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

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."

#science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology

[robertshirley]

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

[georgekomarov]

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 :)

[jochemvanderspek]

"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 :)

[lukeneville]

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

[grayaj]

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

[kobevli]

This series is the absolute best thing in the world right now.

Keep up the great work Dr. Carroll !!!

[kshitishp]

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

[FelixMatathias]

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.

[peterpodgorski]

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

[Amir-vwrk]

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

[esperancaemisterio]

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

[alvarorodriguez]

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?

[venil]

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

[alachance]

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!

[sambarta]

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

[Valdagast]

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.

[davyoooo]

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

[misterkriskooper]

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.

[Mickolas]

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.

[চম্পাকর্মকার]

The biggest ideas in the universe: Applied Mathematics