persistent homology and the stability theorempersistent homology and the stabilitytheorem ulrich...
TRANSCRIPT
Persistent Homology
and the
Stability Theorem
Ulrich Bauer
TUM
February 19, 2018
TAGS – Linking Topology to Algebraic Geometry and Statistics
≅ ?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 1 / 31
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http://ulrich-bauer.org/persistence-talk-leipzig.pdf 1 / 31
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 2 / 31
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 2 / 31
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http://ulrich-bauer.org/persistence-talk-leipzig.pdf 2 / 31
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 2 / 31
What is persistent homology?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 3 / 31
What is persistent homology?
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 3 / 31
What is persistent homology?
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 3 / 31
What is persistent homology?
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 3 / 31
What is persistent homology?
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 3 / 31
What is persistent homology?
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 3 / 31
What is persistent homology?
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 3 / 31
What is persistent homology?
Persistent homology is the homology of a Bltration
A Bltration is a certain diagram K ∶ R→ Top R is the poset category of (R, ≤) A topological space Kt for each t ∈ R An inclusion map Ks Kt for each s ≤ t ∈ R
Consider homology with coe?cients in a Beld (often Z2)
H∗ ∶ Top→ Vect Persistent homology is a diagram M ∶ R→ Vect
(persistence module)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 4 / 31
What is persistent homology?
Persistent homology is the homology of a Bltration
A Bltration is a certain diagram K ∶ R→ Top R is the poset category of (R, ≤) A topological space Kt for each t ∈ R An inclusion map Ks Kt for each s ≤ t ∈ R
Consider homology with coe?cients in a Beld (often Z2)
H∗ ∶ Top→ Vect Persistent homology is a diagram M ∶ R→ Vect
(persistence module)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 4 / 31
What is persistent homology?
Persistent homology is the homology of a Bltration
A Bltration is a certain diagram K ∶ R→ Top R is the poset category of (R, ≤) A topological space Kt for each t ∈ R An inclusion map Ks Kt for each s ≤ t ∈ R
Consider homology with coe?cients in a Beld (often Z2)
H∗ ∶ Top→ Vect Persistent homology is a diagram M ∶ R→ Vect
(persistence module)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 4 / 31
What is persistent homology?
Persistent homology is the homology of a Bltration
A Bltration is a certain diagram K ∶ R→ Top R is the poset category of (R, ≤) A topological space Kt for each t ∈ R An inclusion map Ks Kt for each s ≤ t ∈ R
Consider homology with coe?cients in a Beld (often Z2)
H∗ ∶ Top→ Vect
Persistent homology is a diagram M ∶ R→ Vect(persistence module)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 4 / 31
What is persistent homology?
Persistent homology is the homology of a Bltration
A Bltration is a certain diagram K ∶ R→ Top R is the poset category of (R, ≤) A topological space Kt for each t ∈ R An inclusion map Ks Kt for each s ≤ t ∈ R
Consider homology with coe?cients in a Beld (often Z2)
H∗ ∶ Top→ Vect Persistent homology is a diagram M ∶ R→ Vect
(persistence module)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 4 / 31
Homology inference
Homology inference
Given: Bnite sample P ⊂ Ω of unknown shape Ω ⊂ Rd
Problem (Homology inference)
Determine the homologyH∗(Ω).
Problem (Homological reconstruction)
Construct a shape X with H∗(X) ≅ H∗(Ω).
Idea:
approximate the shape by a thickening Bδ(P) covering Ω
Requires strong assumptions:
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 5 / 31
Homology inference
Given: Bnite sample P ⊂ Ω of unknown shape Ω ⊂ Rd
Problem (Homology inference)
Determine the homologyH∗(Ω).
Problem (Homological reconstruction)
Construct a shape X with H∗(X) ≅ H∗(Ω).
Idea:
approximate the shape by a thickening Bδ(P) covering Ω
Requires strong assumptions:
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 5 / 31
Homology inference
Given: Bnite sample P ⊂ Ω of unknown shape Ω ⊂ Rd
Problem (Homology inference)
Determine the homologyH∗(Ω).
