#science #physics #ideas The Biggest Ideas in the Universe | 13. Geometry and Topology

#science #physics #ideas The Biggest Ideas in the Universe | 13. Geometry and Topology 118,512 viewsJun 16, 2020 Sean Carroll 154K subscribers The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody. This is Idea #13, "Geometry and Topology." Yes that's two ideas, and furthermore they're from math more than from science, but we'll put them to good use. In particular we look at Riemannian (non-Euclidean) geometry, and a kind of topological invariants called "homotopy groups." My web page: http://www.preposterousuniverse.com/ My YouTube channel: https://www.youtube.com/c/seancarroll Mindscape podcast: http://www.preposterousuniverse.com/p... The Biggest Ideas playlist: https://www.youtube.com/playlist?list... Blog posts for the series: http://www.preposterousuniverse.com/b... Background image by RyoThorn at DeviantArt: https://www.deviantart.com/ryothorn/a... #science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology 223 Comments rongmaw lin Add a comment... Jochem van der Spek Jochem van der Spek 1 year ago "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 :) 38 luke Neville luke Neville 1 year ago That was the best description of the Riemann curvature tensor I've seen, these videos are much appreciated 56 Kobev3li Kobev3li 1 year ago This series is the absolute best thing in the world right now. Keep up the great work Dr. Carroll !!! 122 Robert Shirley Robert Shirley 1 year ago It is the greatest gift that some people could spend time to teach, to interact and respond. 62 ROBOT UNIC0RN ROBOT UNIC0RN 1 year ago 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! 6 Álvaro Rodríguez Álvaro Rodríguez 1 year ago (edited) 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? 8 padrick beggs padrick beggs 1 year ago Incredible as always and great timing! I’ve been teaching myself differential geometry in an attempt to ready myself for Riemann Geo and GR : ) 4 Amity Affliction Amity Affliction 1 year ago My favorite series. I appreciate the detail you get into, compared to most. Excellent vids 👌🏻 2 darkruby darkruby 1 year ago I thought that this series would end couple of episodes ago, but the big ideas keep coming!! Awesome! 16 George Farahat George Farahat 1 year ago (edited) 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. 1 unòrsominòre. unòrsominòre. 1 year ago I'm ready to buy the book(s) from this series of lect- ups, of videos. Absolutely stunning material, thanks prof. Carroll 3 bruinflight bruinflight 1 year ago (edited) 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! George Komarov George Komarov 1 year ago 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 :) 3 davyoooo davyoooo 1 year ago Sean! Thank goodness for you, my man! You keeping me (kinda) sane during the lock down. Thanks so much! 5 Haydar Masud Haydar Masud 1 year ago 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. 1 L Dewey MD L Dewey MD 1 year ago 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(?) 1 Charles Norman Charles Norman 1 year ago 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. 1 Jonathan Brown Jonathan Brown 1 year ago These videos mean so much to me! Thank you, Sean! 1 Mark thebldr Mark thebldr 1 year ago Thanks for making something that is waaay over my head a lot easier. Teachers and professor like you should get paid like professional athletes. 1 Michael Hutton Michael Hutton 1 year ago 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? LaserGuidedLoogie LaserGuidedLoogie 1 year ago Thank you for putting out this content, this is very useful! William Bobillet William Bobillet 1 year ago I wish I had such teacher ... clear, so well explained, things kept simple so that the maths are easy if you decide to dig further after that ... Masterpiece! Valdagast Valdagast 1 year ago 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. 23 Sam Barta Sam Barta 1 year ago 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 5 Edmundo de la Garza Edmundo de la Garza 1 year ago (edited) 🤯 Thanks Doc!😷 Phenomenal gift to us all, and a delightfully casual presentation that keeps me coming back for more. Tim Seguine Tim Seguine 1 year ago "These are hard concepts": such an understatement after having just summarized an entire semester of differential geometry in 45 minutes. 1 Sebastian Dierks Sebastian Dierks 1 year ago 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! 2 Patrick James Patrick James 1 year ago Great explanation of a tensor - thank you so much for these lectures - videos. ph ph 1 year ago (edited) @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! Barefoot Barefoot 1 year ago (edited) 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? 