The Biggest Ideas in the Universe | Q&A 13 - Geometry and Topology
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- Опубліковано 20 чер 2020
- 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.
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#science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology - Наука та технологія
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.
it was worth me coming in here today just to read this comment :D :0
Sean Caroll saying ”I probably shouldn’t do this” means that it’s probably going to be somethine you really want to hear. :)
I love it when Sean uploads
@@PetraKann Solid point
@@PetraKann Does anything exist in reality? All we know are what we ourself observe. But what even is ourself, but an observation as well? The observations are meaningless unless we give them meaning. And to do that, we must assign meaning. It need not be coherent nor predictive of future observations and actually often is neiher, but in physics, we try to define a coherent mathematical meaning that predicts future observations. And it actually works. And it works better than any other mechanism to the point it's impossible to imagine another mechanism today.
And it doesn't matter if there are no "perfect" objects. What matters is that an object is close enough to perfect that the defects that do remain are small enough that over the time we make the observation, the effect of the defect on the final observation is so small that we cannot tell a difference between it and the perfect object.
It's not a circus; it's a miracle it is even possible.
I am always happy to have a new slew of concepts to flow through. They're a real treat! Best of all, Sean has actually taught me of concepts I never dreampt I was capable of grasping!😁
00:00:25 Geometry
14:30 so it really seems like on initial viewing the look of the Metric we get is heavily influenced by the way we choose our coordinate system. It seems like we chose a rule for describing S ^2 in IR ^3, that we measure the theta ~0~ co-ord first, and then the phi co-ord, and that really influenced what phi can look like, and therefore what the phi element of the Metric looks like. That is that it has a sin like quality, as it’s big near the equator and small near the poles
00:21:07 Another way to talk about the Riemann Tensor. This part was very interesting but also very hard to follow, so worth a rewatch after the GR episodes
30:05 Topology, more on Homotopy
38:00 what we are really trying to do, is figure out particular (particular because in this case we use Homotopy) invariant characterizations of a space (that can be deformed, smoothly). And so to clear up confusing conflations, note that we need the spaces themselves (that are superficially different based on their smooth deformations) to be invertable. We don’t need the spaces to be invertable to the circles (spheres). We use homeomorphisms of spheres to quantify the homeomorphisms *of spaces* (planes), so we don’t care if the planes are invertable to the spheres.
I also want to add that I really can’t see what Topology is useful for at this point. I can see why Geometry is useful, but not Topology. 43:12 Sean tells us how Topology is relevant for physics as
pi sub 0, 1, 2, correspond to Domain Walls, Cosmic Strings, and Monopoles, but I don’t think I properly grasped the explanation. 53:09 “that’s where the relationship between Topology and Topological Defects [physics] come from; it’s not the Topology of space, it’s the Topology of the space of zero-energy Field configurations.” Yeah. Definitely didn’t get that.
53:28 Cohomology. We know that Homotopy is one way of characterizing the Topology of spaces, and Cohomology is another where we see what spaces are and aren’t equivalent to each other by seeing what spaces can or can’t be integrated/differentiated in a well-defined way
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.
This whole series is fantastic, but this video happens to be my most revelatory yet. Thanks again Dr. Carroll!
This is a great video showing how starting with mathematical concepts scientists are able to relate them to space and the deep cosmos...
People this smart are just built different. Impressive...
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.
Sean you are exceptionally gifted in explaning difficult concepts to layman and layman++. Im just saying.
Layman++ haha, I like that
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...
*They keep getting BIGGER!* Thanks, Sean.
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
I feel very good to hear from you. Thanks a lot
Don't be upset, Sean spends his time learning and teaching. No Time to argue with flat earthers.
Thanks Sean!!
I am beginning to think these ideas are too big to fit inside my head, but I will keep trying.
Great vid. Thanks 👍
Excellent!
Irresistible topics indeed :)
47:33 - Domain Walls. Pretty cool! ^.^
55:18 - Fundamental Theorem of Calculus. ^.^
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
Do you have references for a deeper dive into the topology stuff? BTW: Lov'in the videos. Thanks.
Sean's brain is so plump and juicy with knowledge. My brain is like a dried, shrivelled up raisin in comparison 😔
Sean's only human and you have the same biological equipment as him. Even if you feel ignorant use that as an opportunity to learn more and you can get to where he's at too 🙌🏽
@@chaoticstorm8145 Yeah but WHAT a human!
These are truly huge ideas.
50:40 The potential of the combined field phi_1 and phi_2 you wrote down is not rotationally symmetric. So the vacuum manifold will not be the circle but four isolated points. Did you perhaps mean (phi1^2+phi2^2)^2?
@59:11
"There's the second derivative and the third derivative and so forth..."
... and the question remains for the reverse: for how many measured and then calculated "snaps" can the originating "acceleration" (or even "jerk") be reliably derived as functions themselves?
I get your point. :-)
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.
Trivial group is a group, and it's conventionally called "zero group" . It is called so for mathematical reasons which would probably take at least a few lectures to explain, and it has nothing to do with constructing natural ordinals from sets. There is no such thing as an "empty group", because you can't define addition or multiplication rule for elements without elements. And topology of an empty set is by definition trivial.
