Manifolds, classification of surfaces and Euler characteristic | Differential Geometry 25

Here we give an informal introduction to the modern idea of `manifold', putting aside all the many logical difficulties that are bound up in this definition: difficulties associated with specification, with the use of `infinite sets', with the notions of `functions' etc.

Even those students who aspire to understanding mathematics correctly ought to be at least aware of the standard formulations, and if one is teaching a course at a major university these days one is limited by the curriculum and the orientation of students and other lecturers in the level of directness that one may address these foundational problems.
I will eventually be discussing the difficulties with these concepts in the MathFoundations series.

In this lecture we talk about charts, manifolds, orientation, and then look more carefully at the two dimensional case of compact surfaces, where things are more concrete and explicit, largely through the classification of Dehn and Heegard which utilizes in a major way the Euler characteristic.

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[DanielBeecham]

Hey, just wanted to say that I love your videos, you are a great teacher! Thank you, from a physics student in Sweden.

[sijojosephdr]

Thank you professor, this is what I was expecting for a long time. I consider this as your new year gift to me. I hope that it will enrich my basic understanding of General Relativity.

[jkli]

Thanks! That helps me to know more about manifolds .  
from a student in Hong Kong, China

[ddorman]

Thank you Dr. N.J. Wildberger for your fantastic and detailed description of X, Y, Z, I look forward to working with you on the future of theoretical science and the simple explanations that can be applied to understanding X, Y, Z, peace and love, Doug.

[phumzilemadonsela]

Thank you for a concise definition of manifold

[sridharnatarajan]

Sir, Thank you very much for your videos. Thanks to your videos, I have understood concepts that I have struggled to get for several years

[Potatos]

I love math and I love learning about topology concepts, especially manifolds! I wish we could be taught things like this in high school, but instead we learn about boring trig identities and completing the square. My mom's a mathematician and even she says she's only had to use completing the square once to solve some PDE! The ideas you explain in the video are much more interesting, and probably useful than anything I ever learned in a high school math class(aside from Calculus perhaps). Thanks for putting this up!

[baominjing]

This video is so good for me, but I don’t know how to see the subtitle

[thisisvaze]

Awesome Lecture. I have a learned a lot from your videos.

[parunach]

Very nice lecture. Trying to deduce whether one can use manifolds to talk about mapping between an input and output space in neural networks.

[mathieu]

Wow ! you make it very easy. Thank you !

[TurboLoveTrain]

You should run conventions on how to teach math.

[蕭羽白]

thank you professor, this is just great!

[kevinwu]

this is the Discovery Channel of math.