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Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Topology of Random Cell Complexes
Ben Schweinhart
Princeton University
December 10, 2014
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Examples
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Open Cell Foams
Interesting topology
Very important material in practice, not well understood
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Grain Growth
Show movie
Steady state for which scale-free properties haveconverged? Dependence on initial conditions?
Need metric - a nice one is given by considering the localtopology.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Grain Growth
Show movie
Steady state for which scale-free properties haveconverged? Dependence on initial conditions?
Need metric - a nice one is given by considering the localtopology.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Notes on Topology
These systems have interesting topology, but they havenot yet been studied using topological methods.*
Crystallography doesn’t apply to disordered materials.
Plan: First, I will introduce a new, general method toquantify the topology of cell complexes.
Then I will talk about a case study, and discusscomputational results - strong evidence of existence of asteady state.
If I have time, I will briefly describe two other topologicalmethods.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Notes on Topology
These systems have interesting topology, but they havenot yet been studied using topological methods.*
Crystallography doesn’t apply to disordered materials.
Plan: First, I will introduce a new, general method toquantify the topology of cell complexes.
Then I will talk about a case study, and discusscomputational results - strong evidence of existence of asteady state.
If I have time, I will briefly describe two other topologicalmethods.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Notes on Topology
These systems have interesting topology, but they havenot yet been studied using topological methods.*
Crystallography doesn’t apply to disordered materials.
Plan: First, I will introduce a new, general method toquantify the topology of cell complexes.
Then I will talk about a case study, and discusscomputational results - strong evidence of existence of asteady state.
If I have time, I will briefly describe two other topologicalmethods.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Notes on Topology
These systems have interesting topology, but they havenot yet been studied using topological methods.*
Crystallography doesn’t apply to disordered materials.
Plan: First, I will introduce a new, general method toquantify the topology of cell complexes.
Then I will talk about a case study, and discusscomputational results - strong evidence of existence of asteady state.
If I have time, I will briefly describe two other topologicalmethods.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Notes on Topology
These systems have interesting topology, but they havenot yet been studied using topological methods.*
Crystallography doesn’t apply to disordered materials.
Plan: First, I will introduce a new, general method toquantify the topology of cell complexes.
Then I will talk about a case study, and discusscomputational results - strong evidence of existence of asteady state.
If I have time, I will briefly describe two other topologicalmethods.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Swatches
Key Idea
Quantify the local topology of cell complexes by usingprobability distributions of local configurations.
Used to define a distance on cell complexes. The distancecan be used to, e.g., quantify the variability of cellcomplexes generated in a particular way, compare two cellcomplexes (i.e. experimental results with simulations), ortest convergence to a steady state
Applicable to many different physical systems, computable.
Joint work with Jeremy Mason (Bogazici University) andBob MacPherson (Institute for Advanced Study)
Paper on the arxiv: Topological Similarity of Random CellComplexes, and Applications to Dislocation Networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Swatches
Key Idea
Quantify the local topology of cell complexes by usingprobability distributions of local configurations.
Used to define a distance on cell complexes. The distancecan be used to, e.g., quantify the variability of cellcomplexes generated in a particular way, compare two cellcomplexes (i.e. experimental results with simulations), ortest convergence to a steady state
Applicable to many different physical systems, computable.
Joint work with Jeremy Mason (Bogazici University) andBob MacPherson (Institute for Advanced Study)
Paper on the arxiv: Topological Similarity of Random CellComplexes, and Applications to Dislocation Networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Swatches
Key Idea
Quantify the local topology of cell complexes by usingprobability distributions of local configurations.
Used to define a distance on cell complexes. The distancecan be used to, e.g., quantify the variability of cellcomplexes generated in a particular way, compare two cellcomplexes (i.e. experimental results with simulations), ortest convergence to a steady state
Applicable to many different physical systems, computable.
Joint work with Jeremy Mason (Bogazici University) andBob MacPherson (Institute for Advanced Study)
Paper on the arxiv: Topological Similarity of Random CellComplexes, and Applications to Dislocation Networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Swatches
Key Idea
Quantify the local topology of cell complexes by usingprobability distributions of local configurations.