Problem (Homological reconstruction)
Construct a shape X with H∗(X) ≅ H∗(Ω).
Idea:
approximate the shape by a thickening Bδ(P) covering Ω
Requires strong assumptions:
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 5 / 31
Homology inference
Given: Bnite sample P ⊂ Ω of unknown shape Ω ⊂ Rd
Problem (Homology inference)
Determine the homologyH∗(Ω).
Problem (Homological reconstruction)
Construct a shape X with H∗(X) ≅ H∗(Ω).
Idea:
approximate the shape by a thickening Bδ(P) covering ΩRequires strong assumptions:
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 5 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology reconstruction by thickening
Theorem (Niyogi, Smale,Weinberger 2006)
Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that
Bδ(P) coversΩ, and
δ <√3/20 reach(M).
Then H∗(M) ≅ H∗(B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 6 / 31
Homology inference using persistence
Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)
LetΩ ⊂ Rd. Let P ⊂ Ω, δ > 0 be such that
Bδ(P) coversΩ, and
the inclusionsΩ Bδ(Ω) B2δ(Ω) preserve homology.
Then H∗(Ω) ≅ imH∗(Bδ(P) B2δ(P)).
0.2 0.4 0.6 0.8 1.0 1.2
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 7 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
L K
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
X KL
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
This is not always possible:
L X? K
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
This is not always possible:
L X? K
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
This is not always possible:
L X? K
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
This is not always possible:
L X? K
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Homological realization
This motivates the homological realization problem:
Problem
Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that
H∗(X) = imH∗(L K).
This is not always possible:
L X? K
Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)
The homological realization problem isNP-hard, even inR3.http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31
Stability
Persistence and stability: the big picture
Data point cloud P ⊂ Rd
Geometry function f ∶ Rd → R
Topology topological spaces (Bltration) K ∶ R→ Top
Algebra vector spaces (persistence module) M ∶ R→ Vect
Combinatorics intervals (persistence barcode) B ∶ R→Mch
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 9 / 31
Persistence and stability: the big picture
Data point cloud P ⊂ Rd
Geometry function f ∶ Rd → R
Topology topological spaces (Bltration) K ∶ R→ Top
Algebra vector spaces (persistence module) M ∶ R→ Vect
Combinatorics intervals (persistence barcode) B ∶ R→Mch
distance
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 9 / 31
Persistence and stability: the big picture
Data point cloud P ⊂ Rd
Geometry function f ∶ Rd → R
Topology topological spaces (Bltration) K ∶ R→ Top
Algebra vector spaces (persistence module) M ∶ R→ Vect
Combinatorics intervals (persistence barcode) B ∶ R→Mch
distance
sublevel sets
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 9 / 31
Persistence and stability: the big picture
Data point cloud P ⊂ Rd
Geometry function f ∶ Rd → R
Topology topological spaces (Bltration) K ∶ R→ Top
Algebra vector spaces (persistence module) M ∶ R→ Vect
Combinatorics intervals (persistence barcode) B ∶ R→Mch
distance
sublevel sets
homology
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 9 / 31
Persistence and stability: the big picture
Data point cloud P ⊂ Rd
Geometry function f ∶ Rd → R
Topology topological spaces (Bltration) K ∶ R→ Top
Algebra vector spaces (persistence module) M ∶ R→ Vect
Combinatorics intervals (persistence barcode) B ∶ R→Mch
distance
sublevel sets
homology
structure theorem
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 9 / 31
Stability of persistence barcodes for functions
Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)
Let ∥f − g∥∞ = δ. Then there exists amatching between the
intervals of the persistence barcodes of f and g such that
matched intervals have endpointswithin distance ≤ δ, and
unmatched intervals have length ≤ 2δ.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 10 / 31
Stability of persistence barcodes for functions
Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)
Let ∥f − g∥∞ = δ. Then there exists amatching between the
intervals of the persistence barcodes of f and g such that
matched intervals have endpointswithin distance ≤ δ, and
unmatched intervals have length ≤ 2δ.
δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 10 / 31
Stability of persistence barcodes for functions
Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)
Let ∥f − g∥∞ = δ. Then there exists amatching between the
intervals of the persistence barcodes of f and g such that
matched intervals have endpointswithin distance ≤ δ, and
unmatched intervals have length ≤ 2δ.
δ
2δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 10 / 31
Interleavings
Let δ = ∥f − g∥∞. Write Ft = f −1(−∞, t] and Gt = g−1(−∞, t].
Then the sublevel set Bltrations F,G ∶ R→ Top are
δ-interleaved:
Ft Ft+δ Ft+2δ Ft+3δ
Gt Gt+δ Gt+2δ Gt+3δ
Applying homology (functor) preserves commutativity
persistent homology of f , g yields
δ-interleaved persistence modules R→ Vect
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 11 / 31
Interleavings
Let δ = ∥f − g∥∞. Write Ft = f −1(−∞, t] and Gt = g−1(−∞, t].
Then the sublevel set Bltrations F,G ∶ R→ Top are
δ-interleaved:
Ft Ft+δ Ft+2δ Ft+3δ
Gt Gt+δ Gt+2δ Gt+3δ
Applying homology (functor) preserves commutativity
persistent homology of f , g yields
δ-interleaved persistence modules R→ Vect
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 11 / 31
Interleavings
Let δ = ∥f − g∥∞. Write Ft = f −1(−∞, t] and Gt = g−1(−∞, t].
Then the sublevel set Bltrations F,G ∶ R→ Top are
δ-interleaved:
Ft Ft+δ Ft+2δ Ft+3δ
Gt Gt+δ Gt+2δ Gt+3δ
Applying homology (functor) preserves commutativity
persistent homology of f , g yields
δ-interleaved persistence modules R→ Vect
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 11 / 31
Geometric interleavings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 12 / 31
Geometric interleavings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 12 / 31
Geometric interleavings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 12 / 31
Geometric interleavings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 12 / 31
Geometric interleavings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 12 / 31
Geometric interleavings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 12 / 31
Persistence modules
A persistence module M is a diagram (functor) R→ Vect:
a vector space Mt for each t ∈ R a linear map mt
s ∶ Ms →Mt for each s ≤ t (transition maps)
respecting identity: mtt = idMt
and composition: mts ms
r = mtr
If each dimMt <∞, we say M is pointwise Bnite dimensional.
Amorphism f ∶ M → N is a natural transformation:
a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:
Ms Mt
Ns Nt
mts
fs ft
nts
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 13 / 31
Persistence modules
A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R
a linear map mts ∶ Ms →Mt for each s ≤ t (transition maps)
respecting identity: mtt = idMt
and composition: mts ms
r = mtr
If each dimMt <∞, we say M is pointwise Bnite dimensional.
Amorphism f ∶ M → N is a natural transformation:
a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:
Ms Mt
Ns Nt
mts
fs ft
nts
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 13 / 31
Persistence modules
A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R a linear map mt
s ∶ Ms →Mt for each s ≤ t (transition maps)
respecting identity: mtt = idMt
and composition: mts ms
r = mtr
If each dimMt <∞, we say M is pointwise Bnite dimensional.
Amorphism f ∶ M → N is a natural transformation:
a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:
Ms Mt
Ns Nt
mts
fs ft
nts
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 13 / 31
Persistence modules
A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R a linear map mt
s ∶ Ms →Mt for each s ≤ t (transition maps)
respecting identity: mtt = idMt
and composition: mts ms
r = mtr
If each dimMt <∞, we say M is pointwise Bnite dimensional.
Amorphism f ∶ M → N is a natural transformation:
a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:
Ms Mt
Ns Nt
mts
fs ft
nts
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 13 / 31
Persistence modules
A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R a linear map mt
s ∶ Ms →Mt for each s ≤ t (transition maps)
respecting identity: mtt = idMt
and composition: mts ms
r = mtr
If each dimMt <∞, we say M is pointwise Bnite dimensional.
Amorphism f ∶ M → N is a natural transformation:
a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:
Ms Mt
Ns Nt
mts
fs ft
nts
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 13 / 31
Persistence modules
A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R a linear map mt
s ∶ Ms →Mt for each s ≤ t (transition maps)
respecting identity: mtt = idMt
and composition: mts ms
r = mtr
If each dimMt <∞, we say M is pointwise Bnite dimensional.
