Topology | Math History | NJ Wildberger

This video gives a brief introduction to Topology. The subject goes back to Euler (as do so many things in modern mathematics) with his discovery of the Euler characteristic of a polyhedron, although arguably Descartes had found something close to this in his analysis of curvature of a polyhedron. We introduce this via rational turn angles, a renormalization of angle where a full turn has the value one (very reasonable, and ought to be used more!!) The topological nature of the Euler characteristic was perhaps first understood by Poincare, and we sketch his argument for its invariance under continuous transformations.

We discuss the sphere, torus, genus g surfaces and the classification of orientable, and non-orientable closed 2 dimensional surfaces, such as the Mobius band (which has a boundary) and the projective plane (which does not). The interest in these objects resulted from Riemann's work on surfaces associated to multi-valued functions in the setting of complex analysis.

Finally we briefly mention the important notion of a simply connected space, and the Poincare conjecture, solved recently, according to current accounts, by G. Perelman.

If you enjoy this subject, you can have a look at my video series Algebraic Topology. This series has now also been continued, so if you go to the Playlist MathHistory, you will find more videos on the History of Mathematics.

Video Content
00:00 Topology
04:43 Euler characteristic of a polyhedron
08:51 A polyhedron homeomorphic to a torus
10:11 H. Poincare (1895)
15:20 Descartes/ letter to Leibniz (1676) studied curvature of polyhedron
16:18 Rational angle version to curvature
23:56 Total curvature equals Euler characteristic
27:28 B.Riemann ( 1826-1866)- Complex functions
38:10 Riemann surfaces
43:30 Classification of 2 dimensional surfaces
44:05 List of all compact orientable surfaces

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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained:    / playlist  
list=PL8403C2F0C89B1333
Year 9 Maths:    • Year9Maths  
Ancient Mathematics:    • Ancient Mathematics  
Wild West Banking:    • Wild West Banking  
Sociology and Pure Mathematics:    • Sociology and Pure Mathematics  
Old Babylonian Mathematics (with Daniel Mansfield):    / playlist  
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History:    • MathHistory: A course in the History ...  
Wild Trig: Intro to Rational Trigonometry:    • WildTrig: Intro to Rational Trigonometry  
MathFoundations:    • Math Foundations  
Wild Linear Algebra:    • Wild Linear Algebra  
Famous Math Problems:    • Famous Math Problems  
Probability and Statistics: An Introduction:    • Probability and Statistics: an introd...  
Boole's Logic and Circuit Analysis:    • Boole's Logic and Circuit Analysis  
Universal Hyperbolic Geometry:    • Universal Hyperbolic Geometry  
Differential Geometry:    • Differential Geometry  
Algebraic Topology:    • Algebraic Topology  
Math Seminars:    • MathSeminars  
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And here are the Wild Egg Maths Playlists:

Triangle Centres:    • ENCYCLOPEDIA OF TRIANGLE CENTERS  
Six: An elementary course in pure mathematics:    • Six: An elementary course in Pure Mat...  
Algebraic Calculus One:    • Algebraic Calculus One  
Algebraic Calculus Two:    • Algebraic Calculus Two  
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