Used to define a distance on cell complexes. The distancecan be used to, e.g., quantify the variability of cellcomplexes generated in a particular way, compare two cellcomplexes (i.e. experimental results with simulations), ortest convergence to a steady state
Applicable to many different physical systems, computable.
Joint work with Jeremy Mason (Bogazici University) andBob MacPherson (Institute for Advanced Study)
Paper on the arxiv: Topological Similarity of Random CellComplexes, and Applications to Dislocation Networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Swatches
Key Idea
Quantify the local topology of cell complexes by usingprobability distributions of local configurations.
Used to define a distance on cell complexes. The distancecan be used to, e.g., quantify the variability of cellcomplexes generated in a particular way, compare two cellcomplexes (i.e. experimental results with simulations), ortest convergence to a steady state
Applicable to many different physical systems, computable.
Joint work with Jeremy Mason (Bogazici University) andBob MacPherson (Institute for Advanced Study)
Paper on the arxiv: Topological Similarity of Random CellComplexes, and Applications to Dislocation Networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Regular Cell Complexes
Definition
A Regular Cell Complex is a space built inductively byattaching cells in each dimension (points, line segments, disks,etc). Each cell is glued on by homeomorphisms from theirboundaries into the existing structure.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Graph Representation
Represent a regular cell complex by the adjacency graph ofthe cells, labeled by dimension.
Captures all topological properties.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Graph Representation
Represent a regular cell complex by the adjacency graph ofthe cells, labeled by dimension.
Captures all topological properties.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Swatches
Definition
The swatch at vertex v of radius r is the neighborhood of v ofradius r in the graph distance (every edge has length one).
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Swatch Types
Two swatches have the same swatch type if they representthe same topological configuration.
Sub-swatch of a swatch: swatch of smaller radius at thesame root (central vertex).
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Swatch Types
Two swatches have the same swatch type if they representthe same topological configuration.
Sub-swatch of a swatch: swatch of smaller radius at thesame root (central vertex).
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Cloth
For each r , consider the probability distribution of swatchtypes of radius r at the vertices of a cell complex.
This family of probability distributions is called the clothof the cell complex.
Captures all local topological properties of the cellcomplex.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Cloth
For each r , consider the probability distribution of swatchtypes of radius r at the vertices of a cell complex.
This family of probability distributions is called the clothof the cell complex.
Captures all local topological properties of the cellcomplex.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Cloth
For each r , consider the probability distribution of swatchtypes of radius r at the vertices of a cell complex.
This family of probability distributions is called the clothof the cell complex.
Captures all local topological properties of the cellcomplex.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Tree of Swatches
Connect swatch type of radius r with all subswatch typesof radius r − 1.
The cloth is a weighting on this tree, subject toconsistency conditions.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Tree of Swatches
Connect swatch type of radius r with all subswatch typesof radius r − 1.
The cloth is a weighting on this tree, subject toconsistency conditions.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Distance on Swatches
Distance between two swatches = one over order (# cells)of largest common subswatch, or 0 if they are equal.
d(S1,S2) = 113
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Distance on Swatches
Distance between two swatches = one over order (# cells)of largest common subswatch, or 0 if they are equal.
d(S1,S2) = 113
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Distance on Cell Complexes
Definition
If C1 and C2 are cell complexes, dr (C1,C2) is the EarthMover’s Distance between the probability distributions atradius r induced by distance on swatches.
Earth Mover’s Distance = the infimum of the costs oftransformations between the two probability distributionson swatch types of radius r .
Cost = amount of probability mass moved between swatchtypes, weighted by swatch distance.
Distance can be used to compare different structures, testconvergence to steady state, iteratively modify structuresto reach a desired state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Distance on Cell Complexes
Definition
If C1 and C2 are cell complexes, dr (C1,C2) is the EarthMover’s Distance between the probability distributions atradius r induced by distance on swatches.
Earth Mover’s Distance = the infimum of the costs oftransformations between the two probability distributionson swatch types of radius r .
Cost = amount of probability mass moved between swatchtypes, weighted by swatch distance.
