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The Cardinality
- f a
The Cardinality of a Metric Space Tom Leinster (Glasgow/EPSRC) - - PowerPoint PPT Presentation
The Cardinality of a Metric Space Tom Leinster (Glasgow/EPSRC) - - PowerPoint PPT Presentation
The Cardinality of a Metric Space Tom Leinster (Glasgow/EPSRC) Parts joint with Simon Willerton (Sheffield) Where does the idea come from? enriched categories categories metric spaces Where does the idea come from? enriched
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Where does the idea come from?
finite categories enriched categories metric spaces ⊂ ⊂ every finite category A has a cardinality (or Euler characteristic) |A|
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Where does the idea come from?
finite categories enriched categories metric spaces ⊂ ⊂ every finite category A has a cardinality (or Euler characteristic) |A| But where does this idea come from? See two papers listed on my web page
✛
SLIDE 5
Where does the idea come from?
finite categories enriched categories metric spaces ⊂ ⊂ every finite category A has a cardinality (or Euler characteristic) |A|
SLIDE 6
Where does the idea come from?
finite categories finite enriched categories metric spaces ⊂ ⊂ every finite category A has a cardinality (or Euler characteristic) |A| every finite enriched category A has a cardinality |A|
✯ GENERALIZE
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Where does the idea come from?
finite categories finite enriched categories finite metric spaces ⊂ ⊂ every finite category A has a cardinality (or Euler characteristic) |A| every finite enriched category A has a cardinality |A|
✯ GENERALIZE
every finite metric space A has a cardinality |A|
❄ SPECIALIZE
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Where does the idea come from?
finite categories finite enriched categories finite metric spaces ⊂ ⊂ every finite category A has a cardinality (or Euler characteristic) |A| every finite enriched category A has a cardinality |A|
✯ GENERALIZE
every finite metric space A has a cardinality |A|
❄ SPECIALIZE
SLIDE 9
- 1. The cardinality of a finite metric space
Definition
Let A = {a1, . . . , an} be a finite metric space. Write Z for the n × n matrix with Zij = e−2d(ai,aj).
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- 1. The cardinality of a finite metric space
Definition
Let A = {a1, . . . , an} be a finite metric space. Write Z for the n × n matrix with Zij = e−2d(ai,aj). The cardinality of A is |A| =
- i,j
(Z −1)ij ∈ R.
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- 1. The cardinality of a finite metric space
Definition
Let A = {a1, . . . , an} be a finite metric space. Write Z for the n × n matrix with Zij = e−2d(ai,aj). The cardinality of A is |A| =
- i,j
(Z −1)ij ∈ R.
Remark
In principle, Z is defined by Zij = C d(ai,aj) for some constant C. We’ll see that taking C = e−2 is most convenient.
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- 1. The cardinality of a finite metric space
Definition
Let A = {a1, . . . , an} be a finite metric space. Write Z for the n × n matrix with Zij = e−2d(ai,aj). The cardinality of A is |A| =
- i,j
(Z −1)ij ∈ R.
Remark
In principle, Z is defined by Zij = C d(ai,aj) for some constant C. We’ll see that taking C = e−2 is most convenient.
Warning (Tao)
There exist finite metric spaces whose cardinality is undefined (i.e. with Z non-invertible).
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Reference
‘Metric spaces’, post at The n-Category Caf´ e, 9 February 2008
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- 1. The cardinality of a finite metric space
Definition
Let A = {a1, . . . , an} be a finite metric space. Write Z for the n × n matrix with Zij = e−2d(ai,aj). The cardinality of A is |A| =
- i,j
(Z −1)ij ∈ R.
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- 1. The cardinality of a finite metric space
Example (two-point spaces)
Let A = (• ← d →
- ). Then
Z = e−2·0 e−2·d e−2·d e−2·0
- =
1 e−2d e−2d 1
- ,
SLIDE 16
- 1. The cardinality of a finite metric space
Example (two-point spaces)
Let A = (• ← d →
- ). Then
Z = e−2·0 e−2·d e−2·d e−2·0
- =
1 e−2d e−2d 1
- ,
Z −1 = 1 1 − e−4d
- 1
−e−2d −e−2d 1
- ,
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- 1. The cardinality of a finite metric space
Example (two-point spaces)
Let A = (• ← d →
- ). Then
Z = e−2·0 e−2·d e−2·d e−2·0
- =
1 e−2d e−2d 1
- ,
Z −1 = 1 1 − e−4d
- 1
−e−2d −e−2d 1
- ,
|A| = 1 1 − e−4d (1 − e−2d − e−2d + 1) = 1 + tanh(d) .
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- 1. The cardinality of a finite metric space
Example (two-point spaces)
Let A = (• ← d →
- ). Then
Z = e−2·0 e−2·d e−2·d e−2·0
- =
1 e−2d e−2d 1
- ,
Z −1 = 1 1 − e−4d
- 1
−e−2d −e−2d 1
- ,
|A| = 1 1 − e−4d (1 − e−2d − e−2d + 1) = 1 + tanh(d) . 1 2 |A| d
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- 1. The cardinality of a finite metric space
Cardinality assigns to each metric space not just a number, but a function.
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- 1. The cardinality of a finite metric space
Cardinality assigns to each metric space not just a number, but a function.
Definition
Given t ∈ (0, ∞), write tA for A scaled up by a factor of t. The cardinality function of A is the partial function χA : (0, ∞) − → R, t − → |tA|.
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- 1. The cardinality of a finite metric space
Cardinality assigns to each metric space not just a number, but a function.
Definition
Given t ∈ (0, ∞), write tA for A scaled up by a factor of t. The cardinality function of A is the partial function χA : (0, ∞) − → R, t − → |tA|.
