General Notes
manifolds is the study of geometry of nonlinearity (exponents, etc).
differential == dynamical.

control theory + optimization.

perturbation theory is forced change (stochastic is more random change) used to find approximate solutions to complex problems by starting from the exact solution of a related, simpler problem. So we would use this for building a function and then having a test function that executes it. And if it breaks the entire system we would make small changes until it functions closer to the way we want it to.

differential geometry (is a toolbox)
 complex geometry
 algebraic geometry
 information geometry
 general relativity and guage theory (standard model) are differential geometric

game theory between correction + mutation

Information theory: understanding genetic information and the evolution of genomes

Dynamical Systems Theory: analyze the longterm behavior of evolutionary systems and population dynamics

Topology: studying the structure of fitness landscapes and evolutionary trajectories

differential equations

graph theory

Computational Algebra: for developing efficient algorithms to simulate and analyze mutationselection systems

Partial Stochastic Processes: While the application of mutations is directed, the outcomes may still have some stochastic elements
Self assembling AI
When reasoning about mutations you have this problem of needing a complete mutation of multiple components to even have a functioning, useful mutation.
Euclidean Geometry
Given a line and a point not on it (below or above), exactly one line parallel to the given line can be drawn through the point.
NonEuclidean Geometry
Think about light bending around the curved space of a black hole.
Light always travels in the shortest path in a straight line, it’s just that the space itself is curved.
So if you were looking at the “horizon” you would actually be looking spherically bc of that bend in spacetime. It wouldn’t even be called a horizon.
Unlike Euclidean geometry where if you move an object to the left all the particles move in uniform in the same direction, NonEuclidean particles move in different directions which is very similar to spaghettification around a black hold caused by it’s curved space.
If it was in hyperbolic space then movements could actually caused the object to rip apart.
Hyperbolic geometry
Given a line and a point not on it, infinitely many lines parallel to the given line can be drawn through the point.
The sum of angles are always > 180 degrees
Spherical Geomety
Given a line and a point not on it, no lines parallel to the given line can be drawn through the point.
The sum of angles are always < 180 degrees
Differential Geometry
If you want to do analysis on manifolds you’re looking for diff geometry. Topology is more about studying the object itself — what its properties are, if you can categorize them, finding invariants etc. Like how do you mathematically keep a ball and a torus apart, without parametrizing those spaces.
Diff geom is more about how do I parametrize my space so I can write down my function. A function usually lives in a space. Differential Geometry is the study of the types of differentiable functions that can live on various spaces. E.g. if your underlying space is different so are your functions usually. E.g. you can have differentiable functions without fixed points on a plane but not on a ball. Which is the reason for hurricanes and the like.
Resources
 Into the Wild: Machine Learning In NonEuclidean Spaces
 How One Line in the Oldest Math Text Hinted at Hidden Universes
 An Introduction to Manifolds
 Introduction to Smooth Manifolds (Graduate Texts in Mathematics)
 IEEE SIG PROC MAG 1 Geometric deep learning: going beyond Euclidean data
 What Is Math?
 Hyperbolic Embeddings with a Hopefully Right Amount of Hyperbole