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Math, Math research, 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 priorities:

• Write out my talk A Structural Semiotic Study of How We Use Variables.
• Rewrite my talk on math discovery as a paper Mathematics Expresses Discovery.
• Develop and organize my thoughts for a second paper Mathematics Unfolds Through Ambiguity.

Current research program

• Relate the four classical Lie families to the four levels (metalogics, geometries) in the ways of figuring things out.
  • Understand the four classical Lie families (the choice frameworks, root systems, geometries).
    • Understand the choice frameworks.
    • Understand the most fundamental Lie groups {$A_2$}, then {$A_n$}, and then the other classical families.
      • Understand complexification of compact real forms.
        • Understand real forms.
          • Understand Hopf algebras.
            • Understand tensors and tensor products.
              • Understand tensors in terms of triviality.
• Understand the four levels (metalogics, geometries) in the ways of figuring things out.
  • Understand the ways of figuring things out in physics.
    • Understand the role of measurement in physics, both relative and absolute.
      • Understand tensors and tensor products.
• Understand the significance of the foursome in mathematics and how it relates to the four levels.
  • Challenge: Understand Yoneda's lemma and relate it to the four levels of knowledge: Whether, What, How, Why.
    • Understand Yates index set theorem as a statement about the foursome, and relate it to the Yoneda lemma.
    • Understand the Univalence Axiom in terms of the Yoneda Lemma.
  • Understand how the arithmetical hierarchy relates to the foursome and sevensome.
• Consider whether and how {$F_1$} models four stages of God's going beyond himself.
  • Understand groups as analogues of Hopf algebras.
  • Understand finite fields and what {$F_{1^n}$} could mean.
  • Consider these four stages as four interpretations of the variable q in Gaussian binomial coefficients. Relate that to my theory of variables.
• Relate Bott periodicity and the eight-cycle of divisions of everything.
  • Systematize the concepts related to Bott periodicity.
  • Understand Bott periodicity by understanding the Tenfold way
  • Understand the divisions of everything in terms of exact sequences.
  • Understand the roles of the four classical Lie structures in Bott periodicity.
• Understand the Snake lemma as the eightfold way.

Math Big Picture

• Study and systematize Terrence Tao's notions of What is good mathematics?
• Investigation: Understand mathematics by describing it as an activity.
• Understand what makes mathematics distinct as a branch of knowledge.
  • Understand the nature of mathematical beauty.?
• Express my philosophy's concepts in terms of mathematics.
• Understand how all of mathematics unfolds.
  • Survey the branches of math and how they unfold.
  • Investigation: Overview all of the concepts in math and how they arise
    • Investigation: Organize the concepts in the Math Companion
    • Investigation: Analyze the 100 concepts in the Math Companion and figure out their building blocks
    • Investigation: Classify applications of mathematical arguments
    • Investigation: Classify the most important mathematical constants
    • Investigation: Classify the kinds of mathematical objects, for example, those that arise in combinatorics.
    • Investigation: Classify the most important mathematical functions and mappings
    • Make a glossary of mathematical concepts and see which are premathematical.
    • infinity, symmetry
  • Formulate and appreciate the most basic mathematical principles.
  • Investigation: Organize the various interpretations of combinatorial objects
  • Understand the purpose of abstraction and how mathematical concepts grow in abstraction.
  • Survey the theorems of math and how they unfold.
  • Survey the problems in math and how they unfold.
    • How do math puzzles express math ideas?
  • Investigation: Organize and take up various challenges in math
  • Understand how the ways of figuring things out are involved in the unfolding of mathematics, and what else is involved.
• 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.
  • Investigation: Explore how duality is broken by comparing a category and its opposite category
• 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 three-cycle 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.
  • Investigation: Express the six transformations in terms of the four geometries.
    • How do Möbius transformations express six specifications of four geometries?
      • How can the Möbius transformations be understood in terms of the actions of 2x2 matrices?
  • Investigation: Explain the ways of interpreting multiplication.
  • 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.