Distance can be used to compare different structures, testconvergence to steady state, iteratively modify structuresto reach a desired state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Distance on Cell Complexes
Definition
If C1 and C2 are cell complexes, dr (C1,C2) is the EarthMover’s Distance between the probability distributions atradius r induced by distance on swatches.
Earth Mover’s Distance = the infimum of the costs oftransformations between the two probability distributionson swatch types of radius r .
Cost = amount of probability mass moved between swatchtypes, weighted by swatch distance.
Distance can be used to compare different structures, testconvergence to steady state, iteratively modify structuresto reach a desired state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Distance on Cell Complexes
Definition
If C1 and C2 are cell complexes, dr (C1,C2) is the EarthMover’s Distance between the probability distributions atradius r induced by distance on swatches.
Earth Mover’s Distance = the infimum of the costs oftransformations between the two probability distributionson swatch types of radius r .
Cost = amount of probability mass moved between swatchtypes, weighted by swatch distance.
Distance can be used to compare different structures, testconvergence to steady state, iteratively modify structuresto reach a desired state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence of Cell Complexes
Limit distance= d(C1,C2) = limr→∞ dr (C1,C2).
A sequence {Ci} of cell complexes converges if it is aCauchy sequence in d .
Equivalent to all swatch frequencies converging.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence of Cell Complexes
Limit distance= d(C1,C2) = limr→∞ dr (C1,C2).
A sequence {Ci} of cell complexes converges if it is aCauchy sequence in d .
Equivalent to all swatch frequencies converging.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence of Cell Complexes
Limit distance= d(C1,C2) = limr→∞ dr (C1,C2).
A sequence {Ci} of cell complexes converges if it is aCauchy sequence in d .
Equivalent to all swatch frequencies converging.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Limit
Theorem
(Benjamini-Schramm Graph Limit) A convergent sequence ofcell complexes gives rise to a limit distribution on the space ofcountable, connected cell complexes a root specified. Thedistance can be extended to the space of these distributions.
Also implies convergence of all local topological properties:those that can be defined in terms of maps from a fixedlabeled graph H into the diagram of Ci .
Reference: Large Networks and Graph Limits
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Limit
Theorem
(Benjamini-Schramm Graph Limit) A convergent sequence ofcell complexes gives rise to a limit distribution on the space ofcountable, connected cell complexes a root specified. Thedistance can be extended to the space of these distributions.
Also implies convergence of all local topological properties:those that can be defined in terms of maps from a fixedlabeled graph H into the diagram of Ci .
Reference: Large Networks and Graph Limits
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Limit
Theorem
(Benjamini-Schramm Graph Limit) A convergent sequence ofcell complexes gives rise to a limit distribution on the space ofcountable, connected cell complexes a root specified. Thedistance can be extended to the space of these distributions.
Also implies convergence of all local topological properties:those that can be defined in terms of maps from a fixedlabeled graph H into the diagram of Ci .
Reference: Large Networks and Graph Limits
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Track distance dr to candidate steady state
Discuss other applications of topology to the case study.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Track distance dr to candidate steady state
Discuss other applications of topology to the case study.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Track distance dr to candidate steady state
Discuss other applications of topology to the case study.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Track distance dr to candidate steady state
Discuss other applications of topology to the case study.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Track distance dr to candidate steady state
Discuss other applications of topology to the case study.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Model
Curvature-driven evolution of polygonal curves in T3.
The polygonal curves meet each other at vertices ofdegree three.
The curves are composed of line segments meeting atnodes.
Evolves by energy minimization, assuming constant energyγ per unit length.
This can be viewed as a very simple model of a dislocationnetwork in the process of recovery.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Model
Curvature-driven evolution of polygonal curves in T3.
The polygonal curves meet each other at vertices ofdegree three.
The curves are composed of line segments meeting atnodes.
Evolves by energy minimization, assuming constant energyγ per unit length.
This can be viewed as a very simple model of a dislocationnetwork in the process of recovery.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Model
Curvature-driven evolution of polygonal curves in T3.
The polygonal curves meet each other at vertices ofdegree three.
