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

Wonderful lecture.

The topic was didactically perfectly motivated. Thank you very much!

Great..... lecture.... Its a key to entering in the modern mathematics

Another excellent lecture! Thanks

Nice suit and nice lecture! Thanks.

Thanks, lots of stuff explained in a intuitive way

Can you make a loop that approaches infinity or indeed any surface that approaches the infinities of it's orthogonality plus one?

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)

@enpeacemusic192

2 ай бұрын

I think so, yeah, homotopy of paths is ány continuous deformation of paths afaik

Thank you

Gem

In attempting to use topology in sociological circumstances, are therrighte different winding numbers for thought streams of what are commonly termed the

@John-js2uj

3 ай бұрын

What on earth are you trying to say?

@richardchapman1592

3 ай бұрын

@@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

3 ай бұрын

@@richardchapman1592 You’ve got to be a bot

@richardchapman1592

3 ай бұрын

@@John-js2uj so trained in logic and emotionally damaged couldn't refute that unless you saw me in flesh and blood.

@richardchapman1592

3 ай бұрын

@@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.

at 39:00, when you said f and g are homotopy equivalent, did you mean to say homotopic?

@bengrange

Ай бұрын

and at 53:16, you meant "equivalence classes" not relations. Thank you for the great lectures!!

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.

18:29 surjection=onto= heat everything to image. Onetoone. Man to one. Bikection

17:11

Great suit. Big effort on the outfit. Well done

Great lecture. Camera work needs improvement.

39:59 😂😂

@joshuad.furumele365

6 ай бұрын

I see you, and i raise you 29:03

@turtle926

4 ай бұрын

I raise further with 44:44 😎

Last comment on my editor needed a vector from the centre of a word to the end.

6:10

isnt S^1 x [0,1] the cylinder?

@DogeMcShiba

Ай бұрын

Yes, the annulus is homeomorphic to the surface of a cylinder.