1 ManWhoUsesComputer ManWhoUsesComputer 1 year ago Great! I've heard of this tensor - it's so nice to see it! Thank you! SkateKraft SkateKraft 7 months ago 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. ❤️ Vincent Button Vincent Button 1 year ago 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!! David Wright David Wright 1 year ago 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. Ambrotos Aeternus Ambrotos Aeternus 1 year ago (edited) 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...:) 1 KAĞAN NASUHBEYOĞLU KAĞAN NASUHBEYOĞLU 1 year ago Thank you so much Prof.Carroll for great series. 1 Mitchell E Mitchell E 1 year ago Awesome lecture, thanks so much! Xing Gu Xing Gu 5 months ago 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. SlyFox SlyFox 1 year ago 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! E Carter E Carter 1 year ago 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. Stewart Hayne Stewart Hayne 1 year ago Thank you for going into the “Mathyness” in this pop physics video. So grateful. Thanks! 1 Amir# Amir# 1 year ago Hello everyone 👋 welcome to the biggest ideas in the universe. Im your host sean carrol... Always glad to hear this! You are super charismatic! 14 Sebastian Dierks Sebastian Dierks 1 year ago (edited) 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! ankles iii ankles iii 1 year ago Thank you so much for making these! 5 Martin Buch Martin Buch 1 year ago 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...) 1 Andrey Verbin Andrey Verbin 1 year ago 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! Stumpy Mason Stumpy Mason 1 year ago This was great, love the longer vids, and the new pop-ups edits above your head is appreciated. 1 Tim Seguine Tim Seguine 1 year ago 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. Kowzorz Kowzorz 1 year ago 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. Shalkka Shalkka 1 year ago 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? 1 Steve White Steve White 1 year ago 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 larsyxa larsyxa 1 year ago (edited) 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) ? 1 Michael J Morrison Michael J Morrison 1 year ago Given that space gets "lumpy" and distorted in the vicinity of stars, planets, black holes and other matter, using topology etc, can the entropy of "space" itself be calculated? Might this be possible regardless of the type of matter/energy that space itself is made up of? Could this be true even though we cannot investigate the energy of the planck scale? Does this provide insight into the understanding of gravity? Christine LaBeach Christine LaBeach 1 year ago All these videos are profoundly informative. You aren't going to get this level of knowledge from most other videos on YouTube with maybe the exception of Science Asylum, Veritasium and Ask a Space Man. Great work Sean! 1 R C R C 1 year ago Really enjoying Sean Carroll”s videos!!! monkeypeas monkeypeas 1 year ago It's nice to know there a positively curved universe where the circumference of a circle is exactly 2r 1 Andrew Kemp Andrew Kemp 1 year ago This is one that I've been waiting for.. looking forward to watching this later! Brandon Lewis Brandon Lewis 1 year ago Could you expound a bit more on the concept of "embedding" in spaces in the Q&A episode? Guri Buza Guri Buza 1 year ago (edited) 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 1 markweitzman's wannabe a theoretical physicist school markweitzman's wannabe a theoretical physicist school 1 year ago 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. Ian Prado Ian Prado 1 year ago Hi Dr. Carrol, fantastic lectures. What software are you using to write on a digital blackboard? Stadtpark90 Stadtpark90 1 year ago (edited) 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. Brian Smith Brian Smith 1 year ago Superb lecture. Thank you. 1 Matthew Hondrakis Matthew Hondrakis 1 year ago 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! Mind Foocked Mind Foocked 1 year ago 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! Jeff Bass Jeff Bass 1 year ago 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. Stephen Bryant Stephen Bryant 1 year ago (edited) When mapping a 1-sphere to another 1-sphere at about 1:03, I don’t see the difference between one loop (yellow) and zero (brown). It looks like one could be deformed into the other. What’s missing? Oh, I think I see it. The loops are within the 1-sphere, not in some embedding space (?) Mike Truesdale Mike Truesdale 1 year ago 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. 