Notes
Book
Math 数学
Discovery
Andrius Kulikauskas
 ms@ms.lt
 +370 607 27 665
 My work is in the Public Domain for all to share freely.
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Math, Math exercises
数学笔记本
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
 Investigation: Understand mathematics by describing it as an activity.
 Understand what makes mathematics distinct as a branch of knowledge.
 Express my philosophy's concepts in terms of mathematics.
 Understand how all of mathematics unfolds.
 Investigation: Identify and study the thinking of mathematicians who pursue a broader view of mathematics.
 Investigation: Understand mathematics as the discrimination of a variety of dualities.
 Understand how zero and infinity become distinct, how their equivalence is violated.
 How is love (and life) related to duality, reflections, transformations and other math concepts?
 How does 1 mediate the duality of 0 and infinity? And how is that duality variously broken?
 Investigation: Collect and organize examples of figuring things out in mathematics
 Investigation: Among the ways of figuring things out in mathematics, how does the threecycle extend mathematical structure?
 Relate the four regularities of choice with four geometries, metalogics, foursome, qualities of signs, positive commands.
 Investigation: Understand how variables are variously used in mathematics.
 Express the six transformations in terms of the four geometries.
 Relate pairs of the regularities of choice with the six interpetations of multiplication, the visualizations, the transformations, the negative commands.
 Relate the six transformations to ways of figuring things out.
 Relate the six transformations with the types of variables.
 Relate the six transformations with the six visualizations.
 How do the six transformations relate to symmetry?
 Study how to apply the ways of figuring things out in mathematics.
 What is the relationship between the surface math problem and the deep way of figuring things out?
 How do we discover the right way to figure out a math problem?
 How do we combine several distinct ways of figuring things out?
 How can I apply my results to figure things out in math, the biggest problems?
 Understand math as an activity involving the three languages: argumentation, verbalization and narration.
 Investigation: What is math intution? How does it develop?
 Investigation: Discover patterns in how a mathematical theorem holds mathematical knowledge
 Investigation: Make sense, if possible, of the six methods of mathematical proof.
Math foundations?
Binomial theorem
Physics
Lie theory
Geometry
 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
 Investigation: Describe computer automata, Turing machines, computability theory and the arithmetical hierarchy in terms of category theory.
 Relate the arithmetical hierarchy with the chain of perspectives.

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