algebraic data structures for topological summaries [-1.5ex]ezra/talks/... · fly wings biology...
TRANSCRIPT
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Algebraic data structures for topological summaries
Ezra Miller
Duke University, Department of Mathematics
and Department of Statistical Science
ongoing mathematics with Justin Curry (Duke postdoc)
Ashleigh Thomas (Duke grad)
Surabhi Beriwal (Duke undergrad)
and biology with David Houle (Florida State, Biology)
Mathematical Congress of the Americas
Montréal, Quebeca, Canada
27 July 2017
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Outline
1. Fly wings
2. Biological background
3. Persistent homology
4. Poset modules
5. Encoding
6. Fringe presentation
7. Decomposition
8. Future directions
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fruit fly wings
Normal fly wings [images from David Houle’s lab]:
1
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fruit fly wings
Normal fly wings [images from David Houle’s lab]:
Topologically abnormal veins:
1
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fruit fly wings
photographic image
2
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fruit fly wings
spline
2
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Biological background
What generates topological novelty?[Houle, et al.]: selecting for certain continuous wing vein deformations yields
• skew toward more oddly shaped wings, but also• much higher rate of topological novelty
Hypothesis. Topological novelty arises when directional selection pushescontinuous variation in a developmental program beyond a certain threshold.
Test the hypothesis• "plot" wings in "form space"• determine whether topological variants lie "in the direction of" continuous
shape selected for, and at the extreme in that direction
Goal. Statistical analysis encompassing topological vein variation, givingappropriate weight to new singular points in addition to varying shape
• compare phenotypic distance to genotypic distance; needs• metric specifying distance between topologically distinct wings
To proceed. Statistics with fly wings as data objects statistics withmultiparameter persistence diagrams as data objects
Need. Data structures, algorithms, theoretical guarantees3
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Example: filling brains [w/Bendich, Marron, Pieloch, Skwerer]
5
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4’’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4’’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Persistent homology
Topological space X• Fixed X homology HiX for each dimension i
• Build X step by step: measure evolving topology
Def. Let X•
be a filtered space, meaning ∅ = X0 ⊂ X1 ⊂ · · · ⊂ Xm = X .The persistent homology HiX• is HiX1 → HiX2 → · · · → HiXm, a sequence ofvector space homomorphisms.
Examples
1. Given a function f : X → R, let Xt = f−1
((−∞, t ]
). Choose t0, . . . , tm ∈ R
so HiXt changes from tk to tk+1
2. Any simplicial complex: build it simplex by simplex in some order
History. invented by [Frosini, Landi 1999], [Robins 1999];[Edelsbrunner, Letscher, Zomorodian 2002]: includes efficient computation;
[Carlsson, Zomorodian 2009]: multiparameter persistence;
[Cagliari, Harer, Knudson, Mukherjee,. . . ]: developments in theory, applications
4’’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set
• 2nd parameter: distance from edge set
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set given as points in R2
• 2nd parameter: distance from edge set given as Bézier curves
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set given as points in R2
• 2nd parameter: distance from edge set given as Bézier curves
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set given as points in R2
• 2nd parameter: distance from edge set given as Bézier curves
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set given as points in R2
• 2nd parameter: distance from edge set given as Bézier curves
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set given as points in R2
• 2nd parameter: distance from edge set given as Bézier curves
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set given as points in R2
• 2nd parameter: distance from edge set given as Bézier curves
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set (require distance ≥ −r )
• 2nd parameter: distance from edge set
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set (require distance ≥ −r )
• 2nd parameter: distance from edge set (require distance ≤ s)
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set (require distance ≥ −r )
• 2nd parameter: distance from edge set (require distance ≤ s)
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set (require distance ≥ −r )
