What is...tropical geometry?

[@Juniper-111]

I recommend Madeline Brandt's video on tropical geometry for anyone interested in learning more!

[@VisualMath]

Excellent recommendation, thanks 😀 Here is the link: kzbin.info/www/bejne/l4rUeJSkn8-ImrM

[@Sidionian]

Thanks for the upload! Looking forward to schemes, stacks and Topos Theory :)

[@VisualMath]

Thanks, I am getting to schemes and friends. I am just slow 😅

[@Jaylooker]

I wonder if tropical geometry could be used to prove the Riemann existence theorem which relates function fields to compact Riemann surface or branched coverings of the Riemann sphere C ∪ {∞}. Tropical geometry treats fields like the tropical semiring R ∪ {∞} = S^1 as mentioned at 7:10. A knot is an embedding of S^1 into the three sphere S^3. Can fields be treated like knots?

[@VisualMath]

That sounds like an interesting idea, thanks for the food for thoughts! The question is how would you model the embedding of a knot (which is the main part) on the tropical semiring?

[@Jaylooker]

@@VisualMath At least for Riemann existence theorem I wonder if the tropical algebraic tori (T*)^n = R^n have n dependent on the Krull dimension of the field K (ie function field; number field). This is in relation to the Kapranov’s theorem. Yes, I am interested in the knots of tropical semiring. 3-dimensional real space R^3 can be used to define a knot instead of the sphere S^3 to possibly make it easier.

[@Nibor999]

Looks quite interesting. I didn't fully understand what it was all about, because you didn't define a number of things. But I did like the short video, which sort of gave a "feel" for what TG is. Thanks!

[@VisualMath]

Yeah, I am not sure whether KZbin is the right place to formally define things, so I never do that 🤫 Anyway, as long as you think its interesting and you might have another look and read about it, then I am happy ☺

[@Null_Simplex]

How does this relate to simplicial complexes? Seems like a more general simlicial complex on the surface.

[@VisualMath]

I think simplicial complexes run in parallel to TG. Some of the basic objects in TG are things like simplicial complexes, but the focus of the fields is different 🤔

[@boomerzilean]

Hi, will you be talking more about tropical geometry? Also can you make your facecam a bit bigger?

[@VisualMath]

Yes, I try to cover more at the later videos in the "Algebraic geometry" series. I hope you will like it 🙂 I decided to make the facecam not too big: my face is not that exciting 😅and I do not want to cover a bigger part of the screen. I hope that makes sense.

[@artemetra3262]

very cool!

[@VisualMath]

Thanks, I think so as well 😀 I am glad that you liked the video and topic ☺