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source: James Cook 2016年9月1日
Algebraic Topology: for fun
Here I collect the discussions from a reading course on Algebraic Topology held at Liberty University in Fall 2016. We follow Rotman's text for the most part and assuredly all errors are very much my fault.
L1, fixed pt. Thm showcase, 8-30-16. Rotman, Chapter 0, pages 1-5. 49:12
L2, categorical terms, start of homotopy, 9-1-16, part 1 59:51
L2, categorical terms, start of homotopy , 9-1-16, part 2 6:04
L3, homotopy , 9-6-16 42:07
L4, homotopy and cones, connectedness, 9-8-16, part1 59:51
L4, homotopy and cones, connectedness, 9-8-16, part 2 13:04
L5, affine spaces, 9-13-16, part 1 59:51
L5, affine spaces, 9-13-16, part 2 11:09
L6, homotopy and fundamental group, 9-15-16, part 1 59:51
L6, homotopy and fundamental group, 9-15-16, part 2 18:32
L7, homotopy and fundamental group, 9-20-16, part 3 59:51
L7, homotopy and fundamental group, 9-20-16, part 4 20:07
L8, FTA via homotopy, Free Abelian Groups, 9-22-16, part 1 59:51
L8, FTA via homotopy, Free Abelian Groups, 9-22-16, part 2 12:07
L9, Singular Homology, 9-27-16, part 1 59:51
L9, Singular Homology, 9-27-16, part 2 7:33
L10, homology functor, 9-27-16, part 1 59:51
L10, homology functor, 9-27-16, part 2 8:04
L11, homology, 9-29-16, part 2 10:50
L12, homology, 10-4-16, part 1 59:51
L12, long exact sequences, 10-11-16, part 1 59:51
L12, long exact sequences, 10-11-16, part 2 28:20
Showing posts with label A. (subjects)-Engineering & Physical Sciences-Mathematics-Algebraic Topology. Show all posts
Showing posts with label A. (subjects)-Engineering & Physical Sciences-Mathematics-Algebraic Topology. Show all posts
2017-05-17
Introductory Workshop on Algebraic Topology @ MSRI
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source: LeonhardEuler1 2014年8月5日
Videos from the MSRI introductory workshop on "Algebraic Topology" that took place at MSRI, Berkeley in January 2014. The workshop page (with videos and references and exercises) can be found here: http://www.msri.org/workshops/685
Introduction to Operads (William Dwyer @ MSRI) 59:38 About this talk: http://www.msri.org/workshops/685/sch...
Goodwillie's Calculus of Functors (Michael Ching @ MSRI) 57:23
Morita Theory in Stable Homotopy Theory (Brooke Shipley @ MSRI) 1:02:01
Chromatic Redshift (John Rognes @ MSRI) 54:48
Higher Categories and Algebraic K-Theory (Andrew Blumberg @ MSRI) 58:10
Towards Explicit Models for Higher K-Theories (Christopher Douglas @ MSRI) 1:02:17
Models for Homotopical Higher Categories (Julie Bergner @ MSRI) 55:00
Computations in the Stable Homotopy Groups of Spheres (Mark Behrens @ MSRI) 1:04:00
Computations in Motivic Homotopy Theory (Daniel Isaksen @ MSRI) 53:50
Views on the J-Homomorphism (Craig Westerland @ MSRI) 54:13
Local Structure of Groups and of their Classifying Spaces (Bob Oliver @ MSRI) 52:46
Equivariant Homotopy and Localization (Michael Hopkins @ MSRI) 1:03:17
Homotopy Theory of Kac-Moody Groups (Nitya Kitchloo @ MSRI) 59:57
[private video]
Representation Stability and Applications to Homological Stability (Thomas Church @ MSRI) 1:01:52
Stability of Moduli Spaces of Manifolds (Oscar Randal-Williams @ MSRI) 51:21
[private video]
Loop Groups, TQFTs, and Algebraic Geometry (Constantin Teleman @ MSRI) 1:02:15
source: LeonhardEuler1 2014年8月5日
Videos from the MSRI introductory workshop on "Algebraic Topology" that took place at MSRI, Berkeley in January 2014. The workshop page (with videos and references and exercises) can be found here: http://www.msri.org/workshops/685
Introduction to Operads (William Dwyer @ MSRI) 59:38 About this talk: http://www.msri.org/workshops/685/sch...