• 2nd parameter: distance from edge set (require distance ≤ s)
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set (require distance ≥ −r )
• 2nd parameter: distance from edge set (require distance ≤ s)
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set (require distance ≥ −r )
• 2nd parameter: distance from edge set (require distance ≤ s)
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Multiparameter persistence
Plan. (with Houle, Curry, Thomas, +. . . ) Encode with 2-parameter persistence
• 1st parameter: distance from vertex set (require distance ≥ −r )
• 2nd parameter: distance from edge set (require distance ≤ s)
Sublevel set Wr ,s is near edges but far from vertices
Z2-module:
↑ ↑ ↑
→ Hr−ε,s+δ → Hr,s+δ → Hr+ε,s+δ →↑ ↑ ↑
→ Hr−ε,s → Hr,s → Hr+ε,s →↑ ↑ ↑
→ Hr−ε,s−δ → Hr ,s−δ → Hr+ε,s−δ →↑ ↑ ↑
6
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Examples
A piece of fly wing vein The (r , s)-plane R2
Observations
• stratification alters persistence module
• discretization approximates algebraic
7
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Modules over posets
Filter X by poset Q of subspaces: Xq ⊆ X for q ∈ Q ⇒ persistent homology is a
Def. Q-module:
• Q-graded vector space H =⊕
q∈Q Hq with
• homomorphism Hq → Hq′ whenever q � q′ in Q such that
• Hq → Hq′′ equals the composite Hq → Hq′ → Hq′′ whenever q � q′ � q′′
Examples
• brain arteries: Q = {0, . . . ,m}
• brain arteries: Q = R
• wing veins: Q = Z2
• wing veins: Q = R2
• multifiltration Q = Rn n real filtrations of any topological space
• Q = Zn implies H = Zn-graded k[x1, . . . , xn]-module
– standard combinatorial commutative algebra
– some proofs proceed by reducing to this case
8’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding persistence modules
An R2-module finitely encoded
10
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding persistence modules
An R2-module finitely encoded
10
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding persistence modules
An R2-module finitely encoded
10
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding persistence modules
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding persistence modules
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Encoding persistence modules
Finiteness conditions: • Zn-modules: finitely generated⇔ noetherian• Rn-modules from data analysis↔ ??
Def. H induces isotypic regions in Q: equivalence classes for equivalencerelation generated by a ∼ b whenever • a � b in Q
and • Ha∼−→ Hb.
H is tame if dimk Hq
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u
• downset D ⊆ Q if D =⋃
d∈D Q�d
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Examples• In R2:
U = and = D
• In R3:
U =
semialgebraic
or
piecewise linear
= D
[Andrei Okounkov, Limit shapes, real and imagined, Bulletin of the AMS 53 (2016), no. 2, 187–216]
12
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Examples• In R2:
U = and = D
• In R3:
U =
semialgebraic
or
piecewise linear
= D
[Andrei Okounkov, Limit shapes, real and imagined, Bulletin of the AMS 53 (2016), no. 2, 187–216]
12
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u
• downset D ⊆ Q if D =⋃
d∈D Q�d
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u
• downset D ⊆ Q if D =⋃
d∈D Q�d
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
birth upsets →
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
← death downsets
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
birth upsets →
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
← death downsets
← scalar entries
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
birth upsets →
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
← death downsets
← scalar entries
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
birth upsets →
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
← death downsets
← scalar entries
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
ϕ11
has image k
as long as ϕ11 6= 0.
12’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
ϕ11
has image k
as long as ϕ11 6= 0.
12’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
birth upsets →
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
← death downsets
← scalar entries
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
birth upsets →
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
← death downsets
← scalar entries
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Fringe presentation
Def. Fix a poset Q.• upset U ⊆ Q if U =
⋃
u∈U Q�u k[U] ⊆ k[Q]
• downset D ⊆ Q if D =⋃
d∈D Q�d k[Q]։ k[D]
For any subset S ⊆ Q, set k[S] =⊕
s∈S ks.
Def [w/Curry & Thomas]. A fringe presentation of H is a monomial matrix
birth upsets →
U1...
Uk
D1 · · · Dℓϕ11 · · · ϕ1ℓ
.... . .
...
