tiling-harmonic functions · 2015. 5. 16. · a bounded tiling-harmonic function on the regular...
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Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Tiling-Harmonic Functions
Jacob Klegar
mentored by Prof. Sergiy Merenkov, CCNY-CUNY
Fifth Annual PRIMES ConferenceMay 16, 2015
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
We are studying functions on the vertices of square tilings.A square tiling is defined broadly as a connected set ofsquares in the plane with disjoint interiors and whoseedges are parallel to the coordinate axes.This project works with subsets of the regular squarelattice Z2.
A tiling S is a subtiling of a tiling T if the set of its squaresis a subset of the set of T ’s.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
We are studying functions on the vertices of square tilings.A square tiling is defined broadly as a connected set ofsquares in the plane with disjoint interiors and whoseedges are parallel to the coordinate axes.This project works with subsets of the regular squarelattice Z2.
A tiling S is a subtiling of a tiling T if the set of its squaresis a subset of the set of T ’s.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
We are studying functions on the vertices of square tilings.A square tiling is defined broadly as a connected set ofsquares in the plane with disjoint interiors and whoseedges are parallel to the coordinate axes.This project works with subsets of the regular squarelattice Z2.
A tiling S is a subtiling of a tiling T if the set of its squaresis a subset of the set of T ’s.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
We are studying functions on the vertices of square tilings.A square tiling is defined broadly as a connected set ofsquares in the plane with disjoint interiors and whoseedges are parallel to the coordinate axes.This project works with subsets of the regular squarelattice Z2.
A tiling S is a subtiling of a tiling T if the set of its squaresis a subset of the set of T ’s.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
DefinitionThe oscillation osc(u, t) of a function on a square t is thedifference between the maximum and minimum values on thatsquare.
DefinitionThe energy E(u) of a function on a tiling is the sum over allsquares t in that tiling of (osc(u, t))2.
E(u) =∑
t
(osc(u, t))2
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
DefinitionThe oscillation osc(u, t) of a function on a square t is thedifference between the maximum and minimum values on thatsquare.
DefinitionThe energy E(u) of a function on a tiling is the sum over allsquares t in that tiling of (osc(u, t))2.
E(u) =∑
t
(osc(u, t))2
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Calculating the Energy
Example
gives the energy
E(u) = (5 − 1)2 + (6 − 0)2 + (4 − 0)2 + (6 − 0)2 = 104.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Calculating the Energy
Example
gives the energy
E(u) = (5 − 1)2 + (6 − 0)2 + (4 − 0)2 + (6 − 0)2 = 104.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
DefinitionA function on a finite square tiling is called tiling-harmonicif its energy is minimized among all functions on that tilingwith the same boundary values.A function on an infinite tiling is tiling-harmonic if it istiling-harmonic on all finite subtilings.
RemarkGiven a tiling and a set of boundary values, tiling-harmonicfunctions are not necessarily unique.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
DefinitionA function on a finite square tiling is called tiling-harmonicif its energy is minimized among all functions on that tilingwith the same boundary values.A function on an infinite tiling is tiling-harmonic if it istiling-harmonic on all finite subtilings.
RemarkGiven a tiling and a set of boundary values, tiling-harmonicfunctions are not necessarily unique.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Background
DefinitionA function on a finite square tiling is called tiling-harmonicif its energy is minimized among all functions on that tilingwith the same boundary values.A function on an infinite tiling is tiling-harmonic if it istiling-harmonic on all finite subtilings.
RemarkGiven a tiling and a set of boundary values, tiling-harmonicfunctions are not necessarily unique.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
A Tiling-Harmonic Function
Example
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Another Tiling-Harmonic Function
TheoremThe function f (x , y) = y is tiling-harmonic.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Graph Harmonic Functions
DefinitionA function on a square tiling is called graph-harmonic if thevalue at each vertex is the average of the values of itsneighbors.
This is the discrete analogue to the harmonic functions ofcomplex analysis.Given a set of boundary values, such a function is unique.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Graph Harmonic Functions
DefinitionA function on a square tiling is called graph-harmonic if thevalue at each vertex is the average of the values of itsneighbors.
This is the discrete analogue to the harmonic functions ofcomplex analysis.Given a set of boundary values, such a function is unique.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Graph Harmonic Functions
DefinitionA function on a square tiling is called graph-harmonic if thevalue at each vertex is the average of the values of itsneighbors.
