The following application has motivated us to develop new Computational Geometry and Topology methods, involving Brillouin zones and periodic k-fold persistent homology: We model crystals by (infinite) periodic point sets, i.e. by the union of several translates of a lattice, where every point represents an atom. Two periodic point sets are equivalent if there is a rigid transformation from one to the other. A periodic point set can be represented by a finite cutout s.t. copying this cutout infinitely often in all directions yields the periodic point set. The fact that these cutouts are not unique creates problems when working with them. Therefore, material scientists would like to work with a complete, continuous invariant instead. In this talk, I will present two continuous invariants that are at least generically complete: Firstly, the density fingerprint, computing the probability that a random ball of radius r contains exactly k points of the periodic point set, for all positive integers k and positive reals r. And secondly, the persistence fingerprint, which is the sequence of order k persistence diagrams, newly defined for infinite periodic point sets, for all positive integers k. Joint work with Herbert Edelsbrunner, Alexey Garber, Vitaliy Kurlin, Georg Osang, Janos Pach, Morteza Saghafian, Philip Smith, and Mathijs Wintraecken.

1. Geometric and Topological Fingerprints for Periodic
Crystals
Teresa Heiss (IST Austria), Herbert Edelsbrunner, Alexey Garber,
Vitaliy Kurlin, Georg Osang, János Pach, Morteza Saghafian, Philip
Smith, Mathijs Wintraecken
SIAM MS21, May 26, 2021
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 1 / 27

2. What is a (periodic) crystal?
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 2 / 27

3. What is a (periodic) crystal?
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 2 / 27

4. What is a (periodic) crystal?
“A [periodic] crystal is a solid composed of atoms, ions, or molecules
arranged in a pattern that is periodic in three dimensions.” [ASTM F1241]
i.e. there exist three linearly independent translations, such that each of
them maps the crystal to itself.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 3 / 27

5. What is a (periodic) crystal?
“A [periodic] crystal is a solid composed of atoms, ions, or molecules
arranged in a pattern that is periodic in three dimensions.” [ASTM F1241]
i.e. there exist three linearly independent translations, such that each of
them maps the crystal to itself.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 3 / 27

6. What is a periodic crystal not?
quasi-periodic crystal
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 4 / 27

7. What is our mathematical model for a crystal?
We model a crystal by an infinite periodic point set.
Centers of the atoms = points
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 5 / 27

8. Definition (Periodic point set)
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 6 / 27

9. Definition (Periodic point set)
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 6 / 27

10. Definition (Periodic point set)
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 6 / 27

11. Definition (Periodic point set)
((v1, v2, v3), M)
finite description of S
(CIF file)
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 6 / 27

12. Description not unique
A B
C
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 7 / 27

13. What is our mathematical model for a crystal?
We model a crystal by a periodic point set
.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 8 / 27

14. What is our mathematical model for a crystal?
We model a crystal by a periodic point set
an equivalence class of periodic point sets.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 8 / 27

15. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

16. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

17. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

18. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

19. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

20. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

21. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

22. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

23. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

24. Motivation: Crystal Structure Prediction
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 9 / 27

25. Definition fingerprint
Goal: map every crystal to a point in a nice space
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

26. Definition fingerprint
Goal: map every crystal to a point in a nice space
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

27. Definition fingerprint
Goal: map every crystal to a point in a nice space
A B
C
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

28. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant)
A B
C
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

29. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant)
A B
C
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

30. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant), s.t. the map is
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

31. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant), s.t. the map is injective
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

32. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant), s.t. the map is injective and continuous.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 10 / 27

33. Definition bottleneck distance
dB,2(S, S0
) := inf
γ:S→S0
sup
s∈S
ks − γ(s)k2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 11 / 27

34. Definition bottleneck distance
dB,2(S, S0
) := inf
γ:S→S0
sup
s∈S
ks − γ(s)k2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 11 / 27

35. Pair of periodic point sets close in bottleneck distance
Example α:
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 12 / 27

36. Pair of periodic point sets close in bottleneck distance
Example β:
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 13 / 27

37. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant), s.t. the map is injective and continuous.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 14 / 27

38. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant), s.t. the map is injective and continuous. → “fingerprint”
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 14 / 27

39. Definition fingerprint
Goal: map every crystal to a point in a nice space (we call this a crystal
invariant), s.t. the map is injective and continuous. → “fingerprint” [1]
[1] Edelsbrunner, Heiss, Kurlin, Smith, Wintraecken: The density fingerprint of a periodic point set. In: Proceedings of SoCG,
2021
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 14 / 27

40. Caveat
Goal: map every crystal to a point in Rn, s.t. the map is
invariant under isometries, injective, continuous.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 15 / 27

41. Caveat
Goal: map every crystal to a point in Rn, s.t. the map is
invariant under isometries, injective, continuous.
Problem: There are too many equivalence classes of periodic point sets.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 15 / 27

42. Caveat
Goal: map every crystal to a point in Rn, s.t. the map is
invariant under isometries, injective, continuous.
Problem: There are too many equivalence classes of periodic point sets.
Possible Solution: Make n depend on parameter of crystals
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 15 / 27

43. Why do we want a fingerprint function?
• develop understanding for the space of crystals
• compare crystals by comparing fingerprints
• vector representation → machine learning
• . . .
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 16 / 27

