0:00 - Good morning (& recap) 1:57 - Fundamental Theorem of Algebra as application 15:48 - Brouwer Fixed-Point Theorem in Dimension 2 23:10 - Borsuk-Ulam Theorem 36:47 - Three-Piece Sphere Cover (corollary on antipodal pts) 43:24 - Product of fundamental groups is fundamental group of products 47:58 - (Application) Fundamental group of the n-dimensional torus 50:17 - Induced homomorphisms (morphisms carry over) 57:30 - Functoriality of fundamental group (+ amazing observation by student) 1:00:40 - Fundamental group of S^n trivial for n > 1 by loop contraction argument 1:12:43 - R^2 not homeomorphic to R^m for m other than 2 1:15:00 - Fundamental group(s) of punctured Euclidean space
1:08:00 compactness works, provided that f^-1(x1) is compact. This follows from the fact that f is proper, which in turn is true because I is compact and the sphere is Hausdorff
At 1:14:30 you write that R^n\{0}=S^(n-1)xR due to the polar coordinates. I think it should be R^n\{0}=S^(n-1)x(0,+oo). Thank you for the lectures anyway, you are very clear. I really appreciate :)
Page 94 barrister = n 1 Also called: barrister-at-law. (in England) a lawyer who has been called to the bar and is qualified to plead in the higher courts. Cf. solicitor. 2 (in Canada) a lawyer who pleads in court 3 US. a less common word for lawyer. [C16: from BAR'] Barry is a male name.
Topological holes cannot be shrunk down to zero. Null homotopic implies contraction to a point but non null homotopic implies a topological hole! The circle contains a topological hole, the sphere contains two holes. Non null homotopic implies duality or a second point prevents contraction of a loop to zero. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Points are dual to lines -- the principle of duality in geometry. Length, distance, space is defined by two (dual) points which are boundaries of the line -- space duality. Space is a dual concept. Up is dual to down, left is dual to right, in is dual to out -- space duality. Space is dual to time -- Einstein. Time is a dual concept, the future is dual to past -- time duality. Space duality is dual to time duality. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Duality creates reality. "Always two there are" -- Yoda.
0:00 - Good morning (& recap)
1:57 - Fundamental Theorem of Algebra as application
15:48 - Brouwer Fixed-Point Theorem in Dimension 2
23:10 - Borsuk-Ulam Theorem
36:47 - Three-Piece Sphere Cover (corollary on antipodal pts)
43:24 - Product of fundamental groups is fundamental group of products
47:58 - (Application) Fundamental group of the n-dimensional torus
50:17 - Induced homomorphisms (morphisms carry over)
57:30 - Functoriality of fundamental group (+ amazing observation by student)
1:00:40 - Fundamental group of S^n trivial for n > 1 by loop contraction argument
1:12:43 - R^2 not homeomorphic to R^m for m other than 2
1:15:00 - Fundamental group(s) of punctured Euclidean space
"Good morning... GOOD MORNING..." so old school. I like it. Has to be done with a smile. Bijection of cordiality.
these lectures explain Alan Hatchers book really well
I like this proof of the fun theorem of algebra.
1:08:00 compactness works, provided that f^-1(x1) is compact. This follows from the fact that f is proper, which in turn is true because I is compact and the sphere is Hausdorff
why does compactness of f^-1(x1) tell us that there are only finitely many intervals which map to x1?
is it because you take the intervals which make up f^-1(B) to be the open cover of f^-1(x1), and it has a finite subcover?
thanks nico
The story about Brouwer is great
At 1:14:30 you write that R^n\{0}=S^(n-1)xR due to the polar coordinates.
I think it should be R^n\{0}=S^(n-1)x(0,+oo).
Thank you for the lectures anyway, you are very clear. I really appreciate :)
@@mocktheta Yes you are absolutely right
Actually my point was quite silly :)
c est quoi embedding en francais merci d avance
Plongement
Page 94
barrister = n 1 Also called: barrister-at-law. (in England) a lawyer who has been called to the bar and is qualified to plead in the higher courts. Cf. solicitor. 2 (in Canada) a lawyer who pleads in court 3 US. a less common word for lawyer. [C16: from BAR']
Barry is a male name.
bar sinister = n 1 (not in heraldic usage) another name for bend sinister. 2 the condition or stigma of being of illegitimate birth.
Topological holes cannot be shrunk down to zero. Null homotopic implies contraction to a point but non null homotopic implies a topological hole! The circle contains a topological hole, the sphere contains two holes.
Non null homotopic implies duality or a second point prevents contraction of a loop to zero.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Points are dual to lines -- the principle of duality in geometry.
Length, distance, space is defined by two (dual) points which are boundaries of the line -- space duality.
Space is a dual concept.
Up is dual to down, left is dual to right, in is dual to out -- space duality.
Space is dual to time -- Einstein.
Time is a dual concept, the future is dual to past -- time duality.
Space duality is dual to time duality.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Duality creates reality.
"Always two there are" -- Yoda.