This is the discrete analogue to the harmonic functions ofcomplex analysis.Given a set of boundary values, such a function is unique.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Two Main Conjectures
Conjecture (Liouville’s Theorem for TH Functions)A bounded tiling-harmonic function on the regular lattice grid(Z2) must be constant.
Liouville’s is a major theorem for harmonic functions.This theorem serves as a "simpler version" of the second,more important conjecture.
ConjectureA tiling-harmonic function on the upper half-plane that vanishesalong the x-axis must be proportional to y.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Two Main Conjectures
Conjecture (Liouville’s Theorem for TH Functions)A bounded tiling-harmonic function on the regular lattice grid(Z2) must be constant.
Liouville’s is a major theorem for harmonic functions.This theorem serves as a "simpler version" of the second,more important conjecture.
ConjectureA tiling-harmonic function on the upper half-plane that vanishesalong the x-axis must be proportional to y.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Two Main Conjectures
Conjecture (Liouville’s Theorem for TH Functions)A bounded tiling-harmonic function on the regular lattice grid(Z2) must be constant.
Liouville’s is a major theorem for harmonic functions.This theorem serves as a "simpler version" of the second,more important conjecture.
ConjectureA tiling-harmonic function on the upper half-plane that vanishesalong the x-axis must be proportional to y.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Two Main Conjectures
Conjecture (Liouville’s Theorem for TH Functions)A bounded tiling-harmonic function on the regular lattice grid(Z2) must be constant.
Liouville’s is a major theorem for harmonic functions.This theorem serves as a "simpler version" of the second,more important conjecture.
ConjectureA tiling-harmonic function on the upper half-plane that vanishesalong the x-axis must be proportional to y.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Motivation
The second conjecture may provide an alternative proof ofthe quasisymmetric rigidity of square Sierpinski carpets.Tiling-harmonic functions are also interesting combinatorialobjects in their own right.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Motivation
The second conjecture may provide an alternative proof ofthe quasisymmetric rigidity of square Sierpinski carpets.Tiling-harmonic functions are also interesting combinatorialobjects in their own right.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Harnack’s Inequality
Conjecture (Harnack’s Inequality)On a nonnegative tiling-harmonic function, the ratio of thevalues on two points a fixed distance r apart is bounded.
Proving Harnack’s Inequality would be a major step towardproving Liouville’s Theorem.Harnack’s Inequality is known for graph harmonicfunctions.We have strong experimental evidence that the maximumratio for a tiling-harmonic function is bounded by that of thegraph harmonic function with the same boundary values.A proof of this bound would imply Harnack’s Inequality fortiling-harmonic functions.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Harnack’s Inequality
Conjecture (Harnack’s Inequality)On a nonnegative tiling-harmonic function, the ratio of thevalues on two points a fixed distance r apart is bounded.
Proving Harnack’s Inequality would be a major step towardproving Liouville’s Theorem.Harnack’s Inequality is known for graph harmonicfunctions.We have strong experimental evidence that the maximumratio for a tiling-harmonic function is bounded by that of thegraph harmonic function with the same boundary values.A proof of this bound would imply Harnack’s Inequality fortiling-harmonic functions.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Harnack’s Inequality
Conjecture (Harnack’s Inequality)On a nonnegative tiling-harmonic function, the ratio of thevalues on two points a fixed distance r apart is bounded.
Proving Harnack’s Inequality would be a major step towardproving Liouville’s Theorem.Harnack’s Inequality is known for graph harmonicfunctions.We have strong experimental evidence that the maximumratio for a tiling-harmonic function is bounded by that of thegraph harmonic function with the same boundary values.A proof of this bound would imply Harnack’s Inequality fortiling-harmonic functions.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Harnack’s Inequality
Conjecture (Harnack’s Inequality)On a nonnegative tiling-harmonic function, the ratio of thevalues on two points a fixed distance r apart is bounded.
Proving Harnack’s Inequality would be a major step towardproving Liouville’s Theorem.Harnack’s Inequality is known for graph harmonicfunctions.We have strong experimental evidence that the maximumratio for a tiling-harmonic function is bounded by that of thegraph harmonic function with the same boundary values.A proof of this bound would imply Harnack’s Inequality fortiling-harmonic functions.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Harnack’s Inequality
Conjecture (Harnack’s Inequality)On a nonnegative tiling-harmonic function, the ratio of thevalues on two points a fixed distance r apart is bounded.