Amorphism f ∶ M → N is a natural transformation:
a linear map ft ∶ Mt → Nt for each t ∈ R
morphism and transition maps commute:
Ms Mt
Ns Nt
mts
fs ft
nts
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 13 / 31
Persistence modules
A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R a linear map mt
s ∶ Ms →Mt for each s ≤ t (transition maps)
respecting identity: mtt = idMt
and composition: mts ms
r = mtr
If each dimMt <∞, we say M is pointwise Bnite dimensional.
Amorphism f ∶ M → N is a natural transformation:
a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:
Ms Mt
Ns Nt
mts
fs ft
nts
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 13 / 31
Interval PersistenceModules
Let K be a Beld. For an arbitrary interval I ⊆ R,
deBne the interval persistence module K(I) by
K(I)t =
⎧⎪⎪⎨⎪⎪⎩
K if t ∈ I,0 otherwise,
with transition maps of maximal rank.
Schematic example:
0 0 K K K 0 0
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 14 / 31
Barcodes: the structure of persistence modules
Theorem (Krull–Schmidt; Crawley-Boewey 2015)
Let M be a pointwise Bnite-dimensional persistence module.
Then M is interval-decomposable:
there exists a unique collection of intervals B(M) such that
M ≅ ⊕I∈B(M)
K(I).
B(M) is called the barcode of M.B(M)
The decomposition itself is not unique.
This is why we use homology with coe?cients in a Beld.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 15 / 31
Barcodes: the structure of persistence modules
Theorem (Krull–Schmidt; Crawley-Boewey 2015)
Let M be a pointwise Bnite-dimensional persistence module.
Then M is interval-decomposable:
there exists a unique collection of intervals B(M) such that
M ≅ ⊕I∈B(M)
K(I).
B(M) is called the barcode of M.B(M)
The decomposition itself is not unique.
This is why we use homology with coe?cients in a Beld.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 15 / 31
Barcodes: the structure of persistence modules
Theorem (Krull–Schmidt; Crawley-Boewey 2015)
Let M be a pointwise Bnite-dimensional persistence module.
Then M is interval-decomposable:
there exists a unique collection of intervals B(M) such that
M ≅ ⊕I∈B(M)
K(I).
B(M) is called the barcode of M.B(M)
The decomposition itself is not unique.
This is why we use homology with coe?cients in a Beld.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 15 / 31
Algebraic stability of persistence barcodes
Theorem (Chazal et al. 2009, 2012; B, Lesnick 2015)
If two persistence modules are δ-interleaved,
then there exists a δ-matching of their barcodes:
matched intervals have endpoints within distance ≤ δ,
unmatched intervals have length ≤ 2δ.
Mt Mt+δ Mt+2δ
Nt Nt+δ Nt+2δ
δ
2δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 16 / 31
Barcodes as diagrams
The matching category
Amatching σ ∶ S↛ T is a bijection S′ → T′, where S′ ⊆ S, T′ ⊆ T.
Composition of matchings σ ∶ S↛ T and τ ∶ T ↛ U :
Matchings form a categoryMch objects: sets
morphisms: matchings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 17 / 31
The matching category
Amatching σ ∶ S↛ T is a bijection S′ → T′, where S′ ⊆ S, T′ ⊆ T.
Composition of matchings σ ∶ S↛ T and τ ∶ T ↛ U :
Matchings form a categoryMch objects: sets
morphisms: matchings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 17 / 31
The matching category
Amatching σ ∶ S↛ T is a bijection S′ → T′, where S′ ⊆ S, T′ ⊆ T.
Composition of matchings σ ∶ S↛ T and τ ∶ T ↛ U :
Matchings form a categoryMch objects: sets
morphisms: matchings
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 17 / 31
Barcodes as matching diagrams
We can regard a barcode B as a functor R→Mch:
For each real number t, let Bt be the set of intervals of Bthat contain t, and
for each s ≤ t, deBne the matching Bs ↛ Bt
to be the identity on Bs ∩ Bt .
