I would like suggestions regarding reading about geometry and manifolds

[s00mb]

Hi, I just finished up with Riemann Geometry not to long ago, and did something with complex geometry on kahler manifolds. In your opinion what would be a next logical step for someone to study? I am very interested in manifold theory and differential geometry in general. I'm somewhat familiar with algebraic geometry as well, I notice that get thrown in sometimes with manifold theory. Main reason I'm asking is that now when I look for books to self study they seem to go off in different directions at this level. I would like other peoples opinions on what would be interesting to self study and a good book/paper related to it. Preference towards things that may have physical applications. Thanks!

 

[jedishrfu]

One could investigate Discrete Differential Geometry:

https://www.cs.cmu.edu/~kmcrane/Projects/DDG/
which is somewhat computational but hey you want to see what you're studying right?

@fresh_42 may have some better suggestions here.

 

[fresh_42]

@fresh_42 may have some better suggestions here.

As so often this is a difficult to answer question. It's a bit like asking: I have visited all locations of the British Museum, which museum should I go next? @s00mb was absolutely right as he noticed

they seem to go off in different directions at this level

The only hint you gave us was

Preference towards things that may have physical applications.

I'm not sure whether mathematical textbooks are within this range. E.g. homological algebra could be a next step, but it is probably quite limited in physical applications. The more detailed the mathematics gets, the further away is it from applications - at least in general.

One application which came to mind was: read a book about general relativity for the physical part, and a book about de Sitter and AdS spaces for the mathematical part. A good starting point could be our insight articles on the key words "universe", "black hole(s)", and "(general) relativity".

 

[jedishrfu]

DDG has a lot of physical applications too.

 

[WWGD]

As so often this is a difficult to answer question. It's a bit like asking: I have visited all locations of the British Museum, which museum should I go next? @s00mb was absolutely right as he noticed

The only hint you gave us was

I'm not sure whether mathematical textbooks are within this range. E.g. homological algebra could be a next step, but it is probably quite limited in physical applications. The more detailed the mathematics gets, the further away is it from applications - at least in general.

One application which came to mind was: read a book about general relativity for the physical part, and a book about de Sitter and AdS spaces for the mathematical part. A good starting point could be our insight articles on the key words "universe", "black hole(s)", and "(general) relativity".

Homological algebrar can be used in/for Topological Data Analysis.

 

[fresh_42]

Homological algebrar can be used in/for Topological Data Analysis.

I'm still not convinced that topological data analysis makes any sense at all. To me this is a contradiction in itself and at best a complicated way to say pattern recognition. It sounds like a marketing strategy by McKinsey et al. to sell data mining tools and consultation power under a fancy name, or to generate some master thesis in statistics. In my opinion it is "Viel Rauch um nichts" (German title of "Up in Smoke (1978)) which literally means: a lot of smoke about nothing. The one who "invented" it could probably well have been in the cast of that movie.

 

[WWGD]

How is persistent homology self-contradictory? Features that persist at diffetent levels /gradations are believed to be signal and those that do not are considered noise. How is this self-contradictory?Edit: I have read papers by respected topologists like Robert Ghrist from Penn on it. Not likely someone with a track record would put out several papers on something unfounded.

 

[fresh_42]

It is an euphemism to call it topological. It's good old data mining in a new frame: marketing, not content! Is it something else as regressions for large data with some simplicial complexes instead of straights? And what does it have to do with physics? Or differential geometry?

 

[WWGD]

It is an euphemism to call it topological. It's good old data mining in a new frame: marketing, not content! Is it something else as regressions for large data with some simplicial complexes instead of straights? And what does it have to do with physics? Or differential geometry?

How is it a euphemism? You form an actual chain complex associated with the data and compute and interpret homology groups.Edit: True there are a lot of baseless claims in new areas but I don't see this as one of them. I am addressing your claim that homological algebra has no applications.

 

[WWGD]

But I don't see what you find self-contradictory about it.

 

[fresh_42]

But I don't see what you find self-contradictory about it.

It's off topic to discuss this here. And it is my personal opinion. My understanding on what deserves the predicate topological is likely narrower than commonly used. Some pyramids don't make a topology for me. It's far closer to algebra than it is to topology, but algebraic data analysis doesn't sound as sophisticated. And I know how the big consultancies work. It's McKinsey, not Urysohn.

 

[vanhees71]

Somehow reminds me of "quantum economics" ;-))).

 

[s00mb]

Wow thanks for all the replies, I left this thread for dead after a couple days didn't expect anyone else to notice it. I am looking up the de sitter spaces stuff. I've done some basic homological algebra before but I s'pose I can get back into it. I just like stuff about manifolds in general doesn't really matter what point of view it's from whether it's algebra topology or geometry. What got me interested was things like non euclidean geometry and finding out there were different geometries. Naturally that led to wanting to learn about metrics and riemann geometry which led to manifolds. I guess what I should have posted is "I am at the level of riemann geometry and kahler manifolds, if I want to know all about manifolds, does anyone have a good reading list?". But I didn't of course hindsight being 20/20. Thanks for the suggestions though, if anyone has anymore after this posting please let me know (Also, I did finsler geometry too). Thanks everyone.

 

[George Jones]

What about fibre bundles?

 

[s00mb]

What would be a good book on fibre bundles? I'd be interested in that but I usually find it hard to wrap my head around it.

 

[fresh_42]

What would be a good book on fibre bundles? I'd be interested in that but I usually find it hard to wrap my head around it.

If you like you can read
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/#topwhere I calculated some examples on a specific manifold, the 3-sphere. There are also references, but not all of them contain fiber bundles.

 

[s00mb]

I will start there (actually already started). Seems very well explained too thank you.