The curves are composed of line segments meeting atnodes.
Evolves by energy minimization, assuming constant energyγ per unit length.
This can be viewed as a very simple model of a dislocationnetwork in the process of recovery.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Model
Curvature-driven evolution of polygonal curves in T3.
The polygonal curves meet each other at vertices ofdegree three.
The curves are composed of line segments meeting atnodes.
Evolves by energy minimization, assuming constant energyγ per unit length.
This can be viewed as a very simple model of a dislocationnetwork in the process of recovery.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
The Model
Curvature-driven evolution of polygonal curves in T3.
The polygonal curves meet each other at vertices ofdegree three.
The curves are composed of line segments meeting atnodes.
Evolves by energy minimization, assuming constant energyγ per unit length.
This can be viewed as a very simple model of a dislocationnetwork in the process of recovery.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Movie
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Topological Moves
Edge Flip
Digon Deletion
Edge Intersection
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Topological Moves
Edge Flip
Digon Deletion
Edge Intersection
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Topological Moves
Edge Flip
Digon Deletion
Edge Intersection
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Initial Conditions
Voronoi Graph: modified 1-skeleton of Voronoi tesselationfor random points in the three-torus.
Random Graph: Place random points on the three-torus.Randomly create edges between pairs that are closeenough.
Both evolve to “steady state”.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Initial Conditions
Voronoi Graph: modified 1-skeleton of Voronoi tesselationfor random points in the three-torus.
Random Graph: Place random points on the three-torus.Randomly create edges between pairs that are closeenough.
Both evolve to “steady state”.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Initial Conditions
Voronoi Graph: modified 1-skeleton of Voronoi tesselationfor random points in the three-torus.
Random Graph: Place random points on the three-torus.Randomly create edges between pairs that are closeenough.
Both evolve to “steady state”.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Find candidate steady state for which many scale-freeproperties appear to have converged.
Track distance dr to candidate steady state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Find candidate steady state for which many scale-freeproperties appear to have converged.
Track distance dr to candidate steady state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Find candidate steady state for which many scale-freeproperties appear to have converged.
Track distance dr to candidate steady state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Find candidate steady state for which many scale-freeproperties appear to have converged.
Track distance dr to candidate steady state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Idea of Computation
Simulate curvature-driven evolution of dimension onenetwork in 3-space.
Represent network as one-dimensional cell complex.
Compute cloth (swatch distributions) for radius r < 10
Find candidate steady state for which many scale-freeproperties appear to have converged.
Track distance dr to candidate steady state
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, I
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
xh106
0
0.005
0.01
0.015
0.02
0.025
0.03
Timesteps
Dis
tanc
ehto
hTes
thCas
eh(r
=7)
RandomhGraphhInitialhConditions,h208KhEdges
Track distance to large, steady state test case as systemevolves.
Change coordinates to aid comparison of systems.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, I
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x7105
0
0.005
0.01
0.015
0.02
0.025
0.03
Number7of7Edges
Dis
tanc
e7to
7Tes
t7Cas
e7,r
=78
Random7Graph7Initial7Conditions,7208K7Edges
Track distance to large, steady state test case as systemevolves.
