MathLang DSL: Unified Mathematical Framework

MathLang Code Editor

// MathLang Unified Framework
// Combining all mathematical structures

// Week 2: Hyperbolic Geometry
manifold M : HyperbolicSurface {
    genus: 2,
    fundamental_group: <a,b,c,d | [a,b][c,d] = 1>
}

// Week 3: Topos Theory  
topos T := Sh(M)
subobject_classifier Omega : T -> Set

// Week 4: Character Varieties
variety X := CharacterVariety(pi1(M), PSL2(C))
trace_coords := [tr(a), tr(b), tr(ab)]

// Week 5: Tropical Geometry
tropical_curve TC := tropicalize(X) {
    semiring: (min, +),
    vertices: [v1, v2, v3]
}

// Week 6: Fibered Categories
fibration F : Categories -> S1 {
    base: S1,
    fiber: category(M),
    monodromy: DehnTwist(a)
}

// Week 7: K-Theory
bundle E : VectorBundle(M) { rank: 2 }
ktheory K0 := GrothendieckGroup(E ⊕ F ⊗ L)

// Cross-connections
connection c1 : M -> X via representation
connection c2 : X -> TC via tropicalization  
connection c3 : T -> K0 via internal_semiring

MathLang Syntax:

manifold M : Type { properties }

topos T := Sh(M)

bundle E : VectorBundle(M)

connection c : A -> B via method

Abstract Syntax Tree

Manifold M : HyperbolicSurface
├─ genus: 2
└─ π₁: <a,b,c,d | [a,b][c,d]=1>

Topos T := Sh(M)
├─ base: M
└─ classifier: Ω

CharacterVariety X
├─ group: π₁(M)
├─ target: PSL₂(ℂ)
└─ coordinates: traces

TropicalCurve TC
├─ source: X
├─ semiring: (min,+)
└─ skeleton: PL-graph

Fibration F
├─ base: S¹
├─ fiber: Cat(M)
└─ monodromy: φ*

VectorBundle E
├─ base: M
├─ rank: 2
└─ K-class: [E]

Suggestions:

Add monodromy connection

Internal bundle operations

Amoeba limit process

Live Mathematical Visualization

M
Hyperbolic

Sh(M)
Topos

X
Character

TC
Tropical

E
Bundle

✓ Visualization updated: 5 objects, 4 connections

Cross-Domain Relationships

Week 2
Hyperbolic

Week 3
Topoi

Week 4
Character

Week 5
Tropical

Week 6
Fibered

Week 7
K-Theory

Cross-Connections:

Hyperbolic → Character via π₁ representations

Character → Tropical via logarithmic limit

Topoi → K-Theory via internal semirings

Fibered → Hyperbolic via mapping tori

Hyperbolic Topos

Sheaves on hyperbolic surfaces with internal logic

Tropical Character

Tropicalization of representation varieties

Fibered K-Theory

K-theory of mapping torus bundles

Unified Framework

All structures connected in single program

Status:

Ready