Goodwillie's Calculus of Functors (Michael Ching @ MSRI) 57:23
Morita Theory in Stable Homotopy Theory (Brooke Shipley @ MSRI) 1:02:01
Chromatic Redshift (John Rognes @ MSRI) 54:48
Higher Categories and Algebraic K-Theory (Andrew Blumberg @ MSRI) 58:10
Towards Explicit Models for Higher K-Theories (Christopher Douglas @ MSRI) 1:02:17
Models for Homotopical Higher Categories (Julie Bergner @ MSRI) 55:00
Computations in the Stable Homotopy Groups of Spheres (Mark Behrens @ MSRI) 1:04:00
Computations in Motivic Homotopy Theory (Daniel Isaksen @ MSRI) 53:50
Views on the J-Homomorphism (Craig Westerland @ MSRI) 54:13
Local Structure of Groups and of their Classifying Spaces (Bob Oliver @ MSRI) 52:46
Equivariant Homotopy and Localization (Michael Hopkins @ MSRI) 1:03:17
Homotopy Theory of Kac-Moody Groups (Nitya Kitchloo @ MSRI) 59:57
[private video]
Representation Stability and Applications to Homological Stability (Thomas Church @ MSRI) 1:01:52
Stability of Moduli Spaces of Manifolds (Oscar Randal-Williams @ MSRI) 51:21
[private video]
Loop Groups, TQFTs, and Algebraic Geometry (Constantin Teleman @ MSRI) 1:02:15
2017-05-16
Reimagining the Foundations of Algebraic Topology (@MSRI)
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source: LeonhardEuler1 2014年7月11日
Videos from the MSRI workshop "Reimagining the Foundations of Algebraic Topology" that took place at MSRI, Berkeley in April 2014. The workshop page (with videos and some supplemental material) can be found here: http://www.msri.org/workshops/689
Algebraic Geometry of Topological Field Theories (David Ben-Zvi @ MSRI) 1:03:38 About this talk: http://www.msri.org/workshops/689/sch...
Poincaré/Koszul Duality (David Ayala @ MSRI) 59:14
Poincaré/Koszul Duality and Formal Moduli (John Francis @ MSRI) 56:47
The Formal Theory of Adjunctions, Monads, Algebras, and Descent (Emily Riehl @ MSRI) 1:01:34
What is an Elementary Higher Topos? (Andre Joyal @ MSRI) 1:08:45
Co-Segal Algebras and Deligne's Conjecture (Hugo Bacard @ MSRI) 52:23
Aspects of Differential Cohomology (Thomas Nikolaus @ MSRI) 1:06:35
Modeling Stable 2-Types (Angelica Osorno @ MSRI) 59:03
Redshift and Higher Categories (Clark Barwick @ MSRI) 1:00:46
The Unicity of the Homotopy Theory of Higher Categories (Christopher Schommer-Pries @ MSRI) 1:04:44
Derived Equivariant Algebraic Geometry (Michael Hill @ MSRI) 1:00:14
Duality, Algebro-Homotopically (Vesna Stojanoska @ MSRI) 1:01:51
Calculations in Multiplicative Stable Homotopy Theory at Height 2 (Charles Rezk @ MSRI) 1:04:12
En Genera (Michael Mandell @ MSRI) 1:04:06
Motives Versus Noncommutative Motives (Goncalo Tabuada @ MSRI) 59:04
Thom Spectra and Twisted Umkehr Maps (David Gepner @ MSRI) 1:01:30
A K(Z,4) in Nature (Andre Henriques @ MSRI) 59:46
source: LeonhardEuler1 2014年7月11日
Videos from the MSRI workshop "Reimagining the Foundations of Algebraic Topology" that took place at MSRI, Berkeley in April 2014. The workshop page (with videos and some supplemental material) can be found here: http://www.msri.org/workshops/689
Algebraic Geometry of Topological Field Theories (David Ben-Zvi @ MSRI) 1:03:38 About this talk: http://www.msri.org/workshops/689/sch...