Change coordinates to aid comparison of systems.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, II
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x7105
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Number7of7Edges
Dis
tanc
e7to
7Tes
t7Cas
e7,r
=78
Random7Graph7Initial7Conditions,7208K7Edges
First Case: Random Graph Initial Conditions
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, II
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x7105
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Number7of7Edges
Dis
tanc
e7to
7Tes
t7Cas
e7,r
=78
Random7Graph7Initial7ConditionsM7208K7EdgesVoronoi7Initial7Conditions7M75M7Edges
Second Case: Large Voronoi
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, II
102
103
104
105
106
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Number7of7Edges
Dis
tanc
e7to
7Tes
t7Cas
e7,r
=78
Random7Graph7Initial7ConditionsM7208K7EdgesVoronoi7Initial7Conditions7M75M7Edges
Second Case: Large Voronoi
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, II
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x7105
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Number7of7Edges
Dis
tanc
e7to
7Tes
t7Cas
e7,r
=78
Random7Graph7Initial7ConditionsM7208K7EdgesVoronoi7Initial7Conditions7M75M7Edges
Second Case: Large Voronoi
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, II
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x7105
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Number7of7Edges
Dis
tanc
e7to
7Tes
t7Cas
e7,r
=78
Random7Graph7Initial7ConditionsM7208K7EdgesVoronoi7Initial7Conditions7M75M7EdgesVoronoi7Initial7ConditionsM7200K7Edges
Third Case: Small Voronoi
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Convergence Results, II
0 0O5 1 1O5 2 2O5 3 3O5 4 4O5
x7105
0
0O005
0O01
0O015
0O02
0O025
0O03
0O035
0O04
0O045
0O05
Number7of7Edges
Dis
tanc
e7to
7Tes
t7Cas
e7,r
=78
Random7Graph7Initial7ConditionsM7208K7EdgesVoronoi7Initial7Conditions7M75M7EdgesVoronoi7Initial7ConditionsM7200K7EdgesRG7Int’l7Cond’nsM7448K7EdgesM7Intersections7Off
Fourth Case: Random Graph with Intersections Disabled
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
On Intersections
0 0.5 1 1.5 2 2.5 3 3.5 4
xl105
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
NumberloflEdges
Pro
port
ionl
oflT
opol
ogic
allE
vent
sIntersections
EdgelFlips
DigonlDeletions
Plot preponderance of different topological changes assimulation proceeds.Intersections computationally expensive; can disregard forbetter steady state statistics.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
On Intersections
0 0.5 1 1.5 2 2.5 3 3.5 4
xl105
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
NumberloflEdges
Pro
port
ionl
oflT
opol
ogic
allE
vent
sIntersections
EdgelFlips
DigonlDeletions
Plot preponderance of different topological changes assimulation proceeds.Intersections computationally expensive; can disregard forbetter steady state statistics.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Error Estimation
0 1 2 3 4 5
x)104
0
1
2
3
4
5
6
7
8x)10
−3
Number)of)Edges
Dis
tanc
e)Rr
=7G
Voronoi)Initial)Conditions,)5M)EdgesVoronoi)Initial)Conditions,)200K)EdgesRandom)Graph)Initial)Conditions,)208K)Edges
Hard to estimate statistical error for cloth: complicatedinterdependencies of swatch frequenciesIdea: compare simulation to representative subsamples oftest case
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Error Estimation
0 1 2 3 4 5
x)104
0
1
2
3
4
5
6
7
8x)10
−3
Number)of)Edges
Dis
tanc
e)Rr
=7G
Voronoi)Initial)Conditions,)5M)EdgesVoronoi)Initial)Conditions,)200K)EdgesRandom)Graph)Initial)Conditions,)208K)Edges
Hard to estimate statistical error for cloth: complicatedinterdependencies of swatch frequenciesIdea: compare simulation to representative subsamples oftest case
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Error Estimation, II
0 1 2 3 4 5
x)104
0
1
2
3
4
5
6
7
8x)10
−3
Number)of)Edges
Dis
tanc
e)Rr
=7G
Voronoi)Initial)Conditions,)5M)EdgesVoronoi)Initial)Conditions,)200K)EdgesRandom)Graph)Initial)Conditions,)208K)Edges
System has converged to state represented by test case ifit is statistically indistinguishable from a subsample of it.Within one standard deviation of subsamples: goodevidence of convergence.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Error Estimation, II
0 1 2 3 4 5
x)104
0
1
2
3
4
5
6
7
8x)10
−3
Number)of)Edges
Dis
tanc
e)Rr
=7G
Voronoi)Initial)Conditions,)5M)EdgesVoronoi)Initial)Conditions,)200K)EdgesRandom)Graph)Initial)Conditions,)208K)Edges
System has converged to state represented by test case ifit is statistically indistinguishable from a subsample of it.Within one standard deviation of subsamples: goodevidence of convergence.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
2D Persistent Homology of Objects in 3-space
Consider ε-neighborhoods of X ⊂ R3, Xε.
As ε increases, holes form and disappear.