1 Milligram Milligram 1 year ago This large vs small pattern seem to pop up here and there; many worlds vs collapsing probabilities (which doesn't seem to contradict each other to me), principle of least action vs tracing light (it may be called something else), fields vs particles, declarative vs imperative programming, lambda calculus vs turing machines. This is not my observation, I saw Bartosz Milewski talk about it in his category theory lectures. Just curious if this is something phycisists think about. chad moore chad moore 1 year ago 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 😞 1 Joao Joao 1 year ago A new video from Dr. Sean! Stopping everything and starting to watch! =) 18 Tom Semo Tom Semo 1 year ago This one was a brain melter. Good stuff. Derrick Steed Derrick Steed 1 year ago I find it's easier to follow if I play at 2x speed and only slow down for the really sticky bits. You are in the Feynman class of expositors, Sean. And I hope that you don't take that as an insult. Kshitish P Kshitish P 1 year ago If geometry becomes Euclidean in small scales then how does the parallel transport of the vector change it ??? please anyone answer 2 Torben Møller Torben Møller 1 year ago Could you imagin a space where to “parallel” lines: Does not have a constant distance AND never crosses. So they get closer but it does not convert. Isthat an option for a type of space? acac acac 1 year ago Wow, nice saddle drawing. I didn't know you could do it with a single stroke! 4 Brent Meeker Brent Meeker 1 year ago 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? FirstRisingSouI FirstRisingSouI 1 year ago Trying very hard not to brag about my awesome non-Euclidean geometry visualization skills. 1 fred burns fred burns 5 days ago 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 Rafael von gehlen Rafael von gehlen 11 months ago Fantastic Lecture! Bailey Bartley Bailey Bartley 1 year ago Say we have a gravitational sphere like earth orbit, is the “shortest distance” based on a 2-D manifold using the least amount of necessary force to go from point A to point B or would force be arbitrary when calculating the distance, where we should rather think in a 3D curveless manifold? Does gravity warp a flat spacetime into a curved one where the “shortest distance” is based on the amount of energy/force needed to alter trajectory rather than viewing it from a 3D Euclidean geometry where spacetime is flat? Does this mean we really live in a 2D world that is complied in our heads as a flat 3D world Skorj Olafsen Skorj Olafsen 1 year ago 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? 1 Edgard Neuman Edgard Neuman 1 year ago (edited) 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) Doug Porter Doug Porter 5 months ago 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. George Komarov George Komarov 1 year ago 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? 1 Rob G Rob G 1 year ago 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. 1 Brandon Lewis Brandon Lewis 1 year ago It's okay, you can call them "lectures". We're 13 "episodes" in, if we're still here, the word "lecture" isn't going to scare us away now. 1 David Jordan David Jordan 1 year ago Thanks for helping me out with continuing my education. Piotr Podgórski Piotr Podgórski 1 year ago This series is the best thing that happened to YouTube since Leonard Susskind's "Theoretical Minimum" 6 Denis Nichita Denis Nichita 1 year ago Sean Carroll should write a textbook about everything :) 5 Mauro Cruz Mauro Cruz 3 months ago (edited) 20:58 Metric: infinitesimal length. 24:25 38:31 43:00 46:21 1:00:32 1:08:28 Joel Curtis Joel Curtis 1 year ago (edited) 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. Bellio Trungy Bellio Trungy 1 year ago (edited) Do graph theory I hate math but love when I can shortcut the work and just see the concepts. 1 DaveDashFTW DaveDashFTW 1 year ago I understand QM, entanglement, special relativity, QFT, geometry etc, but I found the topology stuff really hard to follow. Mark G Mark G 1 year ago You really explain things amazingly well........................ Phonzie Relli Phonzie Relli 1 year ago Well, about 15 years ago, I think on Youtube? There was an instructor somewhere that explained from the ground up how Einstein arrived at E=MC^2. The way he did it was step by step and amazingly easy to understand. I can't find it now for the life of me but I got tired of looking a long time ago so it may be there and I just lost the plot? bryangoggin bryangoggin 1 year ago Really like the communication in this one Tom Kaminski Tom Kaminski 10 months ago What courses should one take in university to learn more about these concepts? John Alexiou John Alexiou 1 year ago 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). Tristan Wibberley Tristan Wibberley 1 year ago 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)? mayaknife mayaknife 1 