ϕk1 · · · ϕkℓ
← death downsets
← scalar entries
k[U1]⊕ · · · ⊕ k[Uk ] = F −−−−−−−−−−−−−−−−→ E = k[D1]⊕ · · · ⊕ k[Dℓ]
with image(F → E) ∼= H.
Compare. [Chachólski, Patriarca, Scolamiero, Vaccarino] “monomial presentation”but fringe presentation is new even for finitely generated Z2-modules
Thm [w/Curry & Thomas]. finitely encoded⇔ admits fringe presentation
11’’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Decomposition
Thm [w/Curry & Thomas]. Finitely encoded Rn-modules admit minimalprimary decomopsition: H =
⋂
faces τ Hτ with socτH−→∼
socτHτ
Problem. Encoding, fringe presentation, primary decomposition not canonicalexcept for downsets (compare: Zn-graded modules vs. monomial ideals)
Solution. Births and deaths are functorial QR code• generators ( = births) of each type and shape• cogenerators ( = socle elements = deaths) of each type and shape
– type: infinite vs. bounded directions: primary decomposition
– shape: tangent cone topology of corresponding upset or downset
• module structure on H ↔ linear map: births→ deaths
Finiteness / semialgebraic properties / computability from finite encoding
Goal. Statistics with rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)[Carlsson, Zomorodian 2009]
Proposal. fringe presentation framework to compute rank functions• data structure (level sets of rank function are semialgebraic)
• algorithms (semialgebraic geometry, computational commutative algebra)13
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Future directions
Algebra• QR codes: module-theoretic topologies on generators and cogenerators
• Syzygy conj: Rn-modules have indicator-homological dimension ≤ n
Computation• calculate encoding / fringe presentation from vertices and Bézier curves
• homological algebra with semialgebraic indicator modules
• data structures for rank functions; computation for statistics
Statistics• geometric probability/statistics on stratified spaces
• No-moduli conj: rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)
determines H0 and Hn−1modules for objects in Rn
• Local statistical sufficiency conj: Different fly wings yield nonisomorphic
modules locally; i.e., deformation of fly wing splines⇒ QR code changes
Topology. Biparameter persistence as persistent intersection homology
Biology. The topological novelty hypothesis
14
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Future directions
Algebra• QR codes: module-theoretic topologies on generators and cogenerators
• Syzygy conj: Rn-modules have indicator-homological dimension ≤ n
Computation• calculate encoding / fringe presentation from vertices and Bézier curves
• homological algebra with semialgebraic indicator modules
• data structures for rank functions; computation for statistics
Statistics• geometric probability/statistics on stratified spaces
• No-moduli conj: rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)
determines H0 and Hn−1modules for objects in Rn
• Local statistical sufficiency conj: Different fly wings yield nonisomorphic
modules locally; i.e., deformation of fly wing splines⇒ QR code changes
Topology. Biparameter persistence as persistent intersection homology
Biology. The topological novelty hypothesis
14
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Future directions
Algebra• QR codes: module-theoretic topologies on generators and cogenerators
• Syzygy conj: Rn-modules have indicator-homological dimension ≤ n
Computation• calculate encoding / fringe presentation from vertices and Bézier curves
• homological algebra with semialgebraic indicator modules
• data structures for rank functions; computation for statistics
Statistics• geometric probability/statistics on stratified spaces
• No-moduli conj: rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)
determines H0 and Hn−1modules for objects in Rn
• Local statistical sufficiency conj: Different fly wings yield nonisomorphic
modules locally; i.e., deformation of fly wing splines⇒ QR code changes
Topology. Biparameter persistence as persistent intersection homology
Biology. The topological novelty hypothesis
14
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Future directions
Algebra• QR codes: module-theoretic topologies on generators and cogenerators
• Syzygy conj: Rn-modules have indicator-homological dimension ≤ n