Poincaré/Koszul Duality (David Ayala @ MSRI) 59:14
Poincaré/Koszul Duality and Formal Moduli (John Francis @ MSRI) 56:47
The Formal Theory of Adjunctions, Monads, Algebras, and Descent (Emily Riehl @ MSRI) 1:01:34
What is an Elementary Higher Topos? (Andre Joyal @ MSRI) 1:08:45
Co-Segal Algebras and Deligne's Conjecture (Hugo Bacard @ MSRI) 52:23
Aspects of Differential Cohomology (Thomas Nikolaus @ MSRI) 1:06:35
Modeling Stable 2-Types (Angelica Osorno @ MSRI) 59:03
Redshift and Higher Categories (Clark Barwick @ MSRI) 1:00:46
The Unicity of the Homotopy Theory of Higher Categories (Christopher Schommer-Pries @ MSRI) 1:04:44
Derived Equivariant Algebraic Geometry (Michael Hill @ MSRI) 1:00:14
Duality, Algebro-Homotopically (Vesna Stojanoska @ MSRI) 1:01:51
Calculations in Multiplicative Stable Homotopy Theory at Height 2 (Charles Rezk @ MSRI) 1:04:12
En Genera (Michael Mandell @ MSRI) 1:04:06
Motives Versus Noncommutative Motives (Goncalo Tabuada @ MSRI) 59:04
Thom Spectra and Twisted Umkehr Maps (David Gepner @ MSRI) 1:01:30
A K(Z,4) in Nature (Andre Henriques @ MSRI) 59:46
2017-04-12
Algebraic Topology by Norman J. Wildberger at UNSW
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source: njwildberger 2011年3月9日
This is the full introductory lecture of a beginner's course in Algebraic Topology, given by N J Wildberger at UNSW. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try.
The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties.
The lecturer is Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia, well known for his discovery of Rational Trigonometry, explained in the series WildTrig, the development of Universal Hyperbolic Geometry, explained in the series UnivHypGeom, and for his other YouTube series WildLinAlg and MathFoundations. He also has done a fair amount of research in harmonic analysis and representation theory of Lie groups.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .
Algebraic Topology
A first course in Algebraic Topology, with emphasis on visualization, geometric intuition and simplified computations. Given by Assoc Prof N J Wildberger at UNSW.
The really important aspect of a course in Algebraic Topology is that it introduces us to a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. What does this exactly mean? That is a key question...
The course has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical. We try to use pictures and models to guide our understanding.
