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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 long-term 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 mutation-selection systems

  • Partial Stochastic Processes: While the application of mutations is directed, the outcomes may still have some stochastic elements

  • The Elements of Statistical Learning

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.

Non-Euclidean Geometry

Think about light bending around the curved space of a black hole.

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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 space-time. 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, Non-Euclidean particles move in different directions which is very similar to spaghettification around a black hold caused by it’s curved space.

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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

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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.