00:00 Review of groups, homomorphisms, and isomorphisms 18:45 Return to topology: path homotopy 22:55 Why must two paths with the same endpoints in R2 be homotopic? 30:20 Homotopy is an equivalence relation 42:15 Different equivalence classes of paths in the annulus 45:20 Loops 58:00 definition of the fundamental group
Dear Professor, thank you so much ❤ I began to love Math with your lectures. Please keep up doing that for people who don't have much opportunity to access such materials ❤
45:00 “When I use a word, it means just what I choose it to mean - neither more nor less.” - Humpty Dumpty. You can tell Lewis Carroll was a mathematician.
Counter question: Why would having the same start and end points be enough? Homotopic means that you can continuously deform (intuitively, bend) one of those paths into the other. Try to do that. No matter what you come up with, you will have to go over the hole, which is forbidden, since you must remain inside the annulus
At 24:30, the explicit linear interpolation formula is given for one possible homotopy, to show that there is always a homotopy of paths in R2, correct? The language suggest that this is THE homotopy (ie the one and only)
Can see this pictorially using a 1dim path on a 2dim surface in 3dim. In larger dimensions not sure how an extrapolation is made using an analogy of an n dimensions path on a pdim brane in an sdim space.
In attempting to use topology in sociological circumstances, are therrighte different winding numbers for thought streams of what are commonly termed the
@@John-js2uj have an egoistic humility that my partial understanding can use these precise mathematical concepts in the imprecise social sciences. Worries me tho that mathematics applied to human circumstance can lead to a kind of cyber fascism if AI is taken too far too fast.
@@John-js2uj would ask of you an email address so I could send you a photo that you could possibly accept as not a fraud, but then there are Trojan horses on mails to worry about.
Watching the video again, it is not clear if the lines between s on f(t) are straight in R2. Some explanation of their continuity as s and t vary would help especially in spaces other than R2.
00:00 Review of groups, homomorphisms, and isomorphisms
18:45 Return to topology: path homotopy
22:55 Why must two paths with the same endpoints in R2 be homotopic?
30:20 Homotopy is an equivalence relation
42:15 Different equivalence classes of paths in the annulus
45:20 Loops
58:00 definition of the fundamental group
The topic was didactically perfectly motivated. Thank you very much!
Wonderful lecture.
Dear Professor, thank you so much ❤ I began to love Math with your lectures. Please keep up doing that for people who don't have much opportunity to access such materials ❤
Great..... lecture....
Its a key to entering in the modern mathematics
Another excellent lecture! Thanks
45:00 “When I use a word, it means just what I choose it to mean - neither more nor less.” - Humpty Dumpty. You can tell Lewis Carroll was a mathematician.
I just don’t get the example at 43:01. Wouldn’t f & g be homotopic to each other since they have the same start & end point?
Counter question: Why would having the same start and end points be enough?
Homotopic means that you can continuously deform (intuitively, bend) one of those paths into the other. Try to do that. No matter what you come up with, you will have to go over the hole, which is forbidden, since you must remain inside the annulus
Thanks, lots of stuff explained in a intuitive way
At 24:30, the explicit linear interpolation formula is given for one possible homotopy, to show that there is always a homotopy of paths in R2, correct? The language suggest that this is THE homotopy (ie the one and only)
I think so, yeah, homotopy of paths is ány continuous deformation of paths afaik
Can see this pictorially using a 1dim path on a 2dim surface in 3dim. In larger dimensions not sure how an extrapolation is made using an analogy of an n dimensions path on a pdim brane in an sdim space.
Nice suit and nice lecture! Thanks.
at 39:00, when you said f and g are homotopy equivalent, did you mean to say homotopic?
and at 53:16, you meant "equivalence classes" not relations. Thank you for the great lectures!!
Can you make a loop that approaches infinity or indeed any surface that approaches the infinities of it's orthogonality plus one?
In attempting to use topology in sociological circumstances, are therrighte different winding numbers for thought streams of what are commonly termed the
What on earth are you trying to say?
@@John-js2uj have an egoistic humility that my partial understanding can use these precise mathematical concepts in the imprecise social sciences. Worries me tho that mathematics applied to human circumstance can lead to a kind of cyber fascism if AI is taken too far too fast.
@@richardchapman1592 You’ve got to be a bot
@@John-js2uj so trained in logic and emotionally damaged couldn't refute that unless you saw me in flesh and blood.
@@John-js2uj would ask of you an email address so I could send you a photo that you could possibly accept as not a fraud, but then there are Trojan horses on mails to worry about.
That’s some of the best looking annulus in NA
Great video
18:29 surjection=onto= heat everything to image. Onetoone. Man to one. Bikection
Great suit. Big effort on the outfit. Well done
Thank you
Gem
39:59 😂😂
I see you, and i raise you 29:03
I raise further with 44:44 😎
Great lecture. Camera work needs improvement.
Last comment on my editor needed a vector from the centre of a word to the end.
Watching the video again, it is not clear if the lines between s on f(t) are straight in R2. Some explanation of their continuity as s and t vary would help especially in spaces other than R2.
17:11
isnt S^1 x [0,1] the cylinder?
Yes, the annulus is homeomorphic to the surface of a cylinder.
6:10