persistence theory revisitedfigure 2: example of r-invariants for a circle valued map 4 repr...

Persistence Theory Revisited

Dan Burghelea

Department of mathematicsOhio State University, Columbus, OH

D.Burghelea Persistence Theory Revisited

SUBLEVEL PERSISTENCE for f : X → RSublevels for a real valued map

LaD.Burghelea Persistence Theory Revisited

Sublevel Persistence (Edelsbrunner Letscher Zamorodian)

Analyses changes in homology of SUBLEVELS (birth -death of homology classes )

Records the results as invariants:1 critical values, a,b, · · ·2 bar codes Br (f ) i) finite intervals [a,b)

ii) infinite intervals [a,∞)


LEVEL PERSISTENCE for f : X → RLevels for a real- valued map

:. ";

.,0

' t-

I

ii

/

- t.,l-'atlrirrItt

x

)f


Level Persistence (= zigzag persistence(Carlson and Vin deSilva)

analyses changes in homology of LEVELS (death (right /left), observability (right / left) of homology classes)

records the results as invariants:1 critical values a,b, · · ·2 bar codes of four types:

Bcr (f ) – closed intervals , [a, b]

Bcr (f ) – open intervals , (a, b)

Bc,or (f ) – closed-open intervals, [a, b)

Bc,or (f ) – open-closed intervals, (a, b]


Death (right/left) (y/n?)

Observability (right/left) (y/n?)

Hr (Xt) // Hr (X[t ,t ′]) Hr (Xt ′)oo

Hr (Xt ′)

OO

Xt = f−1(t), X[t ,t ′] = f−1([t , t ′])


LEVEL PERSISTENCE for f : X → S1


Level Persistence for angle valued maps (Burghelea - Dey)Analyses changes in homology of LEVELS(death (right / left) - observability (right / left) - return)

Records the results as invariants:

1 critical angles θ, θ′, · · ·

2 bar codes of four types Bcr (f ), Bc

r (f ), Bc,or (f ), Bc,o

r (f )– equivalence classes of closed intervals , {[a, b]}– equivalence classes of open intervals , {(a, b)}– equivalence classes of closed-open intervals, {[a, b)}– equivalence classes of open-closed intervals, {(a, b]},

3 Jordan cells Jr (f ) – a collection of pairs (λ, k),


\-*il j


A different perspective on bar codes.

PUTTING TOGETHER Bcr (f ) AND Bo

r−1(f )

f : X → R f : X → S1

Bcr (f ) 3 [a, b]→ z = a + ib ∈ C Bc

r (f ) 3 {[a, b]} → z = e(b−a)+ia ∈ C \ 0

Bor−1(f ) 3 (a, b)→ z = b + ia ∈ C Bo

r−1(f ) 3 {(a, b)} → z = e(a−b)+ib ∈ C \ 0

⇓ ⇓Configuration of points in C, Cr (f )(z), Configuration of points in C \ 0, Cr (f )(z)

= =Monic complex polynomialal Pr (z) Monic complex polynomiall Pr (z) (a0 6= 0)


Configuration Cr (f ) for a real-valued map.


Configuration Cr (f ) for an angle-valued map.


REFINEMENT OF BETTI NUMBERS (w. Haller and Dey)

Theorem1 The cardinality of the support of Cr (f ) is equal to :

the Betti number βr (X ) if f is real valued (B - D),the Novikov-Betti number βN

r (X , ξf ) if f is angle valued andξf ∈ H1(X : Z) the cohomology class represented by thehomotopy class of f .

2 If the angle valued maps f ,g are homotopic thenJr (f ) = Jr (g).


REFINEMENT OF POINCARÉ DUALITY (w.Haller)

TheoremSuppose X a closed n−dimensional manifold.

1 If f is real valued Cr (f )(z) = Cn−r (−f )(−z)2 If f is S1−valued then Cr (f )(z) = Cn−r (f )(z−1)


REFINEMENT OF HODGE THEOREM

Theorem1. Suppose f is real valued continuous X compact and Hr (X )equipped with a scalar product (i.e. X is a Riemannian manifoldor X a simplicial complex).There exists a natural assignment z Hr (X )(z) ⊂ Hr (X ) s.t.