Foundations of mathematics

• Do the Peano axioms presume a notion of "minimality" in defining the natural numbers?
• Investigation: Relate my concepts with persistent homology, topological data analysis and homotopy type theory
• Understand type theory in terms of amounts and units

Geometry

• Solved: Defining geometry
  • Investigation: What are the ways that regularity of choice is expressed?
    • Understand how mathematics defines a perspective.
    • Relate perspective with regularity of choice.
    • Come up with a meaningful definition of symmetry.
      • Gain intuition from gauge theory about Lie theory, symmetry, physics.
      • Understand Bott periodicity.
        • Relate Bott periodicity to the eight divisions of everything.
        • Systematize the concepts related to Bott periodicity.
        • Understand Bott periodicity by understanding the Tenfold way.
        • Understand Bott periodicity by understanding the Periodic table of topological insulators and semiconductors.
        • Relate the clock shifts of Clifford algebras to the three operations +1, +2, +3.
        • Understand geometric algebra and its relationship to Clifford algebra.
        • Understand what string theory has to say about perspectives and concepts.
    • Understand algebraic geometry (sheaves, etc.) by analyzing its theorems.
    • Understand topos theory in terms of regularity of choice.
    • Investigation: What distinguishes geometries? In what sense are there four geometries?
      • Express the four geometries in terms of symmetric functions.
    • Compare building spaces bottom up or by deletion top down with choices left or right, etc.
  • Investigation: What questions does geometry ask about choice??
    • Investigation: Map out the main ideas of geometry.
    • Understand geometry in terms of its theorems.
    • Formulate a complete exposition of triangle geometry as an example of a complete mathematical theory.
    • Identify essential illustrations in geometry.
    • What does it mean for figures to intersect?
    • Can a line intersect with itself?
  • Investigation: What is the role of SU(2) in organizing geometry?
    • 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?
    • How is one dimension embedded in other dimensions?
    • What is a line segment? What makes it "straight"?
    • What is a circle?
• Study: What is projective geometry all about?
  • Investigation: Interpret and systematize the key formulas of universal hyperbolic geometry
  • 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.
  • 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)?

Binomial theorem

• Solved: Interpreting the binomial theorem in terms of Young diagrams
• Solved: Identifying Young diagrams with paths in Pascal's triangle
• Challenge: Describe "the fundamental unit of information" in terms of paths in Pascal's triangle
• Solved: Calculate the Weyl groups of the classical root systems
  • Understand Coxeter groups and the conditions by which they become Weyl groups.
• Challenge: Calculate the symmetry groups of the choice templates
• Challenge: Interpret classical Lie algebras in terms of choice templates
• Challenge: Interpret the simplex category
• Challenge: Find and compare q-analogues of the binomial theorem
• Representation theory
  • Challenge: Calculating and interpreting the irreducible representations of the symmetric groups
  • Challenge: Calculating and interpreting the irreducible representations of the general linear groups
    • Question: How do the irreducible representations of a subgroup (of the general linear group or the symmetric group) relate to those of the group?
    • Understand: Why is the set of character functions of the irreducible representations of G orthonormal with respect to the relevant inner product?
  • In what sense are there six natural bases for the symmetric functions?
  • Challenge: Interpret the polynomials {$\binom{X}{n}$} as the diagonals of Pascal's triangle.
  • Challenge: Interpret the sign of the columns of partitions in terms of the paths in Pascal's triangle.
  • Question: What information does the determinant provide about a representation?
  • Question: What is the relationship between the characteristic functions on conjugacy classes and the characters? How are these orthnormal bases related? By what matrix, combinatorially?
  • Investigate: How do the symmetric functions of the eigenvalues of a matrix expand and complete the information provide by the characters (the trace) and the determinant? Is this knowledge sufficient to completely determine a representation?
  • Challenge: Give a combinatorial interpretation of the Pfaffian.
  • Challenge: Relate the standard tableaux numbering (from the corner) to the two natural numberings of a partition (based on the paths of Pascal's triangle). Consider it as a shift in perspective (from unconditional to conditional).
  • Question: What is the Vandermonde determinant for the eigenvalues of a matrix?
• Challenge: Develop a theory of solution of differential equations by modeling substitutional symmetries (for example, consider trigonometric solution as a finite closed system based on the fact that the fourth derivate of a trigonometric function yields itself.
• Investigation: Understand why there are three infinite families of polytopes?
  • Understand the meaning of the center and the totality of a regularity of choice.
  • What does it mean that a vertex is the marked opposite for the empty set?
  • Understand simplicial homology
  • What is the role of infinity in the regularities of choice?
• Investigation: Classify the equivalences of infinite paths
• Learn about the field with one element, {$F_{1}$}.
  • Investigation: Write an elegant combinatorics of the finite field and interpret what is {$F_{1^n}$}.
  • Challenge: Given a finite field, calculate the probability that a matrix in {$GL(F_q)$} is noninvertible (det=0). What happens in the limit that q->1 ? and q->infinity?
  • Understand how finite field and the field with one element represent infinity.
  • Express God's dance in terms of zero, infinity and one.
• How does Euler characteristic relate to homology, structures with holes?
• What is the relationship between Pascal's triangle and the Grassmannian?
• Relate Pascal's triangle and the arithmetical hierarchy.

