Math, Math exercises, Math questions

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.
- Map out the kinds of fixed point theorem.
- Understand the divisions of everything in terms of exact sequences.
- Relate Bott periodicity and the eight-cycle of divisions of everything.
- Understand the Snake lemma as the eightfold way.

- 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.
- Survey the branches of math and how they unfold.
- Investigation: Overview all of the concepts in math and how they arise
- Investigation: Organize the fundamental ideas in math in terms of how they describe an algebra of questions and answers
- Investigation: Organize the various interpretations of combinatorial objects
- Survey the theorems of math and how they unfold.
- Survey the problems in math and how they unfold.
- 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.
- 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.
- 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.

- 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
- 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?
- 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 symplicial 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.

- Investigation: Collect and develop ideas about physics
- Investigation: Collect examples and overview the ways of figuring things out in physics.
- Investigation: Relate the ways of figuring things out in physics and math.
- Investigation: Intuit the deBroglie wave equation in terms of a particle moving back and forth a fixed distance.

- Investigation: Explain why there are four classical Lie groups and algebras
- Draft of explanation
- Investigation: Intuit the four classical root systems
- Investigation: Study the possible constraints on the matrices for the elements of a Lie algebra
- 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
- 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).

- Investigation: Interpret the Cayley-Dickson construction

- Investigation: Understand the exceptional Lie groups as part of a sequence
- Investigation: Interpret the preservation of symplectic area
- 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.
- Investigation: Multiplying upper triangular matrices
- Exercise: Write out Serre's relations for the classical Lie Algebras
- Understand the underlying ideas in algebra.
- Consider the four contexts for equivalence relating countings.
- How do the special linear groups relate to their maximal compact subgroups?

- 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?

- What does projective geometry say about the existence of infinity?
- 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?

Analysis

- Survey the kinds of differential equations, especially in the sciences, and consider them in terms of their symmetry.

Linear algebra

- 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?
- 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 det (e^A) = e^(tr(A)).
- Think of a geometric interpretation of the trace.

- Investigation: Explore how limits describe external relations whereas colimits describe internal structure.
- Investigation: Organize the concepts in category theory to reveal underlying themes
- Investigation: Use category theory to model composition of perspectives
- Express visualizations in terms of category theory and/or set theory
- Challenge: Understand Yoneda's lemma and relate it to the four levels of knowledge: Whether, What, How, Why.
- 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?
- 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.
- 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?
- Express the divisions of everything in terms of exact sequences.

Logic

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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Puslapis paskutinį kartą pakeistas 2019 birželio 17 d., 19:42