- 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.
Math Notebook: 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 (affine, projective, conformal, symplectic) and six transformations (reflection, shear, rotation, dilation, squeeze, translation).
- 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
- the relationship between the concious and the unconscious, duality, logic
- Langlands program, Poincare/Serre duality
- the association of (conscious) questions and (unconscious) answers
- theorems, and in particular, geometry theorems, Galois theory, the Fundamental Theorem of Algebra, its proofs and its centrality
- Understand the divisions of everything
- 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
- for the ways of figuring things out
- Binomial theorem: Polytopes, Coxeter groups, homology, Gauss-Bonet theorem, Euler's characteristic, Grassmanian
- Geometry: Geometries
- Lie theory, Classical Lie groups, Exceptional Lie groups, Triality
- Tensor, Triviality, Linear algebra
- relation between discrete and continuous projective geometry
- circle folding, origami and what their operations mean geometrically
- Clifford algebra, Geometric algebra, visual complex analysis
- Symplectic geometry, Lagrangian mechanics, Hamiltonian mechanics
- Algebraic geometry, sheaves, how Grothedieck's six operations relate to six representations
- Numbers: Reals, Complexes, Quaternions, Octonions, Cayley-Dickson construction, Associativity, duality-breaking
- Network theory, Set theory axioms.
- for modeling structures
- the eightfold way: homology and cohomology
- for the big picture in mathematics
- for entropy
- for the properties of life
- for subconscious
- for understanding an area of math
- Daniel Chan
- Federico Ardila
- Amritanshu Prasad: Representation Theory: A Combinatorial Viewpoint
- Continuous groups in physics
- DoctorPhys: Theoretical Physics
- Fredric Schuller: Lectures on Geometrical Anatomy of Theoretical Physics
- Curtis McMullen - The Geometry of 3 Manifolds
- Continuous Groups for Physicists
- Video named "17". End of general theory of Lie group and Lie algebra.
- "18" Classification of compact semisimple Lie algebras and Lie groups and their representation theory. Complexification. Direct sum of semisimple and solvable Lie algebras.
- "19" root systems
- "20" simple roots
- "22" orthogonal, symplectic Lie algebras, table of their properties
- "23" unitary Lie algebra An, exceptional algebras
- "24" Representation theory, starting spinors
- "25" Spinors
- Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
- Taylor Dupuy Model theory, mathematical logic, Grothendieck, algebraic geometry
- Daniel Murfet: Topos theory
- Nickolas Rollick: Algebraic Geometry
- ML Baker on Elliptic curves and modular forms