year ago 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? klaas terpstra klaas terpstra 4 months ago Great explanation of Riemann curvature tensor 1 Shikhar Amar Shikhar Amar 1 year ago Please share resources for reference, they would be of great help. BazNard BazNard 1 year ago Still the best videos anywhere on the internet Mgenth bjpafa Mgenth bjpafa 1 year ago (edited) 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. ClayZ ClayZ 1 year ago Imagine getting paid to think about parallel lines going on forever and ever or not. The ultimate work from home job. Two lines diverged in the wood. Hmmmm sounds vaguely poetic. Focus Sean, focus. Donald Duck Donald Duck 1 year ago Wait whaat.. I'm still at 7. Quantum Mechanics.. I mean WTF, I'm about a month out, I think my YouTube 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. 👍😄 2 George Steele George Steele 1 year ago Learning math seems to be exactly like learning a new language. The explanations you give are translations from that language into modern American English. Could we "do" mathematics totally with English? No '=' symbol. No meaning to dX. Pseudo Nym Pseudo Nym 1 year ago “There we go, two birds with one stone! It’s an incredibly complicated, abstract stone...” Wd Fusroy Wd Fusroy 1 year ago (edited) I just searched for lecture #15 in this series but couldn't find it. Is it not yet posted? Elias Barrionuevo Tandel Elias Barrionuevo Tandel 1 year ago Computer scientists and logicians also like homotopy, more specifically homotopy type theory Chip Hill Chip Hill 1 year ago This must have something to do with Feynman's least action path integral! Pritam Karmakar Pritam Karmakar 1 year ago The biggest ideas in the universe: Applied Mathematics 3 Tristan Wibberley Tristan Wibberley 1 year ago 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? Boris Petrov Boris Petrov 1 year ago (edited) For a novice -- why exactly is the formula for infinitesimal length = Aa2 + Bab +Cb2 ?? --- if c2=a2 + b2 Mickolas21928 Mickolas21928 1 year ago 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. Brian McConkey Brian McConkey 1 year ago 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! 1 Richard James Winter Richard James Winter 1 year ago Heard Steve Buscemi is going to play you in the movie. 😊. Great stuff, I need to learn some maths though. Chris Wendler Chris Wendler 1 year ago shouldn't the fundamental group of the torus be Z x Z? [Redacted] [Redacted] 1 year ago These are amazing. bmoneybby bmoneybby 1 year ago Random question: Have we covered zero point energy yet? I've watched basically every video but there's a lot now so I can't quite recall. Bryan Hann Bryan Hann 1 year ago This will be good for my teenage son. (As well as me!) MEMIS HASILCI MEMIS HASILCI 1 year ago topology in the books given free in google and other dont teach anything even to me an engineer Of course the material covered has a great potential it pushes too hard the reader to anakyze every step he reads with a sketch for every statement he encounters it looks like we are in space at a distant planet totally unknown We have mr takeida saying every statement you present must have a pictorial presentation and he does not doing abything mr sean does give some piures but he does not give a use for this rigorous analysis and books give information and leave the rest to philology of mayh amath lesson to hamlet Charles St Pierre Charles St Pierre 1 year ago (edited) Is the parallel postulate just a case of the postulate that the angles of all triangles add to (exactly) 180 degrees? Roman Travkin Roman Travkin 1 year ago π _0 is not a group in general (unless the space itself is a group). Gabriele Puppis Gabriele Puppis 1 year ago Isn't the fundamental group of the torus Z x Z, not Z + Z? Naimul Haq Naimul Haq 1 year ago Sean sure qualifies as a mathematician, better than a physicist. Very good presentation.Also interesting. James Patterson James Patterson 1 year ago I love the note tool - is that GoodNotes? lilit vehuni lilit vehuni 1 year ago Does that mean that if v4 is 0 then you have 0 curvature or flat space? Arindam Bhattacharya Arindam Bhattacharya 1 year ago @47:05 I'm "incredibly complicated abstract stone(d)" by these lectures. 1 Csaba Tilki Csaba Tilki 1 year ago (edited) Bolyai was a Hungarian mathematician, and it is pronounced "b-o-y-a-ee". 1 Davide Around Davide Around 3 weeks ago Your generosity is almost incomprehensible Stan Rogers Stan Rogers 1 year ago 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.) M-Class Designs M-Class Designs 4 months ago I keep playing these and passing out. Not because it's boring, but because Sean's voice is soothing. 