- 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.
Challenges in Math to take on
- make sense of the field with one element
- explanation of four classical Lie groups
- express and apply four geometries and six transformations
- express divisions of everything in terms of finite exact sequences
- relate level and metalevel in four ways to provide the basis for logic and geometry
- completely characterize an area of math such as plane geometry or chess
- understand entropy
- a theory of what "equivalence" variously means
- https://gilkalai.wordpress.com/2009/04/04/how-the-g-conjecture-came-about/
Other challenges to consider
- 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
- Duality: Langlands program, Poincare/Serre duality
- Logic
- Geometry: Classical Lie groups, Lie theory, Tensor, Symplectic geometry, Brownian motion?, Linear algebra, Six operations, Bott periodicity
- Finite fields: Field with one element
- Numbers: Reals, Complexes, Quaternions, Octonions, Cayley-Dickson construction
- Category theory: Voevodsky, type theory
- Entropy: Information geometry
- Clifford algebra
- Combinatorics: Symmetric functions
- Foundations: Logic, Set theory, Models, Proof theory, Automata theory
- Network theory
- Functional analysis? Spectral sequence
- Number theory? basics of the Riemann-Zeta function hypothesis.
- Qiaochu Yuan's reading list
Various math videos
Web courses
Areas of math that I want to learn more about.
Field with one element
I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling God's dance.
- Learn what is known about the field with one element. Learn the underlying algebraic geometry.
- Learn about finite fields, especially their combinatorics. Be able to contemplate F1n.
- Learn how the field with one element relates to the Riemann hypothesis.
- In general, learn about the many different kinds of dualities in math, and look for a common theme.
The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q->1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes.
- Learn about generalizations of Pascal's triangle and binomial expansion, such as the Grassmanian, Betti numbers, the Euler characteristic?, and applications such as the Gauss Bonet theorem?. Consider how Pascal's triangle relates to homology.
- Learn about Q analogue and Decategorification.
- Learn about Geometric representation theory, which seems to be related to Pascal's triangle.
- Learn about the Young tableaux (conceived as paths in Pascal's triangle) and Schur functions and their role in the representation theory of the general linear group.
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 conceputal 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 want to understand Network theory and relate my six visualizations.
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, Matrix
Catalan
Math questions I am focusing on
Math I am currently focusing on
- understanding how Pascal's triangle relates to homology and the Gauss-Bonnet theorem and Euler's characteristic
- understanding the different kinds of Pascal triangles and how they relate to Grassmannians
- understanding the Gaussian binomial coefficients in terms of Coxeter generators
- understanding the relationship between the discrete and continuous case of projective geometry