it not yet posted?

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?

11:28 The need for a more powerful way to describe curvature for GR than just hyperbolic or hypobolic, since different areas of spacetime are curved differently 13:25 we can describe curvature using the relation of circles with fixed radii to their circumference. This is a quantitative characterization of the curvature of a surface; C = 2(pi)(r), or C < 2(pi)(r), or C > 2(pi)(r), where you could imagine characterizing it exactly. And think about what it means to have a circumference warping up or down, that would *change* the radius based on how up or down it is, *unless the surface is curved* . That’s why this is such an adept description of curvature. After that, you can imagine defining this relation bit by bit, describing the curvature of a surface slowly, with infinitesimal increments, also being able to capture infinitesimal changes in the curvature, aka calculus, and so now you aren’t limited to describing surfaces with fixed curvature. 21:10 the metric. It is a generalization of Pythagoras’ theorem, where it tells us the the (physical) length of an infinitesimal curve C, from the coordinate related measures a and b. And it takes the form C = (alpha)a^2 + (ß)b^2 + (gamma)ab, since it is related to a quadratic equation [ which generally look like (a + b) (a + b) ]. 24:00 a Manifold is a space that is curved. 28:20 parallel transport. How parallel transporting a vector along 2 different paths will keep the vector pointing in the same direction on a flat surface, but not in a surface with curvature. This is another characterization of a way that curvature shows up in geometry. 33:30 this is a reflection of how separated vectors in a curved space don’t have a unique way of measuring if they are parallel (or the same, since vectors. This relates to cosmologists and galaxies (in our curved spacetime), but I don’t understand how velocities relate to the issue of parallel-ness of vectors in a way that would make the example make sense. 39:00 We’re getting some real payoffs here. Bearing in mind there is no unique way to measure if separated vectors are parallel in curved space, that means that the vector V1 will be pointing in a different direction than V5 where we just go in a loop. This is true for the triangle path on the sphere, and for a little parallelogram loop in a more generalized space with arbitrary curvature. So, with that in mind, we can use our infinitesimal calculus trick to define curvature using the difference between the V1 and V5 vector for our parallelogram, and this is called the Reimann Curvature Tensor. 47:37 The connection between Newtonian vs Least Action and Parallel Transport vs Shortest Distance and Differentiation vs Integration is crazy and I’m not fully grasping it. Could do with replaying this part

Thank you.

Awesome lecture, thanks so much!

“…Gauss was a bit of a dick...” What a great lecturer!

The 1st homotopy group of the circle is ZxZ not Z+Z afaik.

In many spaces there is going to be a unique geodesic path between two points. If there is such a path, can't you define parallelism at those points by parallel transport along the geodesic?

What courses should one take in university to learn more about these concepts?

Wait whaat.. I'm still at 7. Quantum Mechanics.. I mean WTF, I'm about a month out, I think my RU-vid notification is broken.. (or I just missed it, which is very unlikely 😜) Anyway although I'm clearly mostly struggling to keep up, but I honestly think that it's been wonderful that you're continue doing this public lecture. Thanks Sean, and keep on keeping on sir. 👍😄

Isn't the fundamental group of the torus Z x Z, not Z + Z?

why do you call the fundamental group of S¹ -> X as 0 which we use for things which have no values instead of Unit or 1 which we use for things that have precisely one value with no structure?

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.

I love the note tool - is that GoodNotes?

@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 get the topology part: when you go around twice, the lines are crossing: are you allowed to use points in the plane twice? If you do it with sticks and strings, you can only do it, because the real world is 3D, where a crossing of the line actually happens in the 3rd dimension. I can imagine that you can pull a sling through when you don’t have a stick, and the sling becomes a straight rope / string again (- what might have looked like it would form a knot, was actually not a knot), which you can’t, when there is a stick in the sling... - but still: you need 3 dimensions. Or is the 3rd dimension only required, because the rope itself is actually a 3d object? Is that the reason why my imagination breaks down? What is the rule for projecting the circle into the plane with multiple windings? I don’t get it.

For every divergence there is a convergence.

To paraphrase Tom Lehrer, "By the time he was my age, Riemann had been dead for 10 years."

shouldn't the fundamental group of the torus be Z x Z?

Did he just teased for a General Relativity video?

Hi Dr. Carrol, fantastic lectures. What software are you using to write on a digital blackboard?

It is just great

“There we go, two birds with one stone! It’s an incredibly complicated, abstract stone...”

This will be good for my teenage son. (As well as me!)

7:40

His books are just as good

23:00 ... no square on the LHS?

My head just exploded 😆

Finally a meaningless and almost irrelevant bit of pedantry I can jump on - at 23:18, the instrument you'd be using would be an _opisometer_ rather than an odometer. (That's _really_ old-school stuff that's right in this old fart's wheelhouse.)