0.1 0.2 0.4 0.8∆
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 18 / 31
Barcodes as matching diagrams
We can regard a barcode B as a functor R→Mch:
For each real number t, let Bt be the set of intervals of Bthat contain t, and
for each s ≤ t, deBne the matching Bs ↛ Bt
to be the identity on Bs ∩ Bt .
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 18 / 31
Barcodes as matching diagrams
We can regard a barcode B as a functor R→Mch:
For each real number t, let Bt be the set of intervals of Bthat contain t, and
for each s ≤ t, deBne the matching Bs ↛ Bt
to be the identity on Bs ∩ Bt .
0.1 0.2 0.4 0.8δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 18 / 31
Stability via functoriality?
Ft Ft+2δ
Gt+δ Gt+3δ
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 19 / 31
Stability via functoriality?
H∗(Ft) H∗(Ft+2δ)
H∗(Gt+δ) H∗(Gt+3δ)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 19 / 31
Stability via functoriality?
B(H∗(Ft)) B(H∗(Ft+2δ))
B(H∗(Gt+δ)) B(H∗(Gt+3δ))
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 19 / 31
Stability via functoriality?
B(H∗(Ft)) B(H∗(Ft+2δ))
B(H∗(Gt+δ)) B(H∗(Gt+3δ))
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 19 / 31
Non-functoriality of the persistence barcode
Theorem (B, Lesnick 2015)
There exists no functorVectR →MchRsending each persistence
module to its barcode.
Proposition
There exists no functorVect→Mch sending each vector space of
dimension d to a set of cardinality d.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 20 / 31
Non-functoriality of the persistence barcode
Theorem (B, Lesnick 2015)
There exists no functorVectR →MchRsending each persistence
module to its barcode.
Proposition
There exists no functorVect→Mch sending each vector space of
dimension d to a set of cardinality d.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 20 / 31
Induced barcode matchings
Structure of persistence submodules / quotients
Proposition
Let f ∶ M → N be amonomorphism of persistence modules:
each ft ∶ Mt → Nt is injective.
Then f induces an injective map B(M)→ B(N)
mapping each I ∈ B(M) to some J ∈ B(N)
with larger or same left and same right endpoint.
Ms Mt
Ns Nt
B(N)
B(M)
Dually for epimorphisms (left and right exchanged).
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 21 / 31
Induced matchings
For a general morphism f ∶ M → N of persistence modules:
consider epi-mono factorization
M ↠ im f N .
im f N induces injection B(im f ) B(N)
M ↠ im f induces injection B(im f ) B(M)
compose to amatching B(M)↛ B(N):
B(M)
B(N)
B(im f )
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 22 / 31
Induced matchings
For a general morphism f ∶ M → N of persistence modules:
consider epi-mono factorization
M ↠ im f N .
im f N induces injection B(im f ) B(N)
M ↠ im f induces injection B(im f ) B(M)
compose to amatching B(M)↛ B(N):
B(M)
B(N)
B(im f )
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 22 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
δ
B(N•+δ)B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
δ
B(im f )B(N•+δ)
B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
δ
B(im f )B(im n•–δ,•+δ)
B(N•+δ)B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
δ
B(im f )B(im n•–δ,•+δ)
B(N•+δ)B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
δ
B(M)
B(im f )B(im n•–δ,•+δ)
B(N•+δ)B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Stability from interleavings
Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):
Mt
Nt−δ Nt+δ
ftgt−δ
nt−δ ,t+δ
imnt−δ,t+δ im ft Nt+δ
Mt ↠ im ft
δ
B(M)
B(im f )B(im n•–δ,•+δ)
B(N•+δ)B(N)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 23 / 31
Sending persistence into
Hilbert space
Extending the TDA pipeline
Mapping barcodes into a Hilbert space?
desirable for (kernel-based) machine learning methods
and statistics
stability (Lipschitz continuity): important for reliable
predictions
inverse stability (bi-Lipschitz): avoid loss of information
Existing kernels for persistence diagrams:
stability bounds only for 1−Wasserstein distance
Lipschitz constants depend on bound on number and
range of bars
no bi-Lipschitz bounds known
Can we hope for something better?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 24 / 31
Extending the TDA pipeline
Mapping barcodes into a Hilbert space?
