Duality for Algebras of the Connected Planar Wiring Diagrams Operad - - PowerPoint PPT Presentation

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Duality for Algebras of the Connected Planar Wiring Diagrams Operad - - PowerPoint PPT Presentation

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Duality for Algebras of the Connected Planar Wiring Diagrams Operad Owen Biesel Carleton College obiesel@carleton.edu May 23, 2019 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019 Combining Resistance R 1 R 2 R 1 + R 2

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Duality for Algebras of the Connected Planar Wiring Diagrams Operad

Owen Biesel

Carleton College

  • biesel@carleton.edu

May 23, 2019

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Combining Resistance

R1 R2 ∼ = R1 + R2 R1 R2 ∼ = R1R2 R1 + R2

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Combining Resistance

R1 R2 ∼ = R1 + R2 R1 R2 ∼ = 1 R1 + 1 R2 −1

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Conductance is Inverse Resistance

G = 1/R R = 2 Ω resistance

  • G = 0.5 ℧

conductance

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Combining Conductance

G1 G2 ∼ = 1 G1 + 1 G2 −1 G1 G2 ∼ = G1 + G2

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Combining Maximum Flow Rates

F1 F2 ∼ = min(F1, F2) F1 F2 ∼ = F1 + F2

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Combining Minimum Path Lengths

D1 D2 ∼ = D1 + D2 D1 D2 ∼ = min(D1, D2)

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Series and parallel formulas

Series Parallel Resistance R1 + R2 (R−1

1

+ R−1

2 )−1

Conductance (G −1

1

+ G −1

2 )−1

G1 + G2 Max Flow Rate min(F1, F2) F1 + F2 Min Path Length D1 + D2 min(D1, D2)

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Series and parallel connections are dual

Not quite the usual “dual graph.”

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Connected circular planar graphs

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Placing the dual vertices

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Placing the dual edges

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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The dual graph

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Dual Connected Circular Planar Graphs

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Resistance and conductance are “dual”

4 Ω 1 Ω 4 Ω 3 Ω 1 Ω 1 Ω Resistances Effective resistance = 157 29 Ω 4 ℧ 1 ℧ 3 ℧ 1 ℧ 4 ℧ 1 ℧ Conductances Effective conductance = 157 29 ℧

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Max flow rate and min path length are “dual”

4 1 4 3 1 1 Max flow rates Overall max flow rate = 3 4 1 3 1 4 1 Edge lengths Min path length = 3

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Main theorem

Theorem (B.—, 2019)

(Connected circular planar) graphs form an algebra of the operad Plan of “connected planar wiring diagrams.” “Max flow rate,” “min path length,” “effective resistance,” and “effective conductance” are all morphisms between Plan-algebras. Plan has a duality automorphism giving every algebra a “dual” algebra. The algebra of graphs is isomorphic to its dual algebra, and the isomorphism sends a graph to its dual graph. The dual of the “max flow rate” morphism is “min path length,” and the dual of “effective resistance” is “effective conductance.”

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Gluing together circular planar graphs

  • ?

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Gluing with a planar wiring diagram

=

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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A connected planar wiring diagram

This is a morphism in the operad Plan of connected planar wiring diagrams.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Series and parallel wiring diagrams

Series Parallel

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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The operad of connected planar wiring diagrams

Definition (B.—, 2019)

The (symmetric, coloured) operad Plan:

  • bjects are circularly ordered finite sets (X, θ).

X x θ(x)

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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The operad of connected planar wiring diagrams

Definition (continued)

morphisms from (X1, θ1), . . . , (Xn, θn) to (Y , ϕ) are planar wiring diagrams with the Xi on the inside and Y on the outside: X1 X2 X3 Y Every “cable” has ≤ 1 element of Y . Every “face” has ≤ 1 arc of outer circle. Lemma: This really does define an operad!

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Relationships to other operads

Plan has: A forgetful map to Spivak’s operad of all wiring diagrams. Plan inherits several algebras from there, like flows and potentials. However, Spivak’s operad does not have a duality automorphism. An inclusion map to Jones’s “planar algebras” operad. Plan inherits its duality automorphism Jones’s “1-click” automorphism, but Jones’s

  • perad has too many morphisms for circular planar graphs to be an

algebra.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Plan-algebras

A Plan-algebra A assigns: to each circularly ordered set (X, θ) a set A(X, θ), and to each morphism (X1, θ1), . . . , (Xn, θn) → (Y , ϕ) a function

n

  • i=1

A(Xi, θi) → A(Y , ϕ). These describe what may be inserted into the slots of a wiring diagram and how they glue together. Plan

A

− → Op(Set)

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Example Plan algebras: G and G(0,∞)

G(X, θ) = set of connected circular planar graphs with boundary vertices (X, θ). G(0,∞)(X, θ): same, but with edges weighted by positive real numbers. Lemma: G is generated by the single element satisfying a single relation. That makes it easy to describe algebra morphisms out of G!

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Example Plan algebra: Π

Π(X, θ) = set of planar (noncrossing) partitions of (X, θ). =

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Example Plan-algebra: potential sets

A potential on (X, θ) is a function X → R up to overall additive constant: X 2 1 −2 4 = X 102 101 100 98 104 V(X, θ) is the set of potentials on X.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Gluing compatible potentials

1 −1 2 4 1 2 3 2 1 4 1 = ?