0: Introduction to Algebraic Topology 30:01
1: One-dimensional objects 32:19
2: Homeomorphism and the group structure on a circle 52:09
3: Two-dimensional surfaces: the sphere 42:27
4: More on the sphere 40:52
5: Two-dimensional objects--the torus and genus 49:33
6: Non-orientable surfaces---the Mobius band 42:05
7: The Klein bottle and projective plane 39:42
8: Polyhedra and Euler's formula 45:35
9: Applications of Euler's formula and graphs 42:23
10: More on graphs and Euler's formula 47:50
11: Rational curvature, winding and turning 48:19
12: Duality for polygons and the Fundamental theorem of Algebra 45:36
13: More applications of winding numbers 26:59
14: The Ham Sandwich theorem and the continuum 36:26
15: Rational curvature of a polytope 50:23
16: Rational curvature of polytopes and the Euler number 35:28
17: Classification of combinatorial surfaces (I) 50:32
18: Classification of combinatorial surfaces (II) 1:00:21
19: An algebraic ZIP proof of the classification 42:23
20: The geometry of surfaces 43:54
21: The two-holed torus and 3-crosscaps surface 38:53
22: Knots and surfaces I 52:23
23: Knots and surfaces II 38:11
24: The fundamental group 43:05
25: More on the fundamental group 34:56
26: Covering spaces 53:49
27: Covering spaces and 2-oriented graphs 30:42
28: Covering spaces and fundamental groups 46:41
29: Universal covering spaces 48:17
Review: An informal introduction to abstract algebra 48:55
Review2: Introduction to group theory 46:44
Review3: More on commutative groups---isomorphisms, homomorphisms, cosets and quotient groups 32:02
Review4: Free abelian groups and non-commutative groups 50:59
30: An introduction to homology 46:57
31: An introduction to homology (cont.) 41:24
32: Simplices and simplicial complexes 49:09
33: Computing homology groups 41:07
34: More homology computations 42:55
35: Delta complexes, Betti numbers and torsion 48:17
source: njwildberger 2011年3月9日
This is the full introductory lecture of a beginner's course in Algebraic Topology, given by N J Wildberger at UNSW. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try.
The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties.
The lecturer is Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia, well known for his discovery of Rational Trigonometry, explained in the series WildTrig, the development of Universal Hyperbolic Geometry, explained in the series UnivHypGeom, and for his other YouTube series WildLinAlg and MathFoundations. He also has done a fair amount of research in harmonic analysis and representation theory of Lie groups.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .
Algebraic Topology
A first course in Algebraic Topology, with emphasis on visualization, geometric intuition and simplified computations. Given by Assoc Prof N J Wildberger at UNSW.
The really important aspect of a course in Algebraic Topology is that it introduces us to a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. What does this exactly mean? That is a key question...
The course has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical. We try to use pictures and models to guide our understanding.
0: Introduction to Algebraic Topology 30:01
1: One-dimensional objects 32:19
2: Homeomorphism and the group structure on a circle 52:09
3: Two-dimensional surfaces: the sphere 42:27
4: More on the sphere 40:52
5: Two-dimensional objects--the torus and genus 49:33
6: Non-orientable surfaces---the Mobius band 42:05
7: The Klein bottle and projective plane 39:42
8: Polyhedra and Euler's formula 45:35
9: Applications of Euler's formula and graphs 42:23
10: More on graphs and Euler's formula 47:50
11: Rational curvature, winding and turning 48:19
12: Duality for polygons and the Fundamental theorem of Algebra 45:36
13: More applications of winding numbers 26:59
14: The Ham Sandwich theorem and the continuum 36:26
15: Rational curvature of a polytope 50:23
16: Rational curvature of polytopes and the Euler number 35:28
17: Classification of combinatorial surfaces (I) 50:32
18: Classification of combinatorial surfaces (II) 1:00:21
19: An algebraic ZIP proof of the classification 42:23
20: The geometry of surfaces 43:54
21: The two-holed torus and 3-crosscaps surface 38:53
22: Knots and surfaces I 52:23
23: Knots and surfaces II 38:11
24: The fundamental group 43:05
25: More on the fundamental group 34:56
26: Covering spaces 53:49
27: Covering spaces and 2-oriented graphs 30:42
28: Covering spaces and fundamental groups 46:41
29: Universal covering spaces 48:17
Review: An informal introduction to abstract algebra 48:55
Review2: Introduction to group theory 46:44
Review3: More on commutative groups---isomorphisms, homomorphisms, cosets and quotient groups 32:02
Review4: Free abelian groups and non-commutative groups 50:59
30: An introduction to homology 46:57
31: An introduction to homology (cont.) 41:24
32: Simplices and simplicial complexes 49:09
33: Computing homology groups 41:07
34: More homology computations 42:55
35: Delta complexes, Betti numbers and torsion 48:17
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