Physics

• Investigation: Relate the ways of figuring things out in physics and math.
  • Investigation: Collect examples and overview the ways of figuring things out in physics.
• Investigation: Collect and develop ideas about physics
  • Understand Schroedinger's equation.
  • Investigation: Intuit the deBroglie wave equation in terms of a particle moving back and forth a fixed distance.
  • Calculate whether Alpha Centauri could be made of antimatter.
  • Understand the importance of spinors.

Lie theory

• Investigation: Explain why there are four classical Lie groups and algebras
  • Draft of explanation
  • Understand the classical Lie groups and algebras.
• Understand Lie groups
  • Investigation: Understand the progression from forms to geometries
  • Investigation: Understand the progression from forms to Lie algebras
  • Investigation: Meaningfully define the four classical Lie groups
  • Investigation: Understand Lie groups in terms of the inverses of elements
  • Investigation: Understand Lie groups in terms of rotations
  • Understand complexification?
• Understand what the Dynkin diagram's chain says about Lie groups
  • Investigation: Understand the progression from a Lie algebra to a Lie group
  • Understand trivial Lie groups
  • Understand {$A_1$}, a node in a Dynkin diagram.
  • Understand {$A_2$}, a link in a Dynkin diagram.
  • Understand {$A_n$}, a chain in a Dynkin diagram.
  • Understand the relationship between {$A_{n}$}, {$SL(n+1)$} and {$SU(n+1)$}.
  • Explore what and what not makes for a (simple) three-dimensional root system.
• Understand the consequence of the end of the Dynkin diagram's chain
  • Understand the Dynkin diagrams of the odd and even orthogonal Lie algebras
  • What distinguishes odd and even dimensional orthogonal Lie groups?
• Understand the exceptional Lie groups in terms of Dynkin diagrams
  • Investigation: Understand the exceptional Lie groups as part of a sequence
  • Understand the difference between triality and duality and what that means for Dynkin diagrams.
• Understand the constraints on root systems
  • Interpret Geometrically and Combinatorially the Relationship Between a Root System and Its Cartan Matrix
• Understand root systems
  • Investigation: Intuit the four classical root systems
  • Attempt at intuiting the four classical root systems
  • Consider the four contexts for equivalence relating countings.
  • Investigation: Intuit the five exceptional root systems
  • Why does a root system having roots {$e_i$} also have to contain the roots {$e_i - e_j$} and {$e_i + e_j$} ? Consider the product {$[\_,\_]$}. More generally, study what the product {$[\_,\_]$} yields.
  • What does it mean that root systems are positive definite?
• Understand Lie algebras
  • Investigation: Study the possible constraints on the matrices for the elements of a Lie algebra
  • Exercise: Write out Serre's relations for the classical Lie Algebras
• Understand the Lie correspondence
  • Investigation: To what extent can the Lie correspondence be based not on infinitesimals but on the combinatorics of matrix exponentials??
• Understand numbers: real, complex, quaternion, octonion
  • Investigation: Interpret the Cayley-Dickson construction
  • Understand the significance of the octonions.
• Understand geometry
  • Investigation: Interpret the preservation of symplectic area

