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Introduction E9F5FC Understandable FFFFFF Questions FFFFC0 Notes EEEEEE Software 
See: Logic 数学笔记 Complex analysis
Analytic continuation
Choice frameworks
An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But:
Composition algebra.
Counting
Finite fields
Lie theory
Linear algebra
Linear functionals
Polytopes
Symmetric functions
Bott periodicity
Duality examples (conjugates)
Rotation
Physics
Octonions
Euler's manipulations of infinite series (adding up to 1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that. John Baez: 24 = 6 x 4 = An x Bn Dedekind eta function is based on 24. Discriminant of elliptic curve. Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures. Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were. G2 requires three lines to get between any two points (?) Relate this to the threecycle. Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down? Duality
Geometry
Projective geometry
Variables
John Baez on duality in logic and physics Four geometries
Affine geometry
Duality
Each physical force is related to a duality:
So the types of duality should give the types of forces. CayleyDickson construction
Projective geometry
Conformal geometry
The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and 1 and the other axis is the opposites i and j. And they become related 1 to i to 1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to 1. Symplectic geometry
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position. Walks on trees
Symplectic  basis for coupling  coupling of electric and magnetic fields  is what is responsible for the periodic nature of waves  the higher the frequency, the higher the energy, the tighter the coupling  the coupling is across the entire universe. The coupling models looseness  slack. This brings to mind the vacillation between knowing and not knowing. How do symmetries of paths relate to symmetries of young diagrams Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? Note that 2dimensional phase space (as with a spring) is the simplest as there is no 1dimensional phase space and there can't be (we need both position and momentum). What is the connection between symplectic geometry and homology? See Morse theory. See Floer theory. Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"? Benet linkage  keturgrandinis  lygiagretainis, antilygiagretainis
Yoneda lemma  relates to exponentiation and logarithm Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain. Whether (objects), what (morphisms), how (functors), why (natural transformations). Important for defining the same thing, equivalence. If they satisfy the same reason why, then they are the same. Representable functors  based on arrows from the same object. Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic  a phenomenon in symplectic geometry. Six sextactic points. John Baez about observables (see Nr.15) and the paper A topos for algebraic quantum theory about C* algebras within a topos and outside of it. In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one. In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer? Force (and acceleration) is a second derivative  this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors? Study how Set breaks duality (the significance of initial and terminal objects). Show why there is no ncategory theory because it folds up into the foursome. Understand the Yoneda lemma. Relate it to the four ways of looking at a triangle. The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements. Category theory for me: distinguishing what observations are nontrivial  intrinsic to a subject  and what are observations are contentwise trivial or universal  not related to the subject, but simply an aspect of abstraction. "For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory. In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal. In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel. Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries? (Conscious) Learning from (unconscious) machine learning. Topology  getting global invariants (which can be calculated) from local information. Simple examples that illustrate theory. monad = black box? Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups Primena trejybę. Wikipedia: Homotopy groups Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups: {$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $} Vandermonde determinant shows invertible  basis for finite Fourier transform Euclidean space  (algebraic) coordinate systems  define left, right, front, backwards  and this often makes sense locally  but this does not make sense globally on a sphere, for example Vector bundles
Allen Hatcher. Algebraic topology. Explanation Homotopy, homology, cohomology. We will show in Theorem 3.21 that a finitedimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. Example 4.55: Bott Periodicity. Allen Hatcher. Vector Bundles and KTheory. (Halfwritten). Explanation Table of Contents Bottperiodicity. Algebra  geometry duality. (Pullback). Morphism <> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <> Affine Falgebras [Flinear ring homomorphisms F[Y] to F[x]] Logic
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete. Turing machines  inner states are "states of mind" according to Turing. How do they relate to divisions of everything? If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution? Study homology, cohomology and the Snake lemma to explain how to express a gap. Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. Study the Wolfram Axiom and Nand. Mathematical induction  is infinitely many statements that are true  relate to natural transformation, which also relates possibly infinitely many statements. Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign. Are Nand gates (Nor gates) related to perspectives? Study how all logical relations derive from composition of Nand gates. How is a Nor gate made from Nand gates? (And vice versa.) Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly? Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc Scaling is positive flips over to negative this is discrete rotation is reflection Equations are questions Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism Develop looseness  slack  freedom  ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. In {$D_n$}, think of {$x_ix_j$} and {$x_i+x_j$} as complex conjugates. In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable. Choices  polytopes, reflections  root systems. How are the Weyl groups related? Affine and projective geometries. Adding or subtracting a perspective. Such as adding or deleting a node to a Dynkin diagram. (The chain of perspectives.) Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world. Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships. Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved. Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C? Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension. Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas) Differentiating {$AA^{1}=I$} at {$A(0)=I$} we get {$A(A^{1})'+A'A^{1}=0$} and so {$A'=(A^{1})'$} for any element {$A'$} of a Lie algebra. For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}. Lietuvių kalba:
How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly? What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues. What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets? the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. https://en.wikipedia.org/wiki/Andrei_Okounkov Symmetry of axes  Bn, Cn  leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the hyperoctahedral group. Understand the classification of Coxeter groups. Organize for myself the Coxeter groups based on how they are built from reflections. Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense? SU(3)xSU(2)xSU(1)xSU(0) is reminiscent of the omniscope. The conjugate i is evidently the part that adds a perspective. Then R is no perspective. In what sense is SU(3) related to a rotation in octonion space? If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0). {$SL(n)$} is not compact, which means that it goes off to infinity. It is like the totality. We have to restrict it, which yields {$A_n$}. Whereas the other Lie families are already restricted. The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations. Real forms  Satake diagrams  are like being stepped into a perspective (from some perspective within a chain). An odddimensional real orthogonal case is steppedin and evendimensional is stepped out. Complex case combines the two, and quaternion case combines them yet again. For consciousness. Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams. Arnold  "Polymathematics: complexification, symplectification and all that " 1998 video. 18:50 About his trinity, his idea: "This idea, how to apply it, and the examples that I shall discuss even, are not formalized. The theory that I will describe today is not a conjecture, not a theorem, not a definition, it is some kind of religion. I shall show you examples and in these examples, it works. So I was able, using this religion, to find correct guesses, and to find correct conjectures. And then I was able to work years or months trying to prove them. And in some cases, I was able to prove them. In other cases, other people were finally able to prove them. In other cases other people were able to prove them. But to guess these conjectures without this religion would, I think, be impossible. So what I would like to explain to you is just this nonformalized part of it. I am perhaps too old to formalize it but maybe someone who one day finds the axioms and makes a definition from the general construction from the examples that I shall describe." 39:00 Came up with the idea in 1970, while working on the 16th Hilbert problem. A_n defines a linear algebra and other root systems add additional structure Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list? Wave function Smolin says is ensemble, I say bosonic sharing of space and time Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand adbc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc. Think again about the combinatorial intepretation of {$K^{1}K=I$}. Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. The octonions can model the nonassociativity of perspectives. Hurwitz's theorem for composition algebras Complex numbers describes rotations in twodimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? Conjugate = mystery = false. (Hidden distinction). Triality: C at the center, three legs: quaternions, evendimensional reals, odddimensional reals. Fold, fuse, link. i>j is asymmetric, onedirectional. i<>i* is symmetric, twodirectional, breaks antisymmetry, hides antisymmetry (which is i and which is j?) Video: The rotation group and all that Lorenzo Sadun. Videos: Linear Algebra Nr.88 is SO(3) and so(3) {$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. Encyclopedia: Complexification of a Lie group If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skewsymmetric (symplectic)—and these are related by the complex structure (which is the compatibility). Random phenomena organize themselves around a critical boundary. Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics Anthony Judge Spin 1/2 means there are two states separated by a quanta of energy +/ h. So this is like divisions of everything:
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties  charge. DrPhysicsA Exchange particles  gauge bosons. Heisenberg uncertainty principle  the slack in the vacuum  that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). Galois, Grothendieck and Voevodsky  George Shabat The analysis in a Lie group is all expressed by the behavior of the epsilon. Complexity measures for Boolean functions. How is a Boolean function similar to a linear functional? Terrence Tao: Twisted Convolution and the Sensitivity Conjecture Relate methods of proof and discovery, 3 systemic and 3 not. Induction step by step is different than the outcome, the totality, which forgets the gradation. Equality holds for both value and type, amount and unit. Peano axiom. Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line. Axiom of forgetfullness. Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out. Mathematical induction  is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider NavierStokes equations. Kevin Brown collection of expositions of math Ways of discovery in math: Tricki.org. Overview by Timothy Gowers. The Princeton Companion to Mathematics VU Matematikos ir informatikos skaitykla Yoneda Lemma
