Smooth Spaces 1: Introduction To Synthetic Differential Geometry

In this introductory video we outline the coming course in synthetic differential geometry. We explain the advantages to the synthetic approach to differential geometry, which include being able to set up calculus with greater ease, and being able to think about a large variety of spaces more easily. The remainder of the course will be somewhat informal, although we will make efforts to cover the underlying axioms in an exhaustive way, and we will put a lot of effort into describing numerous applications (which include solving physics problems, computing lengths, areas, volumes and curvatures etc.). In this introductory video we introduce important novel ideas, like the notions of function spaces, nilsquare infinitesimals, and the synthetic approach to defining the operations of calculus.
A rigorous approach to this subject, emphasizing the category theoretic foundations, can be found here:
Category Theory For Beginners: Synthetic Differential Geometry
• Category Theory For Be...
More videos about various things to do with synthetic differential geometry can be found within my playlist:
• Synthetic Differential...

[@AlphaCurveMath]

Thanks for always creating the most unique kind of mathematical content so consistently.

[@RichardSouthwell]

Thanks for the encouragement

[@thierryguth2567]

When Nature draws a line it starts from what in the end appears as the centre going to both directions
Think of a pebble thrown into a pond and the resulting waves...

[@HDQuote]

You are a gift to humanity! Thank you a lot. And you always upload the things I am interested in!

[@RichardSouthwell]

Such encouraging words, thank you

[@Ottmar555]

Any time infinitesimals are used, I like.

[@RichardSouthwell]

The course is loosely based on this book, which is easy to read, with lots of excercises www.amazon.co.uk/Primer-Infinitesimal-Analysis-John-Bell-ebook/dp/B001MJ0E5Y

[@alchemistsanonymous6558]

Your introduction just described Alchemy. 👌

[@abenedict85]

bro, what isn't alchemy?

[@sylvainw5319]

Hi, thanks a lot for this amazing content.
I was wondering if you have heard about Olivia Caramello theory that uses toposes as bridges for unifying different parts of the mathematics ?
It seems very interesting but yet a bit hard to deal with for the non-mathematician that I am.

[@sadface7457]

I knew about the dual numbers but it would interesting to see them in this context.

[@StephenPaulKing]

Can Smoothness be an emergent property? How can we balance the non-existence of measures in continuous sets, such as the Reals, with the needs of Analysis? What about approximations ala Ben Green and Normal Wildberger?

[@RichardSouthwell]

These are excellent questions. The short answer is, synthetic differential geometry is not based on traditional real or rational numbers. It's not even based on traditional set theory. Rigorously speaking, the theory happens in a specially designed topos. You can check out my 'category theory for beginners: synthetic differential geometry video' if you want to see all the axioms and underlying assumptions. But the short answer is nilsquare infinitesimals allow one to talk about continuity in an easier way.

[@pocketvelero]

Are you Catholic? Just curious 🙂