Proving Harnack’s Inequality would be a major step towardproving Liouville’s Theorem.Harnack’s Inequality is known for graph harmonicfunctions.We have strong experimental evidence that the maximumratio for a tiling-harmonic function is bounded by that of thegraph harmonic function with the same boundary values.A proof of this bound would imply Harnack’s Inequality fortiling-harmonic functions.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Maximum Modulus Principle
Theorem (Maximum Modulus Principle)On an m x n rectangular grid with m,n ≥ 4, if the maximumvalue occurs on the interior, then the entire set of interior valuesis constant.
There is an analogous theorem for graph-harmonicfunctions.
Example
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Maximum Modulus Principle
Theorem (Maximum Modulus Principle)On an m x n rectangular grid with m,n ≥ 4, if the maximumvalue occurs on the interior, then the entire set of interior valuesis constant.
There is an analogous theorem for graph-harmonicfunctions.
Example
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Maximum Modulus Principle
Theorem (Maximum Modulus Principle)On an m x n rectangular grid with m,n ≥ 4, if the maximumvalue occurs on the interior, then the entire set of interior valuesis constant.
There is an analogous theorem for graph-harmonicfunctions.
Example
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Limiting Property
TheoremThe limit of a sequence of tiling-harmonic functions is itselftiling-harmonic.
Thus the set of tiling-harmonic functions is closed.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Limiting Property
TheoremThe limit of a sequence of tiling-harmonic functions is itselftiling-harmonic.
Thus the set of tiling-harmonic functions is closed.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Results — Oscillating Boundary Values
TheoremFor every boundary square, consider the range of the boundaryvalues on that square. If the intersection of these ranges isnonempty, then the only tiling-harmonic functions with theseboundary values are constant on the interior.
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Tiling vs Graph Harmonic — Similarities
20
40
60
80
Tiling Harmonic
10020
4060
80100
0.4
0.3
0.2
0.1
0
0.5
0.6
0.9
0.7
0.8
1
20
40
60
80
100
Graph Harmonic
20
40
60
80
100
1
0.9
0
0.1
0.2
0.7
0.3
0.4
0.5
0.6
0.8
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Tiling vs Graph Harmonic — Differences
6040
Tiling Harmonic
200
0
10
20
30
40
50
0
10
20
30
40
50
6040
Graph Harmonic
200
0
10
20
50
45
40
35
30
25
20
15
10
5
0
30
40
50
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Tiling vs Graph Harmonic — Random Boundary
50
40
30
20
Tiling Harmonic
100
10
20
30
40
50
0.8
0.2
0.4
0.6
40
20
Graph Harmonic
00
10
20
30
40
50
0.2
0.6
0.8
0.4
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Tiling vs Graph Harmonic — Random Boundary
50
40
30
20
10
Difference
00
10
20
30
40
-0.5
-1
0
0.5
1
50
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Future Goals
Proof of Harnack’s InequalityProof of Maximum Modulus Principle for oscillationsExplore the Boundary Harnack PrincipleAlternative necessary and/or sufficient conditions fortiling-harmonic functions
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Future Goals
Proof of Harnack’s InequalityProof of Maximum Modulus Principle for oscillationsExplore the Boundary Harnack PrincipleAlternative necessary and/or sufficient conditions fortiling-harmonic functions
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Future Goals
Proof of Harnack’s InequalityProof of Maximum Modulus Principle for oscillationsExplore the Boundary Harnack PrincipleAlternative necessary and/or sufficient conditions fortiling-harmonic functions
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Future Goals
Proof of Harnack’s InequalityProof of Maximum Modulus Principle for oscillationsExplore the Boundary Harnack PrincipleAlternative necessary and/or sufficient conditions fortiling-harmonic functions
Background Conjectures Results Tiling vs Graph Harmonic Future Work Conclusion
Acknowledgements
Many thanks to:The MIT PRIMES ProgramProf. Sergiy Merenkov, CCNY-CUNY, my mentorMatt Getz, a CCNY Graduate Student with whom I havebeen working on this projectProf. Tanya Khovanovamy school, Choate Rosemary Hall, especially Dr. MatthewBardoe and Mr. Samuel Doakand My Parents.