1 dim Hr (X )(z) = Cr (f )(z)2 For z 6= z ′ Hr (X )(z) ⊥ Hr (X )(z ′)

2. For an open and dense set of maps in C(M,R)dim Hr (X )(z) ≤ 1.


Denote

N =

{βr (X ) if f real− valuedβN

r (X , ξ) if f angle− valued

C(M,R) the space of all continuous real valued mapsCξ(M,S1) the space of continuous angle-valued maps inthe homotopy class determined by ξ ∈ H1(X ;Z)(both equipped with the compact open topology induced byd(f , g) = suppx∈X d(f(x), g(x))

NoteMonic degree- N polynomials = CN

Monic degree N polynomials with a0 6= 0 = CN−1 × (C \ 0)(both equipped with the standard topology induced by the standard distance D)


A STRONG STABILITY PROPERTY (w.Haller)

TheoremThe assignments

C(M,R) 3 f Pr (f )(z) ∈ CN

C(M,S1) 3 f Pr (f )(z) ∈ CN−1 × (C \ 0)

are continuous.Moreover one has D(Pr (f ),Pr (g)) ≤ 3d(f ,g)


Measure theoretic considerations

• Consider F : R2 → Z≥0

1 For a ≤ a′,b′ ≤ b one has F (a,b) ≤ F (a′,b′)2 For B = (a,b]× [c,d),a < b, c < d one has

µ(B) := F (a, c) + F (b,d)− F (a,d)− F (b, c) ≥ 0

• Define the Dirac measure

δ(a,b) = limB3(a,b)

µ(B)


Linear algebra

To the linear relationV →W ← V

one associates its regular part

Vreg �Wreg � Vreg

α :=�, β =� linear isomorphisms.One calculates the Jordan matrices of β−1 · α


For f : X → R consider sublevels and suplevels.

X

!'l,l


• Consider

Ifa(r) = Img(Hr(f−1(−∞, a])→ Hr(X)

Ibf (r) = Img(Hr(f−1([b,∞))→ Hr(X))

and then

F fr (a,b) = dim(If

a(r) ∩ Ibf (r) and δf

r(a, b).

• DefineCr (f )(z) = δf

r (a,b), z = a + ib) .


For f : X → S1 consider f̃ : X̃ → R.

\-*il j


• Consider F f̃r (a,b) and δ f̃

r (a,b).

• Define Cr (f )(z) := δ f̃r (a,b), z = (b − a) + ia .

• Use

Hr (Xθ) = Hr (X̃θ)→ Hr (X̃[θ,θ+2π] ← Hr (X̃θ+2π) = Hr (Xθ)

and calculate its regular part.

• Define Jr (f ) as the Jordan cells of β−1 · α.


φ

2πθ4θ2θ10

circle 1

circle 3

circle 2

1

2

3

Y0 Y1Y

θ6θ5θ3

map φ r-invariants

circle 1: 3 times around circle 1circle 2: 1 time around 2 and 3 times around 3circle 3: the identity

dimension bar codes Jordan cells0 (1, 1)

(θ6, θ1 + 2π] (3, 1)1 [θ2, θ3] (1, 2)

(θ4, θ5)

Figure 2: Example of r-invariants for a circle valued map

4 Representation theory and r-invariantsThe invariants for the circle valued map are derived from the representation theory of quivers. The quiversare directed graphs. The representation theory of simple quivers such as paths with directed edges wasdescribed by Gabriel [8] and is at the heart of the derivation of the invariants for zigzag and then levelpersistence in [4]. For circle valued maps, one needs representation theory for circle graphs with directededges. This theory appears in the work of Nazarova [14], and Donovan and Ruth-Freislich [10]. The readercan find a refined treatment in Kac [15].Let G2m be a directed graph with 2m vertices, x1, x1, · · · x2m. Its underlying undirected graph is a

simple cycle. The directed edges in G2m are of two types: forward ai : x2i−1 → x2i, 1 ≤ i ≤ m, andbackward bi : x2i+1 → x2i, 1 ≤ i ≤ m − 1, bm : x1 → x2m.

x2

b1a2

b2

x3

x2m−1

x2m−2

x4

a1

bm

am

x2m

x1

We think of this graph as being residing on the unit circle cen-tered at the origin o in the plane.A representation ρ on G2m is an assignment of a vector space

Vx to each vertex x and a linear map Ve : Vx → Vy for each orientededge e = {x, y}. Two representations ρ and ρ′ are isomorphic if foreach vertex x there exists an isomorphism from the vector space Vx

of ρ to the vector space V ′x of ρ′, and these isomorphisms intertwine

the linear maps Vx → Vy and V ′x → V ′

y . A non-trivial representa-tion assigns at least one vector space which is not zero-dimensional.A representation is indecomposable if it is not isomorphic to thesum of two nontrivial representations. It is not hard to observe thateach representation has a decomposition as a sum of indecompos-

able representations unique up to isomorphisms.

6


persistence theory revisitedfigure 2: example of r-invariants for a circle valued map 4 repr...

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