@@georgekomarov4140 In the video he plays fast and loose with the distinction with type distinction of group and integer. Statements like "zero has a member" can't be literal already on the grounds that integers don't have members by the virtue of being integers (they can have successors or sums but not really memebers). Thus a lot of implicit type casting is going on. And somebody lacking the background skills could make the castings wrong. Using "zero" instead of "zero group" makes for even more guesswork needed.
About the potential V(phi):
I think Sean intended to draw something like this:
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).
Geometry and theology by milo is a great song btw
29:32 My brain hurts.
Valdagast totally relate 🤣😂
Yeah...it's a quick and dirty explanation of tensors. The whole fact that indexes can be in either the top or bottom spot (but seemingly nowhere else?) is also quite weird, and I think it comes out of rules that the elements of the tensor must describe a group that has a co-group. If you're okay with learning the maths, the first few videos in ua-cam.com/video/KticNB8zAIQ/v-deo.html will eventually get you to tensors and indexed notation and then you'll sort of be able to follow the explanation.
@@hhaavvvvii thank you!
Geometric Unity 👍🏻
At about 7:45 this is parametric equations: D
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?
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!
If the universe has a ball, then it has events (particle vibrations) and time coordinatizes when they occur. Even a single particle can spin and this can be coodinatized by time. If you are talking about an empty universe then there are no events and time becomes trivial. This supports the notion that time is an emergent property.
But the vacuum in our universe is not empty space. It is a maelstrom of virtual particles manifesting and annihilating which take time to appear and disappear.
I understand that, and it is true but my imaginary universe has a ball as a single stable particle (or even multiple) with absolute zero temperature or any condition(or force upon) that may cause that vibration. I understand there is no such universe and the real universe has temperature and is in motion. But the reason for the way the reality is what it is, I belive, is not Time but the reason is temperature and diversity of particles and existance of forces. It is the forces that drives them not the time and so we measure that ‘events’ with the help of Time!
Also I agree that the time is an ‘emergent property’ but emergent property is an interpretation of the reality not the reality itself I believe. For example all mathematics exist because we created one number(that number can be any given number but logically it was the 1. The whole mathematics is an emergent property of that number as an interpretation (even though that number 1 itself does not exist) but the math is not real in that sense. Likewise our consciousness exists as an emergent property of the molecular relation between our neurons. The concept of emergent property , I believe , does not make it any real if we define the real as ‘texture of the universe!’ And what creates the ‘emergent properties’ is the coordination and relation of ‘things!’ to each other. Now I do understand that God does not play with dice but we are the one giving meaning to any face of the dice! We evolved to understand by giving meaning and defining them. Therefore, if resistance for electricity R=V/I, can be said time is sort of resistance as T=Distance/Velocity, can be said same thing about entropy. I mean time is resistance and emergent from texture of the reality of space!
@@robertshirley2645 (#1 vs. zero/non-zero)
I agree with you in many senses that time is less fundamental than many physists believe. I guess physists are so blazè about using infinitly large metaphysical spaces (like the Hilbert space) that they are not cautious enough in analyzing the dimensionality of reality. Everything may very well have happened "all at once" and it is only our consciousness that is forcing a timelike view of reality upon us.
Nevertheless, the definition of a physicist is someone who derives and calculates the evolution of a physical system over time, so I guess we can't expect much help from them.😁
Not to mention cosmology without time is really boring.
😃😂 that’s right. You know what? I am waiting for these videos every week and when they are late I don’t feel well about physics anymore 😃 . Thanks to Dr. Sean Carroll, he is really a good physicist and I do truly admire him and his work.
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
Haha was going to comment the same thing.
Did you really end that with the word "fun"? How about "yikes"?
33:20 No, it looked like "desses" which you helpfully corrected to clesses.
But the S' circle is not truly 1 dimensional, because you have to give 2 values for a direct position {R and theta}
Erik Dahlgren that’s why he said in R^2... 😑
It depends on how you treat a circle. A circle itself as a space needs just one value, which is the angle from the starting point. A circle on a plane needs two, but just because in some sense everything on a plane needs two.
It can be confusing to a novice when a physicist switches seamlessly between a 1D analysis and a 2D analysis of the same 1D object. They generally do not confuse each other when they do so.
When we talk about R & theta we are analysing a circle embedded in a 2D plane using polar coordinates, but the circle itself does not include the 2D point that is the origin of that coordinate system (0,0 is not on the circle).
If we define a circle with R = 180/π (~57.3) millimeters then it's circumference is exactly 360 mm.
Using 2D analysis we can pick a reference point on the circle, say R=180/π, theta=0 (equivalently x=~57.3, y=0 in Cartesian coordinates). Each degree of theta corresponds to 1mm along the circle. For a fixed R we can describe any point on the circle with one number, theta.
In a 1D analysis to an inhabitant of the circle, this is home, coordinate 0, and he can pick a direction, start walking, and any point on the circle can be described by one number, the travel distance to the point.
To your point, this 1D circle certainly differs from the number line! Our inhabitant reaches home after traveling 360 mm and the most distant region of the circle can have a coordinate of either 180 or -180 (for travel in the opposite direction). I personally would like to promote the circle to 1D+, but I'm sure the mathematicians disagree. They describe the + using homotopy, homology, cohomology, etc.
The r coordinate is only needed if you are working in a plane and need to identify where the points of the circle live in the plane. If you're only considering the circle itself (i.e. not as embedded in the plane), then the single coordinate theta is enough to uniquely label each point, which means it is truly one-dimensional.
R is fixed so theta is the only value needed to uniquely find all points