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Gluing compatible potentials

1 −1 2 3 1 2 2 1 4 1 = 3 2 1 4 1 −1 1

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Gluing potential sets

Not all potentials can be glued, so V is only a relational Plan-algebra. But P(V) : (X, θ) → the set of subsets of V(X, θ) is an actual Plan-algebra! Send a collection of potential sets to the set of potentials obtained by gluing compatible members.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Min path length: G(0,∞) → P(V)

Weights on a graph distances potential set: D {(a, b) ∈ R2 | |a − b| ≤ D}

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Min path length: G(0,∞) → P(V)

D1 D2 D3 {(a, b) ∈ R2 | ∃x ∈ R such that |a − x| ≤ D1, |x − b| ≤ D2, |x − b| ≤ D3} = {(a, b) ∈ R2 | |a − b| ≤ D1 + min(D2, D3)} “Min path length” is an algebra morphism G(0,∞) → P(V)!

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Example Plan-algebras: Flow Sets

The algebra P(F) of flow sets: A flow on (X, θ) is a sum-zero function X → R: X 3 0.5 −1 −2 −0.5 F(X, θ) =the set of flows on X.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Gluing compatible flows

−1 −1 −1 3 −2 1 1 −1 1 1 −2 1 = 1 2 −2 1 −1 −1

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Gluing flow sets

Not all flows can be glued, so F is only a relational Plan-algebra. But P(F) : (X, θ) → the set of subsets of F(X, θ) is an actual Plan-algebra! Send a collection of flow sets to the set of flows obtained by gluing compatible members.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Max flow rate: G(0,∞) → P(F)

Weights on a graph possible flows: F {(a, −a) ∈ R2 | |a| ≤ F}

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Max flow rate: G(0,∞) → P(F)

F1 F2 F3 {(a, b) ∈ R2 | ∃x, y, z ∈ R such that |x| ≤ F1, |y| ≤ F2, |z| ≤ F3, x = a, y + z = b, x + y + z = 0} = {(a, −a) ∈ R2 | |a| ≤ min(F1, F2 + F3)} “Max flow rate” is a Plan-algebra morphism G(0,∞) → P(F)!

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Resistance: G(0,∞) → P(V × F)

Weights on a graph voltage-current relationships: R

  • V1

V2

  • ,

I −I

  • ∈ R2 × R2
  • (V1 − V2) = IR
  • In general, send a weighted graph to the set of pairs (boundary voltages,

induced boundary currents). “Effective resistance” is a Plan-algebra morphism G(0,∞) → P(V × F)!

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Duality automorphism of Plan

Plan has a “duality” automorphism ∗ : Plan → Plan Compose with any algebra A : Plan → Op(Set) to get a “dual” algebra A∗ : Plan

∗

− → Plan

A

− → Op(Set). A∗∗ ∼ = A.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Duality automorphism of Plan: on objects

∗ : X

  • X ∗

∗ : Y

  • Y ∗

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Duality automorphism of Plan: on morphisms

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Duality automorphism of Plan: on morphisms

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Duality automorphism of Plan: on morphisms

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Duality automorphism of Plan: on morphisms

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Series and parallel wiring diagrams are dual

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Dual of G is G

G is isomorphic to its own dual: the isomorphism sends a graph to its dual.

  • The same holds for G(0,∞).

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Dual of V is F

Theorem (B.—, 2019)

The dual of the algebra of potentials is the algebra of flows: V∗ ∼ = F. Take successive differences of potential values to obtain a flow: X 102 102 101 105 103 potential

  • X ∗

−1 4 −2 −1 flow Corollary: P(V)∗ ∼ = P(F) and P(V × F)∗ ∼ = P(V × F) as well.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Dual of min path length is max flow rate

Theorem (B.—, 2019)

The dual of the “min path length” morphism is the “max flow rate” morphism: the square G∗

(0,∞)

∼ = G(0,∞) (min path length)∗ ↓ ↓ (max flow) P(V)∗ ∼ = P(F) commutes.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Dual of min path length is max flow rate

Proof sketch: only have to check for single weighted edges! D ↔ D

  • {(a, −a) : |a| ≤ D} =

{(a, b) | |a − b| ≤ D} ↔ {(a − b, b − a) : |a − b| ≤ D}

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Dual of resistance is conductance

Theorem (B.—, 2019)

The dual of the “effective resistance” morphism is “effective 1/resistance”: the square G∗

(0,∞)

∼ = G(0,∞) (effective resistance)∗ ↓ ↓ (effective resistance) P(V × F)∗ ∼ = P(V × F) commutes, where the isomorphism G∗

(0,∞) ∼

= G(0,∞) dualizes the graph and inverts the edge weights.

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019

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Dual of resistance is conductance

Proof sketch: R

  • V1

V2

  • ,

I −I

  • (V1 − V2) = IR
  • ↔

c + I c

  • ,

V1 − V2 V2 − V1

  • (V1 − V2) = IR
  • =

1/R

  • V ′

1

V ′

2

  • ,

I ′ −I ′

  • (V ′

1 − V ′ 2) = I ′/R

  • Owen Biesel (Carleton)

Duality for Planar Wiring Diagrams May 23, 2019

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Thank you!

Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019