00:31:32 the book is Algebra and Galois theories by Adrien Douady, who was a mathematician of Grothendieck's caliber, and is behind a great deal of major mathematical results of the XXth century. You dont hear about him because his name does not appear in the articles, but he is behind a lot of Serre's results, Wiles, Yoccoz, Atiyah, Connes, Weil and Grothendieck... He had a nack for visualizing the steps conducting to the proofs of several conjectures, sharing his ideas without proving them. He has major results of his own in Complex Dynamics.
The “topologist’s sine curve” is semi-locally simply connected but not locally simply connected (nor is it locally path-connected). Consider an open ball around a point within the space on the y-axis. This neighborhood is not path-connected, but any loop within the neighborhood is entirely contained either on the y-axis or in the “curvy” part of the space with positive x-coordinates, and any such loop is null-homotopic.
I'm pretty sure its the book by Hatcher on algebraic topology (he mentions the name in various ocations). It can be found here: pi.math.cornell.edu/~hatcher/AT/AT.pdf
I think he was writing path products (path concatenation) backwards when proving the conjugacy of the subgroups induced by different basepoints. It is standard to write a path product from left to right. We should write: p2_*([gamma^-1 • a • gamma]) = [a^-1 • b^-1 • a • b • a] = p2_*([gamma])^-1 • [a] • p1_*([gamma]), where [f] |-> [gamma^-1 • f • gamma] is the isomorphism from pi_1(X~, x~_1) to pi_1(X~, x~_2) along the path gamma. In general, the map: [g] |-> [b • a]^-1 • [g] • [b • a] for all [g] in pi_1(X, x) is an inner automorphism taking H_1 to H _2.
Injective is dual to surjective synthesizes bi-jective or isomorphism! Equivalence, similarity = duality. Duality: two equivalent descriptions of the same thing (homeomorphism) -- Leonard Susskind, physicist.
00:31:32 the book is Algebra and Galois theories by Adrien Douady, who was a mathematician of Grothendieck's caliber, and is behind a great deal of major mathematical results of the XXth century. You dont hear about him because his name does not appear in the articles, but he is behind a lot of Serre's results, Wiles, Yoccoz, Atiyah, Connes, Weil and Grothendieck... He had a nack for visualizing the steps conducting to the proofs of several conjectures, sharing his ideas without proving them. He has major results of his own in Complex Dynamics.
The “topologist’s sine curve” is semi-locally simply connected but not locally simply connected (nor is it locally path-connected).
Consider an open ball around a point within the space on the y-axis. This neighborhood is not path-connected, but any loop within the neighborhood is entirely contained either on the y-axis or in the “curvy” part of the space with positive x-coordinates, and any such loop is null-homotopic.
Nice lecture
31:23 Can anyone say which book he means? I couldn't find the book by searching the name he gives.
Might be Galois Groups and Fundamental Groups by Szamuely
I'm pretty sure its the book by Hatcher on algebraic topology (he mentions the name in various ocations). It can be found here: pi.math.cornell.edu/~hatcher/AT/AT.pdf
I think he was writing path products (path concatenation) backwards when proving the conjugacy of the subgroups induced by different basepoints. It is standard to write a path product from left to right.
We should write:
p2_*([gamma^-1 • a • gamma]) = [a^-1 • b^-1 • a • b • a] = p2_*([gamma])^-1 • [a] • p1_*([gamma]),
where [f] |-> [gamma^-1 • f • gamma] is the isomorphism from pi_1(X~, x~_1) to pi_1(X~, x~_2) along the path gamma.
In general, the map:
[g] |-> [b • a]^-1 • [g] • [b • a] for all [g] in pi_1(X, x)
is an inner automorphism taking H_1 to H _2.
I think so too
Injective is dual to surjective synthesizes bi-jective or isomorphism!
Equivalence, similarity = duality.
Duality: two equivalent descriptions of the same thing (homeomorphism) -- Leonard Susskind, physicist.