Persistent Homology tracks these holes as ε increases.
Look at figures in Mathematica
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
2D Persistent Homology of Objects in 3-space
Consider ε-neighborhoods of X ⊂ R3, Xε.
As ε increases, holes form and disappear.
Persistent Homology tracks these holes as ε increases.
Look at figures in Mathematica
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
2D Persistent Homology of Objects in 3-space
Consider ε-neighborhoods of X ⊂ R3, Xε.
As ε increases, holes form and disappear.
Persistent Homology tracks these holes as ε increases.
Look at figures in Mathematica
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
2D Persistent Homology of Objects in 3-space
Consider ε-neighborhoods of X ⊂ R3, Xε.
As ε increases, holes form and disappear.
Persistent Homology tracks these holes as ε increases.
Look at figures in Mathematica
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistent Homology, ct’d
Philosophy: use persistent homology to study geometry offixed object.
Paper - “Measuring Shape with Topology” (Journal ofMathematical Physics, joint with R. MacPherson)
Holes in Xε correspond to voids in X
Find voids in materials
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistent Homology, ct’d
Philosophy: use persistent homology to study geometry offixed object.
Paper - “Measuring Shape with Topology” (Journal ofMathematical Physics, joint with R. MacPherson)
Holes in Xε correspond to voids in X
Find voids in materials
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistent Homology, ct’d
Philosophy: use persistent homology to study geometry offixed object.
Paper - “Measuring Shape with Topology” (Journal ofMathematical Physics, joint with R. MacPherson)
Holes in Xε correspond to voids in X
Find voids in materials
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistent Homology, ct’d
Philosophy: use persistent homology to study geometry offixed object.
Paper - “Measuring Shape with Topology” (Journal ofMathematical Physics, joint with R. MacPherson)
Holes in Xε correspond to voids in X
Find voids in materials
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistent Homology, Continued
Plot time when hole appears vs time when it disappears
Points on x=y line are noise
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistent Homology, Continued
Plot time when hole appears vs time when it disappears
Points on x=y line are noise
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
1-skeleton of Voronoi Decomposition
Persistent Homology recovers the cells.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Steady State of CDE Simulation
More surprising voids!
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Minimal Cycles
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistence Tree
Usual interpretation of Persistent Homology representsthis as two points for two voids
Correct structure: Persistence Tree
Adjacency of minimal cycles?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistence Tree
Usual interpretation of Persistent Homology representsthis as two points for two voids
Correct structure: Persistence Tree
Adjacency of minimal cycles?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Persistence Tree
Usual interpretation of Persistent Homology representsthis as two points for two voids
Correct structure: Persistence Tree
Adjacency of minimal cycles?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Unknotted Networks
Many embedded graphs from physical systems appear tobe unknotted.
How to measure this?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Unknotted Networks
Many embedded graphs from physical systems appear tobe unknotted.
How to measure this?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Minimal Cycles
0 1 2 3 4 5
xK105
0
0.5
1
1.5
2
2.5
3
3.5
4xK10
−3
NumberKofKEdges
cKo
fKMin
imal
KLoo
psKth
atKa
reKK
notte
d
RGKwithKIntersections,KGraphKDistanceRGKwithKIntersections,KEmbeddedKDistance
Look at shortest cycle containing each edge. Is it knotted?
Start with knotted network, track # of knotted minimalcycles as system evolves.
Curvature-driven evolution appears to unknot networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Minimal Cycles
0 1 2 3 4 5
xK105
0
0.5
1
1.5
2
2.5
3
3.5
4xK10
−3
NumberKofKEdges
cKo
fKMin
imal
KLoo
psKth
atKa
reKK
notte
d
RGKwithKIntersections,KGraphKDistanceRGKwithKIntersections,KEmbeddedKDistance
Look at shortest cycle containing each edge. Is it knotted?
Start with knotted network, track # of knotted minimalcycles as system evolves.
