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
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 gconjecture about an hvectors of a simplicial polytope and the triangulation of a sphere.
 P vs. NPcomplete
 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, GaussBonet theorem, Euler's characteristic, Grassmanian
 Geometry: Geometries, Classical Lie groups, Symplectic geometry Lie theory, Tensor, Linear algebra, Clifford algebra, relation between discrete and continuous projective geometry
 Numbers: Reals, Complexes, Quaternions, Octonions, CayleyDickson construction
 Network theory, Set theory axioms.
 for modeling structures
 the eightfold way: homology and cohomology
 for modeling God going beyond himself
 for the big picture in mathematics
 for entropy
 for the properties of life
 for subconscious
 for understanding an area of math
Math Videos
General
Collections
Geometry
 Federico Ardila
 Amritanshu Prasad: Representation Theory: A Combinatorial Viewpoint
 Continuous groups in physics
 Fredric Schuller: Lectures on Geometrical Anatomy of Theoretical Physics
 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
 Nickolas Rollick: Algebraic Geometry
 ML Baker on Elliptic curves and modular forms
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

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