Computation• calculate encoding / fringe presentation from vertices and Bézier curves
• homological algebra with semialgebraic indicator modules
• data structures for rank functions; computation for statistics
Statistics• geometric probability/statistics on stratified spaces
• No-moduli conj: rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)
determines H0 and Hn−1modules for objects in Rn
• Local statistical sufficiency conj: Different fly wings yield nonisomorphic
modules locally; i.e., deformation of fly wing splines⇒ QR code changes
Topology. Biparameter persistence as persistent intersection homology
Biology. The topological novelty hypothesis
14
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Future directions
Algebra• QR codes: module-theoretic topologies on generators and cogenerators
• Syzygy conj: Rn-modules have indicator-homological dimension ≤ n
Computation• calculate encoding / fringe presentation from vertices and Bézier curves
• homological algebra with semialgebraic indicator modules
• data structures for rank functions; computation for statistics
Statistics• geometric probability/statistics on stratified spaces
• No-moduli conj: rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)
determines H0 and Hn−1modules for objects in Rn
• Local statistical sufficiency conj: Different fly wings yield nonisomorphic
modules locally; i.e., deformation of fly wing splines⇒ QR code changes
Topology. Biparameter persistence as persistent intersection homology
Biology. The topological novelty hypothesis
14
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Future directions
Algebra• QR codes: module-theoretic topologies on generators and cogenerators
• Syzygy conj: Rn-modules have indicator-homological dimension ≤ n
Computation• calculate encoding / fringe presentation from vertices and Bézier curves
• homological algebra with semialgebraic indicator modules
• data structures for rank functions; computation for statistics
Statistics• geometric probability/statistics on stratified spaces
• No-moduli conj: rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)
determines H0 and Hn−1modules for objects in Rn
• Local statistical sufficiency conj: Different fly wings yield nonisomorphic
modules locally; i.e., deformation of fly wing splines⇒ QR code changes
Topology. Biparameter persistence as persistent intersection homology
Biology. The topological novelty hypothesis
Thank You14
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Future directions
Algebra• QR codes: module-theoretic topologies on generators and cogenerators
• Syzygy conj: Rn-modules have indicator-homological dimension ≤ n
Computation• calculate encoding / fringe presentation from vertices and Bézier curves
• homological algebra with semialgebraic indicator modules
• data structures for rank functions; computation for statistics
Statistics• geometric probability/statistics on stratified spaces
• No-moduli conj: rank function Rn × Rn → N(a � b) 7→ rank(Ha → Hb)
determines H0 and Hn−1modules for objects in Rn
• Local statistical sufficiency conj: Different fly wings yield nonisomorphic
modules locally; i.e., deformation of fly wing splines⇒ QR code changes
Topology. Biparameter persistence as persistent intersection homology
Biology. The topological novelty hypothesis
Thank You14
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Topology of probability distributions
Approximation by bandwidth r expansion of size n sample• geometry: sample Sn from space M ⇒ M ≈ Br (S) =
⋃
x∈S Br (x)
• statistics: sample µn from measure µ⇒ µ ≈ Ur (µn) =1n
∑
x∈S1
vol(r)Ur (x)
Ur (x) = uniform measure on Br (x) and vol(r) = volume of ball of radius rIn general: Ur Kr arbitrary kernel and µn ν arbitrary measure
Def. µ has density F ⇒ Br (µ) has density Kr ∗ F (x) =∫
MKr (y − x)dµ(y).
Def. support at sensitivity s:• ν has density F ⇒ ν≥1/s =
{x ∈ M | F (x) ≥ 1
s
}.
• expansion of µ to bandwidth r and sensitivity s is Br (µ)≥1/s ⊆ M.
Note:{
Br (µ)≥rd/s | r ∈ R≥0 and s ∈ R≥1}⊆ M nested as r and s increase
Def. µ has i th bipersistent homology H rsi (µ) = Hi(Br (µ)≥rd/s
), an invariant of µ
Conj. If µ is stratified—a mixture of smooth measures supported on the strataof a Whitney stratification—then its bipersistent homology is finitely encoded.
algebra, geometry, combinatorics of H rs∗ (µ)↔ statistics of µ.