😂😂😂😂 dawangai dawangai 1 year ago Officer, I was not driving. I was parallel transporting my velocity vector. I don't need a license for that. 2 life42theuniverse life42theuniverse 2 months ago 1:14:00 unless the S1 - 2P is a 2-torus in disguise? Change Gamer Change Gamer 1 year ago (edited) Shouldn‘t the title of the series rather be: ‚Biggest Ideas DESCRIBING The Universe‘? Great series! 🙏 1 Olórin Olórin 7 days ago I think Gauss might have given a bit of inspiration to the character of Dr. Rodney McKay from SG Atlantis. haha Pavlos Papageorgiou Pavlos Papageorgiou 1 year ago 48:45 And that is where spherical cows come from! #science #physics #ideas The Biggest Ideas in the Universe | Q&A 13 - Geometry and Topology 35,339 viewsJun 21, 2020 707 DISLIKE SHARE DOWNLOAD CLIP SAVE Sean Carroll 154K subscribers The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody. This is the Q&A video for Idea #13, "Geometry and Topology." We use the excuse to dig into some details of embeddings, how to express the Riemann tensor, what maps are involved in specifying homotopy groups, plus a bit about topological defects. My web page: http://www.preposterousuniverse.com/ My YouTube channel: https://www.youtube.com/c/seancarroll Mindscape podcast: http://www.preposterousuniverse.com/p... The Biggest Ideas playlist: https://www.youtube.com/playlist?list... Blog posts for the series: http://www.preposterousuniverse.com/b... Background image by RyoThorn at DeviantArt: https://www.deviantart.com/ryothorn/a... #science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology 60 Comments rongmaw lin Add a comment... Alon Nissan-Cohen Alon Nissan-Cohen 1 year ago Great series, thank you very much Prof. Carroll! Regarding the last point you raised in the video: there are indeed infinitely many ways to "probe" spaces. These are called Generalized Cohomology Theories, and they are arranged in a category (actually, infinity-category, meaning that it has higher and higher levels of morphisms) called the category of Spectra. Each spectrum is a machine that eats up spaces, and spits out an invariant of the spaces, in a way that behaves similarly to how we expect cohomology theories to behave. Important examples are the Eilenberg-Maclane spectrum, which gives us ordinary cohomology, the K-Theory spectrum, which probes a space by considering all the different vector bundles on it, and the Sphere spectrum, which - guess what - gives us homotopy groups! (well, not exactly... it actually gives the so-called "stable" homotopy groups, but it's pretty close). And there are infinitely more different such spectra, arranged in a complex and beautiful array. The study of this array is called Chromatic Homotopy Theory, and it's one of the "hottest" areas of research currently in homotopy theory. 16 protoword protoword 1 year ago Thank you professor! I know already many things from this episode, but when I listen you, it become refreshing and enjoyable to me again! You are such a great teacher... 1 Nick B Nick B 1 year ago Sean’s Book “The Big Picture” is a true work of art 🖼. Probably the most comprehensive and mind-blowing physics book out there, and I’ve read some great ones. Mihir Borkar Mihir Borkar 1 year ago (edited) Wow! These geometry and topology videos really explained the Riemann curvature tensor and the connection really well. One of the best explanations I've seen! I finally can intuitively understand the Riemann curvature tensor. Thanks so much @Sean Carroll ! P.S. I'm an undergrad at Caltech and saw you at Chandler once. Unfortunately couldn't stop by to say hi. Geert VS Geert VS 1 year ago About the potential V(phi): I think Sean intended to draw something like this: https://www.researchgate.net/profile/Eduardo_Guendelman/publication/258374367/figure/fig2/AS:297562977390594@1447955950328/Scalar-potential-V-ph-with-domain-wall-between-two-false-vacuum-state.png Since his symmetrical potential would not result in a difference between V(+phi) and V(-phi). 1 mezza205 mezza205 1 year ago 32:25 I must say you are a great speaker to give everyone the flavour of physics with math concepts that guide us to the prize :D 1 George Farahat George Farahat 1 year ago This is a great video showing how starting with mathematical concepts scientists are able to relate them to space and the deep cosmos... Shalkka Shalkka 1 year ago 36:15 There are o,0 and symbol for empty set. The line distinguises the letter form the number and the line stays wihtin the circle for number and goes way over it for the empty set symbol. The trivial group is probbably likened to be the same as winding number zero from the integeres. However that trivial group has a class. Thus the set of classes is not empty but has a member. When the integers are constructed from sets it is conventional that {} the empty set stands for zero and {zero} aka {{}} stands for number 1. {} has no members {{}} has a member (namely {}). I would imagine a space with a single point would be incredibly boring. However even more boring than that would be the space of no points. The topology of that space is likely different from the trivial one and I am unsure whether you could make a class as there could be no paths thus the membership number would be different. Thus the empty set could actually signify a different thing than the class 0. The video series opens up the concepts for a lot of people. But it would be a shame if somebody started saying that "zero has 1 member" out of context or was confused by it in other contexts. 