Einstein field equations  energy stress tensor  is 4+6 equations. I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view. Circle (threecycle) vs. Line (link to unconditional)  sixsome  and real forms CurryHowardLambek correspondence of logic, programming and category theory Pascal's triangle  the zeros on either end of each row are like Everything at start and finish of an exact sequence. Edward Frenkel. Langlands program, vertex algebras related to simple Lie groups, detailed analysis of SU(2) and U(1) gauge theories. https://arxiv.org/abs/1805.00203 Study orthogonal groups and Bott periodicity. In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? Consider the classification of Lie groups in terms of the objects for which they are symmetries. NavierStokes equations: Reynolds number relates time symmetric (high Reynolds number) and time asymmetric (low Reynolds number) situations. Differential Forms in Algebraic Topology, Bott & Tu {$A^TA$} is similar to the adjoint functors  they may be inverses (in the case of a unitary matrix) or they may be similar. Terrence Tao problem solving https://books.google.lt/books/about/Solving_Mathematical_Problems_A_Personal.html?id=ZBTJWhXD05MC&redir_esc=y In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$1$} become conjugates. Similarly, {$XYYX = \overline{(YXXY)}$}. Eduardo's Yoneda Lemma diagram is the foursome. Yates Index Theorem  consider substitution. Onedimensional proteins are wound up like the chain of a multidimensional Lie group. Physics is measurement. A single measurement is analysis. Algebra gives the relationships between disparate measurements. But why is the reverse as in the ways of figuring things out in mathematics? Terrence Tao: It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Mathematics Subject Classification wiki How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries N J Wildberger Compare finite field behavior (division winding around) with complex number behavior (winding around). What is the relation between the the chain of Weyl group reflections, paths in the root system, the Dynkin diagram chain, and the Lie group chain. Talk with Thomas
Does the Lie algebra bracket express slack? Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\Delta_i  \Delta_j$} is {$30^{\circ}$} {$\Delta_i  \Delta_j$} {$e^{\sum k_i \Delta_i}$} Has inner product iff {$AA^?=I$}, {$A{1}=A^?$} Killing form. What is it for exceptional Lie groups? The Cartan matrix expresses the amount of slack in the world. {$A_n$} God. {$B_n$}, {$C_n$}, {$D_n$} human. {$E_n$} n=8,7,6,5,4,3 divisions of everything. 2 independent roots, independent dimensions, yield a "square root" (?) Symplectic matrix (quaternions) describe local pairs (Position, momentum). Real matrix describes global pairs: Odd and even? Root system is a navigation system. It shows that we can navigate the space in a logical manner in each direction. There can't be two points in the same direction. Therefore a cube is not acceptable. The determinant works to maintain the navigational system. If a cube is inherently impossible, then there can't be a trifold branching. {$A_n$} is based on differences {$x_ix_j$}. They are a higher grid risen above the lower grid {$x_i$}. Whereas the others are aren't based on differences and collapse into the lower grid. How to understand this? How does it relate to duality and the way it is expressed. In {A_1}, the root {$x_2x_1$} is normal to {$x_1+x_2$}. In {A_2}, the roots are normal to {$x_1+x_2+x_3$}. If two roots are separated by more than {$90^\circ$}, then adding them together yields a new root. {$cos\theta = \frac{a\circ b}{\left \ a \right \\left \ b \right \}$} {$120^\circ$} yields {$\frac{1}{\sqrt{2}\sqrt{2}}=\frac{1}{2}$} Given a chain of composition {$\cdots f_{i1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards, stepping in or climbing out. There is the possiblity of switching adjacent functions at each dot. So each dot corresponds to a node in the Dynkin diagram. And the duality is affected by what happens at an extreme.
In solving for eigenvalues {$\lambda_i$} and eigenvectors {$v_i$} of {$M$}, make the matrix {$M\lambda I$} degenerate. Thus {$\text{det}(M\lambda I)=0$}. The matrix is degenerate when one row is a linear combination of the other rows. So the determinant is a geometrical expression for volume, for collinearity and noncollinearity. Terrence Tao: Each prime p wants to have weight ln p. Compare with Zipf's law, ZipfMandelbrot law, YuleSimon distribution, Preferential attachment, Matthew effect, Pareto's principle. Emily Riehl. Category Theory in Context Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to categories. Cambridge (1997) Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation toApplied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007) How is substitution, as a method of proof, related to lamba calculus, and construction? Walks from A to B in category theory are morphisms and they get mapped to the morphisms from A to B. Relate this to walks on trees. Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another? What can graph theory (for example, random graphs, or random order) say about category theory? Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1. In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same? The Elliptic Umbilic Diffraction Catastrophe. Optics, Bott periodicity? In the category Set, how can you distinguish between a countable and uncountable set? Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here? What would the category of Lists look like? And what would the Yoneda Lemma look like if the functor mapped into the category of Lists? What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.? 
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