Linear algebra

• Understand matrices as the expression of external relationships.
• Investigation: Understand tensor combinatorics as a generalization of matrix combinatorics
• 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?
  • Investigation: Define the determinant of an n x n matrix in terms of an operation on the area defined by two n-dimensional vectors (rows).
  • How to understand a matrix as a system of equations?
  • What is the role of matrices in setting up Galois theory?
• What are the symmetric functions of the eigenvalues of various specializations of matrices?
• Give a combinatorial interpretation for calculating the inverse of a matrix.
• Give a combinatorial interpretation of {$\mathrm{det} (e^A) = e^{\mathrm{tr}(A)}$}
• Think of a geometric interpretation of the trace.
• Investigation: Multiplying upper triangular matrices
• Understand the underlying ideas in algebra.
  • Challenge: Identify a matrix with a geometric collection of eigenvalues and eigenvectors

Complex analysis

• How does the imaginary number determine the nature of complex analysis and related geometry?
• Using complex numbers, interpret {$d/dz \: e^z$}
• Interpret the Mandelbrot set.
  • Give 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?
  • Check what happens if I plug in different values into the Catalan power series.
• Develop a notion of analytic symmetry related to solving the equations {$f^{(n)}=f$}.
• What is the mathematical and philosophical significance of circle folding?

Analysis?

• Map out the kinds of fixed point theorem to understand perspectives.
• Survey the kinds of differential equations, especially in the sciences, and consider them in terms of their symmetry.

Statistics

• Understand what statistics says about all aspects of knowledge, including ignorance, information, randomness, choice, entropy.

Category theory

• Understand category theory in terms of the obstacles to learning it.
• Understand category theory concepts in terms of an algebra of requirements.
• Investigation: Explore how limits describe external relations whereas colimits describe internal structure.
  • Understand the relationship between left adjoints and right adjoints, colimits and limits, and internal structure and external relationships.
    • Understand why there are three definitions of adjunction.
• Investigation: Organize the concepts in category theory to reveal underlying themes
  • Investigation: Specify the category theory that grounds Grothendieck's six operations
  • How do derived functors express the threesome?
• Classify the kinds of sameness in mathematics and elsewhere.
  • Classify examples of equivalence and relate them to adjunctions.
  • Classify the kinds of adjoint strings.
  • What is the significance of associativity and nonassociativity?
  • Investigation: What is the relation between external equality (refered to in associativity and identity) and internal equality (between objects)? Note the ambiguity (between the algebra of actions and the algebra of consequences) inherent in the internal equality.
  • Understand the Univalence Axiom and Voevodsky's homotopy type theory in terms of my philosophical concepts.
  • Challenge: Compare a free group on one generator and a cyclic group (defined by one relation). How does category theory distinguish them? How does that distinction relate to the equivalence imposed by the relation?
• Use combinatorics to understand finite categories.
• Investigation: Use category theory to model composition of perspectives.
• Express visualizations in terms of category theory and/or set theory.
• Understand the relationship between a product and an exponential object
• Understand universal and existential quantifiers in terms of presheaves
• Challenge: What do elementary symmetric functions generate in category theory?
• Challenge: What do the natural bases of symmetric functions generate in category theory?
• Challenge: What does it mean in category theory to have symmetric functions of eigenvalues?
• Express the divisions of everything in terms of exact sequences.

