The magic of group completion

This is the second video out of three on the definition of algebraic K-theory! It is dedicated to the K-theory space of affine algebraic varieties (more generally, all rings).

Made with: Peter Haine

Thumbnail: made by Asama Lekbua

Comments:

K-theory of finite fields was computed by Quillen!

References:

1) Lecture by Thomas Nikolaus where the definition of K-theory space via group completion is discussed:

2) Lecture notes by Marc Hoyois where more details on group comlpetion of spaces are provided:

3) Quillen's paper where K(F_q) was computed:
"On the cohomology and K-theory of the general linear groups over a finite field", Annals of Mathematics, 1972

4) For those interested in learning more about K-theory, check out the remaining lectures of that minicourse by Thomas Nikolaus!

0:00 teaser
0:11 intro
0:47 invariants as epithets
2:01 higher K-groups
4:53 the magic of group completion
6:11 K-theory of finite fields
7:51 outro

[temitope]

Dear Mura (Maria) I want to say a big thank you for all that you do. I had a very hard time understanding the point of mathematics when I was younger, and your videos played a big role in getting me back into it.

I think I failed every math exam I took in secondary school. Most people sort of got it and I didn't, and I hated that, so I hated mathematics.

But, something weird happened: I developed a Jones for maths. I abandoned my humanities background and I am currently pursuing a career in Data Science. Your channel and enthusiasm have been a major force in my journey back to maths.

Keep doing what you do and keep the interviews and projects going. Oh and long live nerds!

[math-life-balance]

please be kind and don't judge the glitter, okay? wish I had ContraPoints' looks, but I don't dare

[PeakMathLandscape]

Yes! Great stuff! Too long has YouTube suffered from a lack of group completion.

[bockmancheung]

Thanks for the video! I am an incoming undergraduate student interested in algebraic topology. I am going to rewatch your videos a few times to hopefully understand better. You two did a wonderful job! Don't worry, we will be very nice and if there are any negative comments, I will defend your videos from them :D

[jesseburstrom]

Will AI like ChatGPT ever be useful in this context?! I hope so!