Curvature-driven evolution appears to unknot networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Minimal Cycles
0 1 2 3 4 5
xK105
0
0.5
1
1.5
2
2.5
3
3.5
4xK10
−3
NumberKofKEdges
cKo
fKMin
imal
KLoo
psKth
atKa
reKK
notte
d
RGKwithKIntersections,KGraphKDistanceRGKwithKIntersections,KEmbeddedKDistance
Look at shortest cycle containing each edge. Is it knotted?
Start with knotted network, track # of knotted minimalcycles as system evolves.
Curvature-driven evolution appears to unknot networks.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Minimal Cycles
0 1 2 3 4 5
xK105
0
1
2
3
4
5
6xK10
−3
NumberKofKEdges
cKo
fKMin
imal
KLoo
psKth
atKa
reKK
notte
d
RGKwithKIntersections,KGraphKDistanceRGKwithKIntersections,KEmbeddedKDistanceRGKw/oKIntersections,KGraphKDistanceRGKw/oKIntersections,KEmbeddedKDistance
Look at shortest cycle containing each edge. Is it knotted?
Start with knotted network, track # of knotted minimalcycles as system evolves.
Curvature-driven evolution appears to unknot networks.
Network unknots even faster if intersections turned off.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Unknottedness
Unknotted networks can contain knots.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Defining Unknottedness
Definition
An embedded graph is unknotted if its complement has thehomotopy type of a graph.
Physical interpretation: equivalent to existence of dualnetwork (important for, i.e., open cell foams).
I’m working on several theoretical questions surroundingthis definition.
Is it equivalent to π1 of the complement being free?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Defining Unknottedness
Definition
An embedded graph is unknotted if its complement has thehomotopy type of a graph.
Physical interpretation: equivalent to existence of dualnetwork (important for, i.e., open cell foams).
I’m working on several theoretical questions surroundingthis definition.
Is it equivalent to π1 of the complement being free?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Defining Unknottedness
Definition
An embedded graph is unknotted if its complement has thehomotopy type of a graph.
Physical interpretation: equivalent to existence of dualnetwork (important for, i.e., open cell foams).
I’m working on several theoretical questions surroundingthis definition.
Is it equivalent to π1 of the complement being free?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Defining Unknottedness
Definition
An embedded graph is unknotted if its complement has thehomotopy type of a graph.
Physical interpretation: equivalent to existence of dualnetwork (important for, i.e., open cell foams).
I’m working on several theoretical questions surroundingthis definition.
Is it equivalent to π1 of the complement being free?
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Knotted Materials
Knotted materials may have interesting, useful properties.
Metal-organic frameworks: new, exotic, sometimes knotted
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Knotted Materials
Knotted materials may have interesting, useful properties.
Metal-organic frameworks: new, exotic, sometimes knotted
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Conclusion
Interesting topology abounds in materials science andphysics.
Topology could be useful for understanding structures thatare currently not well-understood using any methods.
Many applications to work on, many new methods to bedeveloped.
Perhaps the study of these applications will also providenew ideas for topology.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Conclusion
Interesting topology abounds in materials science andphysics.
Topology could be useful for understanding structures thatare currently not well-understood using any methods.
Many applications to work on, many new methods to bedeveloped.
Perhaps the study of these applications will also providenew ideas for topology.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Conclusion
Interesting topology abounds in materials science andphysics.
Topology could be useful for understanding structures thatare currently not well-understood using any methods.
Many applications to work on, many new methods to bedeveloped.
Perhaps the study of these applications will also providenew ideas for topology.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Conclusion
Interesting topology abounds in materials science andphysics.
Topology could be useful for understanding structures thatare currently not well-understood using any methods.
Many applications to work on, many new methods to bedeveloped.
Perhaps the study of these applications will also providenew ideas for topology.
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Presentation Outline
1 Examples
2 Swatches
3 A Case Study
4 Computational Results
5 Persistent Homology
6 Knotting
7 Conclusion
8 Collaborators
Topology ofRandom CellComplexes
BenSchweinhart
Examples
Swatches
A Case Study
ComputationalResults
PersistentHomology
Knotting
Conclusion
Collaborators
Robert MacPherson (Institute for Advanced Study) - PhDAdvisor
Jeremy Mason (Bogazici University) - Swatches
