I feel like 'so... DO IT!' is your catchphrase. I love it. These lectures do skirt over some formalities (although that's what books are for anyway) but they're an absolutely fantastic thing to watch to build intuition and refresh one's knowledge. Teaching is an underrated skill, and you're clearly brilliant at it. I'm having so much fun watching these and following along.
I'm glad you're enjoying them! And you're right, I am sacrificing some formal rigor in favor of giving more intuition - largely because I am trying to cover quite a bit in a 24 lecture course. I hope this gives one a working knowledge of Algebraic Topology and sets one up well to read a text such as Hatcher's to get the fuller treatment.
29:30 Group of homeomorphisms of covering space preserving image of each point and orientation of curve(??? I don’t get why the orientation is preserved) is isomorphic to quotient group of fundamental group of base space by fundamental group of covering space
I could have been clearer on this point: a deck transformation is a homeomorphism that preserves the projection map onto the base space. This entails both that a point that maps to a basepoint is sent to another point that is sent to the basepoint as well as orientation being preserved.
@@MathatAndrews Deck transformations are dual to permutations. Rotations are dual to reflections. Symmetry is dual to anti-symmetry -- permutation groups. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence. Injection is dual to surjection synthesizes bijection or isomorphism -- Hegel. Real is dual imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual.
great lectures! what semester is this being taught to? also, what do you think about the book introduction to topological manifolds by john M lee for algebraic topology?
Deck transformations are dual to permutations. Rotations are dual to reflections. Symmetry is dual to anti-symmetry -- permutation groups. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence.
actually, subgroups of the galois group (when the field extension is a galois extension) are to intermediate subfieds the EXACT same way subgroups of the Deck group (when the covering is regular/normal/galois) are to intermediate covering spaces
@@theheckl Categories (form, syntax) are dual to sets (substance, semantics) -- category theory. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. The tetrahedron is self dual -- just like the integers. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids are dual. Domains are dual to codomains - groups. Sheaves (homology) are dual to co-sheaves (co-homology). Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Riemann geometry contains a hidden dual basis -- curvature is dual. Lie groups (multiplication) are dual to Lie algebras (addition). Exponentials (Lie groups) are dual to logarithms (Lie algebras). Addition is dual to subtraction (additive inverses) -- Abstract algebra. Multiplication is dual to division (multiplicative inverses) -- Abstract algebra. Integration (syntropy) is dual to differentiation (entropy). Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Structure (syntax) is dual to function (semantics) -- protein folding. Protein shape or structure determines its function or purpose -- teleological. Protein folding is dual -- all life is dual. Knowledge is dual according to Immanuel Kant -- synthetic a priori knowledge.
@@theheckl Science or knowledge is based upon empirical measurement, physics -- "a posteriori". Whereas the lack of knowledge (entropy) is "a priori" or before measurement. A priori (noumenal) is dual to a posteriori (phenomenal) -- Immanuel Kant. There is therefore a dual process to that of increasing entropy namely syntropy -- the 4th law of thermodynamics! Gathering knowledge or the scientific method is a syntropic process -- teleological. What you know is syntropic, what you do not know is entropic! -- duality!
I feel like 'so... DO IT!' is your catchphrase. I love it. These lectures do skirt over some formalities (although that's what books are for anyway) but they're an absolutely fantastic thing to watch to build intuition and refresh one's knowledge. Teaching is an underrated skill, and you're clearly brilliant at it. I'm having so much fun watching these and following along.
I'm glad you're enjoying them! And you're right, I am sacrificing some formal rigor in favor of giving more intuition - largely because I am trying to cover quite a bit in a 24 lecture course. I hope this gives one a working knowledge of Algebraic Topology and sets one up well to read a text such as Hatcher's to get the fuller treatment.
@@MathatAndrews Yes. I'm glad you chose breadth over (tedious) depth! The brief teaser for geometric group theory in the last lecture was fun too.
...I mean, _this_ lecture!
@@MathatAndrews I like this decision. Honestly, your lectures make following Hatcher wayyyy easier
Wonderful Presentation. Best ever for my self taught efforts, in rounding out my understanding of several topics all at once!
This helped me in my life. You have earned a subscriber :)
Excellent course. Thank You!
29:30 Group of homeomorphisms of covering space preserving image of each point and orientation of curve(??? I don’t get why the orientation is preserved) is isomorphic to quotient group of fundamental group of base space by fundamental group of covering space
I could have been clearer on this point: a deck transformation is a homeomorphism that preserves the projection map onto the base space. This entails both that a point that maps to a basepoint is sent to another point that is sent to the basepoint as well as orientation being preserved.
@@MathatAndrews Deck transformations are dual to permutations.
Rotations are dual to reflections.
Symmetry is dual to anti-symmetry -- permutation groups.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
Injection is dual to surjection synthesizes bijection or isomorphism -- Hegel.
Real is dual imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
great lectures! what semester is this being taught to? also, what do you think about the book introduction to topological manifolds by john M lee for algebraic topology?
Deck transformations are dual to permutations.
Rotations are dual to reflections.
Symmetry is dual to anti-symmetry -- permutation groups.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
actually, subgroups of the galois group (when the field extension is a galois extension) are to intermediate subfieds the EXACT same way subgroups of the Deck group (when the covering is regular/normal/galois) are to intermediate covering spaces
@@theheckl Categories (form, syntax) are dual to sets (substance, semantics) -- category theory.
Syntax is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
The tetrahedron is self dual -- just like the integers.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids are dual.
Domains are dual to codomains - groups.
Sheaves (homology) are dual to co-sheaves (co-homology).
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Riemann geometry contains a hidden dual basis -- curvature is dual.
Lie groups (multiplication) are dual to Lie algebras (addition).
Exponentials (Lie groups) are dual to logarithms (Lie algebras).
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (syntropy) is dual to differentiation (entropy).
Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics!
Structure (syntax) is dual to function (semantics) -- protein folding.
Protein shape or structure determines its function or purpose -- teleological.
Protein folding is dual -- all life is dual.
Knowledge is dual according to Immanuel Kant -- synthetic a priori knowledge.
@@theheckl Science or knowledge is based upon empirical measurement, physics -- "a posteriori".
Whereas the lack of knowledge (entropy) is "a priori" or before measurement.
A priori (noumenal) is dual to a posteriori (phenomenal) -- Immanuel Kant.
There is therefore a dual process to that of increasing entropy namely syntropy -- the 4th law of thermodynamics!
Gathering knowledge or the scientific method is a syntropic process -- teleological.
What you know is syntropic, what you do not know is entropic! -- duality!
Amazing ,every Thursday one episode is coming, are you taking one class per week in your university.
That's right!
32:24 can you please explain this a little bit better
Can’t wait for homology