15
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Topology of probability distributions
Approximation by bandwidth r expansion of size n sample• geometry: sample Sn from space M ⇒ M ≈ Br (S) =
⋃
x∈S Br (x)
• statistics: sample µn from measure µ⇒ µ ≈ Ur (µn) =1n
∑
x∈S1
vol(r)Ur (x)
Ur (x) = uniform measure on Br (x) and vol(r) = volume of ball of radius rIn general: Ur Kr arbitrary kernel and µn ν arbitrary measure
Def. µ has density F ⇒ Br (µ) has density Kr ∗ F (x) =∫
MKr (y − x)dµ(y).
Def. support at sensitivity s:• ν has density F ⇒ ν≥1/s =
{x ∈ M | F (x) ≥ 1
s
}.
• expansion of µ to bandwidth r and sensitivity s is Br (µ)≥1/s ⊆ M.
Note:{
Br (µ)≥rd/s | r ∈ R≥0 and s ∈ R≥1}⊆ M nested as r and s increase
Def. µ has i th bipersistent homology H rsi (µ) = Hi(Br (µ)≥rd/s
), an invariant of µ
Conj. If µ is stratified—a mixture of smooth measures supported on the strataof a Whitney stratification—then its bipersistent homology is finitely encoded.
algebra, geometry, combinatorics of H rs∗ (µ)↔ statistics of µ.
15
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Bandwidth r expansion / kernel density estimation
images taken from Confidence sets for persistence diagrams,by Fasy, Lecci, Rinaldo, Wasserman, Balakrishnan, Singh,
Annals of Statistics 42 (2014), no. 6, 2301–2339.
16
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Topology of probability distributions
Approximation by bandwidth r expansion of size n sample• geometry: sample Sn from space M ⇒ M ≈ Br (S) =
⋃
x∈S Br (x)
• statistics: sample µn from measure µ⇒ µ ≈ Ur (µn) =1n
∑
x∈S1
vol(r)Ur (x)
Ur (x) = uniform measure on Br (x) and vol(r) = volume of ball of radius rIn general: Ur Kr arbitrary kernel and µn ν arbitrary measure
Def. µ has density F ⇒ Br (µ) has density Kr ∗ F (x) =∫
MKr (y − x)dµ(y).
Def. support at sensitivity s:• ν has density F ⇒ ν≥1/s =
{x ∈ M | F (x) ≥ 1
s
}.
• expansion of µ to bandwidth r and sensitivity s is Br (µ)≥1/s ⊆ M.
Note:{
Br (µ)≥rd/s | r ∈ R≥0 and s ∈ R≥1}⊆ M nested as r and s increase
Def. µ has i th bipersistent homology H rsi (µ) = Hi(Br (µ)≥rd/s
), an invariant of µ
Conj. If µ is stratified—a mixture of smooth measures supported on the strataof a Whitney stratification—then its bipersistent homology is finitely encoded.
algebra, geometry, combinatorics of H rs∗ (µ)↔ statistics of µ.
15’
-
Fly wings Biology Persistence Poset modules Encoding Fringe presentation Decomposition Future
Topology of probability distributions
Approximation by bandwidth r expansion of size n sample• geometry: sample Sn from space M ⇒ M ≈ Br (S) =
⋃
x∈S Br (x)
• statistics: sample µn from measure µ⇒ µ ≈ Ur (µn) =1n
∑
x∈S1
vol(r)Ur (x)
Ur (x) = uniform measure on Br (x) and vol(r) = volume of ball of radius rIn general: Ur Kr arbitrary kernel and µn ν arbitrary measure
Def. µ has density F ⇒ Br (µ) has density Kr ∗ F (x) =∫
MKr (y − x)dµ(y).
Def. support at sensitivity s:• ν has density F ⇒ ν≥1/s =
{x ∈ M | F (x) ≥ 1
s
}.
• expansion of µ to bandwidth r and sensitivity s is Br (µ)≥1/s ⊆ M.
Note:{
Br (µ)≥rd/s | r ∈ R≥0 and s ∈ R≥1}⊆ M nested as r and s increase
Def. µ