# Book: MathNotebook

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

Current research program

Lie theory

• Solved: Defining geometry
• 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?

Computability theory

Parsiųstas iš http://www.ms.lt/sodas/Book/MathNotebook
Puslapis paskutinį kartą pakeistas 2019 balandžio 18 d., 10:27