desirable for (kernel-based) machine learning methods
and statistics
stability (Lipschitz continuity): important for reliable
predictions
inverse stability (bi-Lipschitz): avoid loss of information
Existing kernels for persistence diagrams:
stability bounds only for 1−Wasserstein distance
Lipschitz constants depend on bound on number and
range of bars
no bi-Lipschitz bounds known
Can we hope for something better?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 24 / 31
Extending the TDA pipeline
Mapping barcodes into a Hilbert space?
desirable for (kernel-based) machine learning methods
and statistics
stability (Lipschitz continuity): important for reliable
predictions
inverse stability (bi-Lipschitz): avoid loss of information
Existing kernels for persistence diagrams:
stability bounds only for 1−Wasserstein distance
Lipschitz constants depend on bound on number and
range of bars
no bi-Lipschitz bounds known
Can we hope for something better?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 24 / 31
No bi-Lipschitz feature maps for persistence
Theorem (B, Carrière 2018)
There is no bi-Lipschitzmap from the persistence diagrams
(with the interleaving or any p−Wasserstein distance)
into any Bnite-dimensionalHilbert space,
evenwhen restricting to bounded range or number of bars.
Theorem (B, Carrière 2018)
If therewas such a bi-Lipschitzmap into someHilbert space,
the ratio of the Lipschitz constantswould have to go to∞
togetherwith the bounds on number or range of bars.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 25 / 31
No bi-Lipschitz feature maps for persistence
Theorem (B, Carrière 2018)
There is no bi-Lipschitzmap from the persistence diagrams
(with the interleaving or any p−Wasserstein distance)
into any Bnite-dimensionalHilbert space,
evenwhen restricting to bounded range or number of bars.
Theorem (B, Carrière 2018)
If therewas such a bi-Lipschitzmap into someHilbert space,
the ratio of the Lipschitz constantswould have to go to∞
togetherwith the bounds on number or range of bars.
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 25 / 31
History
When was persistent homology invented?
[Edelsbrunner/Letscher/Zomorodian 2000]
[Robbins 1999]
[Frosini 1990]
[Leray 1946]?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 26 / 31
When was persistent homology invented?
[Edelsbrunner/Letscher/Zomorodian 2000]
[Robbins 1999]
[Frosini 1990]
[Leray 1946]?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 26 / 31
When was persistent homology invented?
[Edelsbrunner/Letscher/Zomorodian 2000]
[Robbins 1999]
[Frosini 1990]
[Leray 1946]?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 26 / 31
When was persistent homology invented?
[Edelsbrunner/Letscher/Zomorodian 2000]
[Robbins 1999]
[Frosini 1990]
[Leray 1946]?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 26 / 31
When was persistent homology invented Brst?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 27 / 31
When was persistent homology invented Brst?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 27 / 31
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JL Kelley, E Pitcher Annals of Mathematics, 1947 JSTOR
The developments of this paper stem from the attempts of one of the authors to deduce relations between homology groups of a complex and homology groups of a complex which is its image under a simplicial map. Certain relations were deduced (see [EP 1] and [EP 2] ...
Cited by 46 Related articles All 3 versions Cite Save More
Exact homomorphism sequences in homology theory
R Bott Bulletin of the American Mathematical Society, 1980 ams.org
American Mathematical Society. Thus Morse grew to maturity just at the time when the subject of Analysis Situs was being shaped by such masters2 as Poincaré, Veblen, LEJ Brouwer, GD Birkhoff, Lefschetz and Alexander, and it was Morse's genius and destiny to ...
Cited by 24 Related articles All 4 versions Cite Save More
Marston Morse and his mathematical works
M Morse, CB Tompkins Duke Math. J, 1941 projecteuclid.org
1. Introduction. We are concerned with extending the calculus of variations in the large to multiple integrals. Theproblem of the existenceof minimal surfaces of unstable type contains many of the typical difficulties, especially those ofa topological nature. Having studied this ...
