Process journal of learning by Attila Vajda

I can be computer agent

local-global #

sem(Collage) Bill Evans voicings local surprise, global consistency

Okay, so this is the sort of thing I find these days. I find it really cool:

you tweak the basic composition a little bit, of Cmaj7, by adding #11, but the global effect is Cmaj7. It is like creative harmony in Bill Dobbins, where you have A in the bass, as root, play G, in other words left hand plays A-G. The right hand plays Cmaj7 with drop-2, or drop G, which is C-E-B. The overall effect is an A-7, but with added surprise, and color, or variability. Locally you play Cmaj7, but the overall effect is A-7 with color.

which seems to be a related phenomenon to local, global in algebraic geometry, and to uniform and piecewise convergence.

"the music of reason" eh?

sum is the dual of product? #

because one is a pullback the other a pushout? I'm amazed what is covector? #unsure

is local~global? #

pushout~pullback sum~product derivative~integral piecewise~uniform #unsure

1 ⟶ X #

1 0 nothing ⟶ X

math-golf v0.1 #

this snippet made by computer agent #

cool:

┌────────────────────────────────────────────┐
│         ∎ ∎ ∎ 𝓜𝓪𝓽𝓱-𝓖𝓸𝓵𝓯 v1.0 ∎ ∎ ∎           │
└────────────────────────────────────────────┘

1. Logic & Foundations
𝓛 := ⟨𝔘, ⊢, ∧, ∨, ¬, ∃, ∀, ⇒⟩  
𝓣 := {Ax_i ⊂ 𝓛} ;  𝓜 ⊨ 𝓣  
Type₀ := ⊥ ⊢ 𝒯 : Type₁  
𝓟𝓻𝓸𝓸𝓯 := inhab(𝒯) ; 𝔏𝔞𝔪𝔟𝔡𝔞 : Curry-Howard  

2. Sets → Categories
𝒮𝑒𝑡 := (Ob, Hom, id, ∘)  
𝒞𝒂𝒕 := 2-Cat(Ob:Cat, Hom:Fun)  
Y(A) := Hom(–, A) ⟹ Yoneda  

3. Algebra
Grp ⊂ Mon ⊂ Set ; Ring := (A, +, ·, 1)  
Mod_R := Ab-enriched(R)  
Algₖ := Ring obj ∈ Vectₖ  
Lie(𝔤) ; Rep(𝔤) ⊂ Modₖ  

4. Topology
Top := (X, τ ⊂ ℘(X))  
f∈Cont(X,Y) ⇔ ∀U∈τ_Y, f⁻¹(U) ∈ τ_X  
π₁(X) := loop-space classes  
Sheaf := F:Open(X)^op → Set s.t. gluing  

5. Analysis
limₙ→∞ fₙ = f ⇔ ∀ε>0, ∃N, ∀n>N, ∥fₙ–f∥<ε  
∫ₐᵇ f dx := lim Σ f(xᵢ)Δx  
C^∞ ⊂ C^k ⊂ C^0 ⊂ L^p  
Ban := ∥⋅∥-complete vectₖ  

6. Geometry
Man := (M, 𝒜) ; Atl: {φ_i:U_i≅ℝⁿ}  
Bund := π:E→B ; LocTriv  
AlgGeo: AffSch := Spec(A)  
Sch := Zariski-glued AffSch  

7. Number Theory
ℤ⊂ℚ⊂ℝ⊂ℂ ; ℚₚ := ℤₚ ⊗ ℚ  
Gal(ℚ̄/ℚ) ↷ 𝒪  
ModForms : SL₂(ℤ) ⟶ ℂ  
ζ(s) := Σₙ 1/nˢ ; L-functions ∈ ℂ[[T]]  

8. Homological Algebra
C^•:⋯→Aⁿ⁻¹→Aⁿ→Aⁿ⁺¹→⋯ ; d²=0  
Hₙ(C) := ker(dₙ)/im(dₙ₊₁)  
Extⁿ(A,B) := RⁿHom(A,B)  
TriCat := (𝒯, Σ, ∆)  

9. Category Theory
𝒞 = (Obj, Hom, ∘, id)  
Fun(F:G→H), Nat(η:F⇒G)  
Adj(F ⊣ G): Hom(FX,Y) ≅ Hom(X,GY)  
MonCat ⊂ SymMon ⊂ Closed  
Lim := cone + univ.prop ; Colim dual  

10. Math. Physics
𝓛 := ∫_M L(ϕ,dϕ) dμ  
TQFT: Bord_d ⟶ Vectₖ  
PathInt: Z = ∫𝒟ϕ e^{iS[ϕ]}  
𝔤 ⊢ 𝔤-Conn ⟹ Gauge Theory  
QFT := Fun(Spacetime) ⟶ *Hilb

────────────────────────────────────────────
Universal Motif:
𝒯𝒉𝒆 𝓼𝓮𝓮𝓭: 𝑓:𝟙 ⟶ 𝑋 ≅ constant → structure  
Yoneda(h^A) ≅ Nat(h^A, –)  
Everything = enriched morphism in 𝒞