Math 数学


Andrius Kulikauskas

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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0



Math, Math exercises


I'm starting to write up the results that I'm getting as I investigate the root of mathematics.

Current research program

  • Consider whether and how {$F_1$} models four stages of God's going beyond himself.
  • Consider these four stages as four interpretations of the variable q in Gaussian binomial coefficients. Relate that to my theory of variables.
  • Relate the foursome and the Yoneda lemma.
  • Understand Yates index theorem as a statement about the foursome, and relate it to the Yoneda lemma.
  • Understand the role of the four levels (metalogics, geometries) in the ways of figuring things out.
  • Understand the four classical Lie structures (the choice frameworks, root systems, geometries) in terms of the foursome.
  • Understand the roles of the four classical Lie structures in Bott periodicity.
  • Understand the divisions of everything in terms of exact sequences.
  • Understand the Snake lemma as the eightfold way.

Math Big Picture

Math foundations?

Binomial theorem


Lie theory


  • Solved: Defining geometry
    • What questions does geometry ask about choice?
    • How is one dimension embedded in other dimensions?
    • What is a line segment? What makes it "straight"?
    • What is a circle?
    • What does it mean for figures to intersect?
    • Can a line intersect with itself?
  • Investigation: Map out the main ideas of geometry.
    • Challenge: Relate the definition of geometry as "the regularity of choice" with Grothendieck's machinery.
  • Investigation: What are the four geometries?
  • Study: What is projective geometry all about?
    • What does projective geometry say about the existence of infinity?
      • Do all lines (through plane) meet at infinity at a common point? Or at a circle? Or do the ends of the lines not meet? Or do they go to a circle of infinite length?
  • Study: Understand the basics of symplectic geometry.
  • Compare the four geometries.
    • Express the four geometries in terms of symmetric functions.
    • Consider how infinity, zero and one are defined in the various geometries. How do these concepts fit together?
      • How do they involve the viewer and their perspective? How might that relate to Christopher Alexander's principles of life and the plane of the viewer?
      • How are they related to the twelve topologies?
  • Investigate orientation. What is the notion of orientation for points, lines, etc? grounding different geometries? considering building spaces bottom up (adding lines) and removing spaces top down (removing hyperplanes)?
  • Compare building spaces bottom up or by deletion top down with choices left or right, etc.
  • Study Norman Wildberger's book and videos as an expression of geometry and try to express it all systematically, for example, using symmetric functions.
    • List out the results of universal hyperbolic geometry and state them in terms of symmetric functions.
  • Understand algebraic geometry (sheaves, etc.) by analyzing its theorems.
  • Understand Bott periodicity.
    • Relate Bott periodicity to the eight divisions of everything and to the three operations.

Complex analysis

  • Using complex numbers, interpret {$d/dz \: e^z$}
  • Study the Catalan numbers and the Mandelbrot set
    • Check what happens if I plug in different values into the Catalan power series.
    • What is a combinatorial interpretation of P - P(n), the generators of the Mandelbrot set, in terms of the Catalan numbers? Get help to generate the difference. What is the best software for that?

Linear algebra

  • How to multiply polar decomposed matrices?
  • Understand Eigenvector decomposition.
    • What matrices have a full set of eigenvectors?
    • What kinds of eigenvectors and eigenvalues do rotation matrices have?
    • How to understand the coordinate system for an eigenvector? Does each nondegenerate matrix have a natural coordinate system?
    • How to understand a matrix as a system of equations?
    • What is the role of matrices in setting up Galois theory?

Category theory

Computability theory


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Puslapis paskutinį kartą pakeistas 2019 vasario 14 d., 09:12