5 larsyxa larsyxa 1 year ago Sean you are exceptionally gifted in explaning difficult concepts to layman and layman++. Im just saying. 1 EarlWallaceNYC EarlWallaceNYC 1 year ago Do you have references for a deeper dive into the topology stuff? BTW: Lov'in the videos. Thanks. fburton8 fburton8 1 year ago These are truly huge ideas. Battle Hamster Battle Hamster 5 months ago Don't be upset, Sean spends his time learning and teaching. No Time to argue with flat earthers. Mal-2 KSC Mal-2 KSC 1 year ago I am beginning to think these ideas are too big to fit inside my head, but I will keep trying. 1 Jai Bellare Jai Bellare 1 year ago I love it when Sean uploads 22 MrFedX MrFedX 1 year ago Sean Caroll saying ”I probably shouldn’t do this” means that it’s probably going to be somethine you really want to hear. :) 1 Peter Max Friis Jensen Peter Max Friis Jensen 1 year ago Irresistible topics indeed :) Coochicoo Coochicoo 1 year ago Sean's brain is so plump and juicy with knowledge. My brain is like a dried, shrivelled up raisin in comparison 😔 9 Nisha Tiwari Nisha Tiwari 1 year ago I feel very good to hear from you. Thanks a lot 2 Imager Imager 1 year ago I always gave up when this was discussed on PhysicsForums.com. Now I can take another step in my education, not bad for someone on Social Security retirement. Thank you!0 2 D.T. Moore D.T. Moore 1 year ago Thanks Sean!! 3 Gilbert Anderson Gilbert Anderson 1 year ago (edited) 47:30 What if you could make causally disconnected regions "fall" the same way? If there was a fundamental sonic mode across the universe could this be imprinted on the CMB? What ARE b-modes anyway? Robert Shirley Robert Shirley 1 year ago I do believe that we cannot treat the Time as such coordinate because it is not fundamental. I mean ‘time only appears when there is an event’. And by event I mean some sort of motion. And by that I mean when there is literally motion in the forms of speed or acceleration, frequency and of course entropy! When something does not lose /gain mass or energy, or does not move /decompose/ combine, there is no time! Time is not fundamental to me for that reason I just mentioned. If you put a ball at the middle of universe with none of those ‘events’ happening, it would mean nonsense to apply any understanding of time to it. That ball would not feel time at all even though it does not move at the speed of light. Briefly, what I mean is that time cannot be possibly part of the fabric of space, but it rather is present in every event. I know philosophically my opinion can be argued that that ball feels time but eventless blah blah blah :)) I believe time can be interpreted as the resistance of the fabric of space(space itself as a continuous matter) to events(such as motion, entropy change,..) in some sense (not common sense though) but to take the Time as coordinate is not because it is there, but because it defines the events and their relations! 2 Yevhenii Diomidov Yevhenii Diomidov 1 year ago 44:25 "You can get strings and monopoles in 3d space" Citation needed! :P 1 Alex Tritt Alex Tritt 1 year ago 4 ^ 4 = 256, not 128. (Ie 2*4 = 8, not 7) I’m sorry I’ve been dealing with low level-ish programming stuff so I can’t help but pick up on it 3 KeNsHiRo MuRuGi KeNsHiRo MuRuGi 1 year ago Great vid. Thanks 👍 3 Erik Dahlgren Erik Dahlgren 1 year ago But the S' circle is not truly 1 dimensional, because you have to give 2 values for a direct position {R and theta} Soggy Gamer Soggy Gamer 11 months ago People this smart are just built different. Impressive... Math adventures Math adventures 1 year ago At about 7:45 this is parametric equations: D AndiRAin1 AndiRAin1 1 year ago Geometric Unity 👍🏻 1 Andrei V Andrei V 1 year ago (edited) Geometry and theology by milo is a great song btw Gilbert Anderson Gilbert Anderson 1 year ago 33:20 No, it looked like "desses" which you helpfully corrected to clesses. Valdagast Valdagast 1 year ago 29:32 My brain hurts. 7 Alex Doll Alex Doll 1 year ago Did you really end that with the word "fun"? How about "yikes"? 2 2 deepak gode deepak gode 5 months ago