44. The density fingerprint and the persistence fingerprint
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 17 / 27

45. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 17 / 27

46. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 17 / 27

47. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 17 / 27

48. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 17 / 27

49. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 18 / 27

50. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 18 / 27

51. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 18 / 27

52. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 18 / 27

53. Persistent homology
What is the “shape” of the point set?
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

54. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

55. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

56. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

57. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

58. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

59. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

60. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

61. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

62. Persistent homology
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

63. Persistent homology
What is the “shape” of the point set?
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 19 / 27

64. k-fold covers
Cover1(X, r)
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 20 / 27

65. k-fold covers
Cover1(X, r)
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 20 / 27

66. k-fold covers
Cover1(X, r)
Coverk(X, r) := {p ∈ R3 : p ∈ Br (x) for at least k points x ∈ X}
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 20 / 27

67. k-fold covers
Cover2(X, r)
Coverk(X, r) := {p ∈ R3 : p ∈ Br (x) for at least k points x ∈ X}
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 20 / 27

68. k-fold covers
Cover3(X, r)
Coverk(X, r) := {p ∈ R3 : p ∈ Br (x) for at least k points x ∈ X}
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 20 / 27

69. k-fold covers
Cover4(X, r)
Coverk(X, r) := {p ∈ R3 : p ∈ Br (x) for at least k points x ∈ X}
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 20 / 27

70. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

71. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

72. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

73. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

74. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

75. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

76. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

77. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

78. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

79. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

80. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

81. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

82. k-fold persistence
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

83. k-fold persistence
Dgmk(X) :=
Dgm((Coverk(X, r))r∈R)
. . . k-fold persistence [2]
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
[2] Herbert Edelsbrunner and Georg Osang. The Multi-cover Persistence of Euclidean Balls. In 34th International Symposium on
Computational Geometry (SoCG 2018), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages
34:1–34:14, Dagstuhl, Germany, 2018.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

84. k-fold persistence
Dgmk(X) :=
Dgm((Coverk(X, r))r∈R)
. . . k-fold persistence [2]
(Dgmk(X))k∈N ∈ DgmN
. . . persistence fingerprint
where Dgm is the set of persistence
diagrams
Fix k, let r increase:
Coverk(X, r1) ⊆ Coverk(X, r2) ⊆ · · · ⊆ Coverk(X, rn)
In the figures above: k=2
[2] Herbert Edelsbrunner and Georg Osang. The Multi-cover Persistence of Euclidean Balls. In 34th International Symposium on
Computational Geometry (SoCG 2018), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages
34:1–34:14, Dagstuhl, Germany, 2018.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 21 / 27

85. Definition (Persistence fingerprint function ϕ)
P . . . set of equivalence classes of periodic point sets in R3
Definition
Define the persistence fingerprint function as
ϕ : P → DgmN
[S]' 7→ (Dgmk(S))k∈N
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 22 / 27

86. Definition (Persistence fingerprint function ϕ)
P . . . set of equivalence classes of periodic point sets in R3
Definition
Define the persistence fingerprint function as
ϕ : P → DgmN
[S]' 7→ (Dgmk(S))k∈N
To avoid multiplicities = ∞, define
Dgmk(S) := lim
j→∞
Dgmk(S ∩ Uj )
Vol(Uj )
,
where Uj consists of j × j × j copies of the unit cell U and
where the division and the limit is applied to the multiplicity of each
persistence pair individually.
Note, this definition yields non-integer multiplicities.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 22 / 27

87. The density fingerprint and the persistence fingerprint
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ψ0
A ψ1
A ψ2
A ψ3
A ψ4
A ψ5
A ψ6
A ψ7
A ψ8
A
Radius of Balls
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 23 / 27

88. Future Work
• fine-tune definition of persistence fingerprint
• prove invariance, continuity and generic completeness for persistence
fingerprint (easy)
• prove completeness without genericity conditions?
• generalize to weighted point sets with atomic weights (should be
straight-forward)
• continuity with respect to a different distance instead bottleneck
distance?
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 24 / 27

89. Pairs of periodic point sets that should get a small
distance assigned
Example γ:
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 25 / 27

90. Alternative distance between crystals
An alternative dissimilarity that may be more relevant in practice considers
affine transformations, τ, that minimize the bottleneck distance:
dAT(A, Q) = inf
τ
max{min{dB,2(A, τ(Q)), dB,2(τ(A), Q)}, | log s1|, | log s3|},
in which s1 ≥ s2 ≥ s3 are the three singular values of the matrix of τ.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 26 / 27

91. Herbert Edelsbrunner, Teresa Heiss, Vitaliy Kurlin, Phil Smith, and
Mathijs Wintraecken.
The density fingerprint of a periodic point set.
In Proceedings of SoCG 2021, 2021.
Herbert Edelsbrunner and Georg Osang.
The Multi-cover Persistence of Euclidean Balls.
In 34th International Symposium on Computational Geometry (SoCG
2018), volume 99 of Leibniz International Proceedings in Informatics
(LIPIcs), pages 34:1–34:14, Dagstuhl, Germany, 2018. Schloss
Dagstuhl–Leibniz-Zentrum fuer Informatik.
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 27 / 27

92. Space of crystals
Teresa Heiss (IST Austria) Fingerprints for Crystals SIAM MS21, May 26, 2021 27 / 27