Notes
Book
Math 数学
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.
学习数学
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 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
 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, CayleyDickson construction, Associativity, dualitybreaking
 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
Math Videos
General
Collections
Geometry
 Daniel Chan
 Federico Ardila
 Amritanshu Prasad: Representation Theory: A Combinatorial Viewpoint
 Continuous groups in 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
Foundations
Web courses

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