Cited by 19 Related articles All 2 versions Cite Save
Unstable minimal surfaces of higher topological structure
U Bauer 2011 Citeseer
The goal of this thesis is to bring together two different theories about critical points of a scalar function and their relation to topology: Discrete Morse theory and Persistent homology. While the goals and fundamental techniques are different, there are certain ...
Cited by 8 Related articles All 4 versions Cite Save More
[PDF] Persistence in discrete Morse theory
MR Hestenes American Journal of Mathematics, 1945 JSTOR
1. Introduction. Let f (x)=(. z,**, xn) be a function defined on a neighborhood of a closed set S in euclidean nspace. Suppose that the points of S at which f, 0 0 are interior to S and that the determinant I is different from zero at these points. A point at which f, 0 is called a ...
Cited by 3 Related articles All 2 versions Cite Save
A theory of critical points
DG Bourgin Annali di Matematica Pura ed Applicata, 1955 Springer
Summary The properties of direct limits of Cech homology groups for a non cofinal collection of compact pairs are explicitly stated with a view towards applications. The usual axioms including full excision are satisfied under obvious restrictions. Some of the Morse rank and ...
Cited by 2 Related articles All 2 versions Cite Save
Arrays of compact pairs
M Morse, M Heins Annals of Mathematics, 1946 JSTOR
Paper II of this series* starts with a theorem relating the number of logarithmic poles and essential critical points of a harmonic or pseudoharmonic function F (x, y). Cf.? 2. II. The function F was defined on the closure G of a finite region bounded by v Jordan curves.
Cited by 2 Related articles All 2 versions Cite Save
Topological Methods in the Theory of Functions of a Single Complex Variable: III. CasualIsomorphisms in the Theory of PseudoHarmonic Functions
D Pope Pacific J. Math, 1962 msp.org
Introduction^ A basic feature of most of the methods used for the numerical calculation of a variational problem is the reduction of the infinite dimensional problem to a finite dimensional problem by some kind of approximation. One of the most natural ...
Cited by 1 Related articles All 3 versions Cite Save More
[PDF] On the approximation of function spaces in the calculus of variations
M Ozawa Kodai Mathematical Seminar Reports, 1958 jlc.jst.go.jp
Let W and W be two closed Riemann surfaces of the same genus g Ξ> 2. Let § be a preassigned homotopy classs of topological mappings T: W+ Wf and ξ)(q) be a subclass of ξ), each member of which carries a given fixed point J) o on W to a point q on W. Let T (qp) ...
Related articles All 4 versions Cite Save
On extremal quasiconformal mappings
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My Citations
ed.ac.uk [PDF]
ams.org [PDF]
psu.edu [PDF]
msp.org [PDF]
projecteuclid.org [PDF]
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 27 / 31
When was persistent homology invented Brst?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 27 / 31
When was persistent homology invented Brst?
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 27 / 31
Morse’s functional topology
Key aspects:
early precursor of persistence and spectral sequences
uses Vietoris homology with Beld coe?cients
applies to a broad class of functions on metric spaces
(not necessarily continuous)
inclusions of sublevel sets have Bnite rank homology
(q-tame persistent homology)
focus on controlled behavior in pathological cases
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 28 / 31
Motivation and application: minimal surfaces
(from Dierkes et al.: Minimal Surfaces, Springer 2010)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 29 / 31
Motivation and application: minimal surfaces
(from Dierkes et al.: Minimal Surfaces, Springer 2010)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 29 / 31
Motivation and application: minimal surfaces
(from Dierkes et al.: Minimal Surfaces, Springer 2010)
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 29 / 31
Existence of unstable minimal surfaces
Using persistent homology:
Number of є-persistent critical points (minimal surfaces)
is Bnite for any є > 0 Morse inequalities for є-persistent critical points
Theorem (Morse, Tompkins 1939)
There is aC1 curve bounding an unstable minimal surface
(an index 1 critical point of the area functional).
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 30 / 31
Thanks for your attention!
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 31 / 31
Thanks for your attention!
http://ulrich-bauer.org/persistence-talk-leipzig.pdf 31 / 31