Logic

• Investigation: Understand and interpret all logical functions in terms of the self relations of a complete function.
• Investigation: Think through the conceptual significance of the Curry-Howard-Lambek correspondence and incorporate statistics

Biology

• Understand entropy, probability, statistics, nondeterminism, the house of knowledge, biology and other concepts in terms of information theory.

Computability theory?

• Understand knowledge and ignorance in terms of the concept of information as relevant in mathematics, computer science and physics.
• Interpret the automata hierarchy.
  • How can the main kinds of automata in the Chomsky hierarchy be expressed in lambda calculus?
• Investigation: Describe computer automata, Turing machines, computability theory and the arithmetical hierarchy in terms of category theory.
  • Challenge: Understand Yates' Index Theorem and relate it to the four levels of knowledge: Whether, What, How, Why.
• Relate the arithmetical hierarchy with the chain of perspectives.
• What is the significance for me of the P vs. NP problem?
• Understand the Riemann hypothesis as a statement about the lack of superfluous patterns.

Quantum physics

• What is the physical significance of the combinatorial interpretations of the orthogonal polynomials?
  • Investigation: Give a physical interpretation of the role that the Hermite polynomials play in the quantum oscillator

Discovery

• Map the ways of figuring things out in chess.

Additional

• Math Research Kaip iš matematikos žinojimo rūmų išsivysto matematikos šakos, sąvokos, klausimai, įrodymai ir tvirtinimai?
  • Map of Math Kaip išsiplėtoja matematikos šakos?
• Patikslinti matematikos žinojimo rūmus.
  • Nurašyti savo pranešimą apie kintamųjų semiotiką.
  • Išsiaiškinti kodėl yra keturios klasikinės Lie šeimos.
    • Suprasti sąryšį tarp kompleksinės Lie grupės ir jos realiųjų pavidalų.
    • Suprasti sąsajas tarp šaknų sistemos {$A_1$}, Lie algebrų {$\mathfrak{su}(2)$}, {$\mathfrak{sl}(2,\mathbb{C})$}, ir Lie grupių {$SU(2)$} bei {$SL(2,\mathbb{C})$}.
      • Susieti su Mobijaus transformacijomis ir su nuotaikų permainomis.
    • Suprasti sąsajas tarp šaknų sistemos {$A_2$}, Lie algebrų {$\mathfrak{su}(3)$}, {$\mathfrak{sl}(3,\mathbb{C})$}, ir Lie grupių {$SU(3)$} bei {$SL(3,\mathbb{C})$}.
    • Suprasti sąsajas tarp šaknų sistemos {$A_{n-1}$}, Lie algebrų {$\mathfrak{su}(n)$}, {$\mathfrak{sl}(n,\mathbb{C})$}, ir Lie grupių {$SU(n)$} bei {$SL(n,\mathbb{C})$}.
    • Suprasti sąsajas tarp šaknų sistemų {$A_{n}$}, {$B_{n}$}, {$C_{n}$} bei {$D_{n}$} ir atitinkamų Lie algebrų bei Lie grupių.
      • Pasimokyti Hopf algebrų.
    • Suprasti kaip būtent kompaktinės Lie grupės grindžia geometrijas.
      • Kas yra geometrija? Ir kaip ją išreiškia Mobijaus transformacijos?
    • Savo pranešimą apie klasikinių Lie šeimų kombinatoriką išversti į anglų kalbą, vis papildyti naujais atradimais, ir jo pagrindu parengti straipsnį.
  • Išmąstyti, kaip tirti vienanarį lauką {$F_1$} ir susieti su Dievo šokiu.
  • Suvokti ketverybės svarbą statistikoje.
  • Sustatyti statistikos žinojimo rūmus. Remtis Norman Anderson vadovėliu.