Notes

Book

Math

Discovery

Andrius Kulikauskas

  • ms@ms.lt
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

Lietuvių kalba

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

  • Make a diagram of concrete mathematical structures that I want to learn about, and related branches of mathematics.
  • Survey combinatorics, especially Stanley's book and Wikipedia, for the various kinds of combinatorial constructions.

Investigations

  • Map of math How do math's branches, concepts and results unfold?
  • Binomial theorem The heart of mathematics. In particular, the reason why there are 4 infinite families of classical Lie groups/algebras.

Challenges in Math to take on

  • Understand the ways of figuring things out
    • understand the binomial theorem
    • explain the four classical Lie groups
    • express and apply four geometries and six transformations
    • relate level and metalevel in four ways to provide the basis for logic and geometry
    • understand the use of variables
  • Understand how math results unfold
  • Understand the divisions of everything
    • express divisions of everything in terms of finite exact sequences
    • understand and interpret Bott periodicity
  • Model how God goes beyond himself
  • Understand entropy as the basis for prayer
  • Completely characterize an area of math such as plane geometry or chess

Open math challenges to consider

  • A theory of what "equivalence" variously means
  • The g-conjecture about an h-vectors of a simplicial polytope and the triangulation of a sphere.
  • P vs. NP-complete
  • a unifying perspective on cohomology (or has Lurie already achieved this?)
  • higher order homotopy groups for sphere

Overviews of Math and Its History

Math to Learn

Math Videos

General

Collections

Geometry

Foundations

Web courses


Areas of math that I want to learn more about.

Automata theory

I want to understand how the characteristic q can be identified with infinity. This may relate to automata theory.

Geometry

I am trying to describe the cognitive significance of 4 geometries (affine, projective, conformal, symplectic) and 6 transformations between them (reflection, shear, rotation, dilation, squeeze, translation).

  • Study these different kinds of geometry. Collect the main theorems of geometry and organize them into these different kinds to illustrate them.
  • I want to understand the basics of Geometric algebra, Clifford algebra and visual complex analysis. I think they are related to symplectic geometry.
  • I want to understand Bott periodicity, which can be understood in terms of Clifford algebra and its clock shifts. I want to be able to relate Bott periodicity to the eight divisions of everything, if possible.
  • I want to understand the theorem distinguishing the reals, complexes, quaternions, octonions and why there is nothing higher. I want to understand the basics of working with (noncommutative) quaternions and (nonassociative) octonions.
  • I want to have a basic understanding of circle folding and origami and what their operations mean geometrically.
  • I want to understand Grothedieck's six operations and how they might relate or not to my six transformations. I want to understand the related algebraic geometry (such as sheaves) and category theory.

Foundations of Mathematics

I want to understand some of the conceptual Foundations of Mathematics.

  • I want to understand the basics of Homotopy type theory and Category theory. I want to understand how equivalences are considered in math. I want to relate that to my theory of variables.
  • I want to show how the possible relationship between logic and metalogic is given by the four geometries

I am trying to make a map of how all of mathematics unfolds. I want to understand the basics of Lie groups and Lie algebras because I think they play a central role in linking analysis and geometry.

  • In particular, I want to understand intuitively the 4 classical groups.
  • I want to understand tensors and be able to calculate them. I think they define triviality.
  • I want to understand how to prove the Fundamental Theorem of Algebra in various ways and understand its importance in organizing mathematics.
  • I want to review Galois theory.

Other assorted concepts: Tetrahedron, Triality, Associativity, Unit

StudyMath


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Puslapis paskutinį kartą pakeistas 2018 rugsėjo 13 d., 17:15
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