I was literally toying around with some of these ideas today. I had the thought that you could take some time series data, compute the velocities and positions over time, and use that to try and learn the metric tensors of it, assuming the time series data is following a geodesic on some manifold for the most part. And from there you could compute the symmetries and curvature of the system. I thought you could use that to try and define an energy for the system or some other conserved quantities, and then use that as an anomaly detector (change in the energy of the system means there was some outside interference).

I don't have a clue about geometry and maths. BUT you've made it clearer than my entire educational system. thank you.

Btw, It's fascinating how simmetry can be applied in such a practical way.

Thank you for contributing this course.

That said, its legibility would be helped by some proofreading. Please accept these suggestions for improvements:

Don't randomly format words in bold, such as the word "vaguely".

"greek" -> "Greek"

"russian" -> "Russian"

"an hungarian" -> "a Hungarian"

reference: https://www.berlitz.com/es-mx/blog/gentilicios-en-ingles

Same rule for toponyms:

"greece" -> "Greece"

Greek proper names have different forms in English than in Spanish:

"Euclide" -> "Euclid"

reference: https://en.wikipedia.org/wiki/Euclid

Join sentence fragments:

"packing of particles. It was" -> "packing of particles, it was"

Capitalization, inconsistent spelling:

"euclidean", "Euclidian" -> "Euclidean"

"non-euclidian" -> non-Euclidean

Incorrect word choice:

replace: "This fifth postulate posteriorly defined any attempt to try to prove it from other postulates of geometry."

with: "This fifth postulate subsequently defied any attempt to try to derive it from the other postulates of geometry."

"19 th" -> "19th"

Unnecessary capitalization:

"Alternative proposed Postulate" - > "Alternative proposed postulate"

Incorrect proposition:

"Academia rejection to the proposal" -> "Academic rejection of the proposal:"

Typographical error:

"universitary" -> "university"

"Different persons came to the same idea from differents backgrounds" -> "Different people arrived at the same idea from different backgrounds."

"Standar" -> "Standard"

Contrast with the Spanish word "estandar".

"erlangen programme" -> "Erlangen program"

"EXISTENTS" -> "existence"

"lipschitz continuous" -> "Lipschitz continuous"

"elemen" -> "element"