Example
A = (• ← 1 →
- ):
1 2 χA(t) t
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- 1. The cardinality of a finite metric space
Cardinality assigns to each metric space not just a number, but a function.
Definition
Given t ∈ (0, ∞), write tA for A scaled up by a factor of t. The cardinality function of A is the partial function χA : (0, ∞) − → R, t − → |tA|.
Generic example
- no. points of A
χA(t) t
- wild
- increasing
SLIDE 23
- 2. Some geometric measure theory
Ref: Schanuel, ‘What is the length of a potato?’
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- 2. Some geometric measure theory
Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm:
1
. A half-open interval is good:
- 1 cm
∪ •
- 1 cm
= •
- 2 cm
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- 2. Some geometric measure theory
Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm:
1
. A half-open interval is good:
- 1 cm
∪ •
- 1 cm
= •
- 2 cm
A closed interval is not so good:
- 1 cm
∪ •
- 1 cm
= •
- 2 cm
+ •
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- 2. Some geometric measure theory
Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm:
1
. A half-open interval is good:
- 1 cm
∪ •
- 1 cm
= •
- 2 cm
A closed interval is not so good:
- 1 cm
∪ •
- 1 cm
= •
- 2 cm
+ • So we declare: size([0, 1]) = 1 cm + 1 point = 1 cm1 + 1 cm0 = 1 cm + 1.
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- 2. Some geometric measure theory
Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm:
1
. A half-open interval is good:
- 1 cm
∪ •
- 1 cm
= •
- 2 cm
A closed interval is not so good:
- 1 cm
∪ •
- 1 cm
= •
- 2 cm
+ • So we declare: size([0, 1]) = 1 cm + 1 point = 1 cm1 + 1 cm0 = 1 cm + 1. In general, size([0, ℓ]) = ℓ cm + 1.
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- 2. Some geometric measure theory
Examples
- Size of rectangle
ℓ k is (k cm + 1)(ℓ cm + 1) = kℓ cm2 + (k + ℓ) cm + 1.
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- 2. Some geometric measure theory
Examples
- Size of rectangle
ℓ k is (k cm + 1)(ℓ cm + 1) = k
- area
ℓ cm2 + (k
- 1
2×perimeter
+ ℓ) cm +
- Euler char
1.
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- 2. Some geometric measure theory
Examples
- Size of rectangle
ℓ k is (k cm + 1)(ℓ cm + 1) = k
- area
ℓ cm2 + (k
- 1
2×perimeter
+ ℓ) cm +
- Euler char
1.
- Size of hollow triangle k
ℓ m is (k cm + 1) + (ℓ cm + 1) + (m cm + 1) − 3 = (k + ℓ + m) cm
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- 2. Some geometric measure theory
Examples
- Size of rectangle
ℓ k is (k cm + 1)(ℓ cm + 1) = k
- area
ℓ cm2 + (k
- 1
2×perimeter
+ ℓ) cm +
- Euler char
1.
- Size of hollow triangle k
ℓ m is (k cm + 1) + (ℓ cm + 1) + (m cm + 1) − 3 = (k +
- perimeter
ℓ + m) cm +
- Euler char
0.
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- 2. Some geometric measure theory
Examples
- Size of rectangle
ℓ k is (k cm + 1)(ℓ cm + 1) = k
- area
ℓ cm2 + (k
- 1
2×perimeter
+ ℓ) cm +
- Euler char
1.
- Size of hollow triangle k
ℓ m is (k cm + 1) + (ℓ cm + 1) + (m cm + 1) − 3 = (k +
- perimeter
ℓ + m) cm +
- Euler char
0.
- Similarly, can compute sizes of
, , , . . .
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- 2. Some geometric measure theory
Fix n ∈ N. What ‘measures’ can be defined on the ‘nice’ subsets of Rn?
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- 2. Some geometric measure theory
Fix n ∈ N. What ‘measures’ can be defined on the ‘nice’ subsets of Rn?
Example
When n = 2, have three measures: Euler characteristic perimeter area.
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- 2. Some geometric measure theory
Fix n ∈ N. What ‘measures’ can be defined on the ‘nice’ subsets of Rn? Hadwiger’s Theorem says that there are essentially n + 1 such measures. They are called the intrinsic volumes, µ0, µ1, . . . , µn, and µd is d-dimensional: µd(tA) = tdµd(A).
Example
When n = 2, have three measures: Euler characteristic perimeter area.
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- 2. Some geometric measure theory
Fix n ∈ N. What ‘measures’ can be defined on the ‘nice’ subsets of Rn? Hadwiger’s Theorem says that there are essentially n + 1 such measures. They are called the intrinsic volumes, µ0, µ1, . . . , µn, and µd is d-dimensional: µd(tA) = tdµd(A).
Example
When n = 2, have three measures: µ0 = Euler characteristic µ1 = perimeter µ2 = area.
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- 3. The cardinality of a compact metric space
Idea
Given a compact metric space A, choose a sequence A0 ⊆ A1 ⊆ · · ·
- f finite subsets of A, with
i Ai dense in A.
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- 3. The cardinality of a compact metric space
Idea
Given a compact metric space A, choose a sequence A0 ⊆ A1 ⊆ · · ·
- f finite subsets of A, with
i Ai dense in A. Try to define
|A| = lim
i→∞ |Ai|.
SLIDE 39
- 3. The cardinality of a compact metric space
Idea
Given a compact metric space A, choose a sequence A0 ⊆ A1 ⊆ · · ·
- f finite subsets of A, with
i Ai dense in A. Try to define
|A| = lim
i→∞ |Ai|.
Theorem
Let A = [0, ℓ] and take any sequence (Ai) as above. Then lim
i→∞ |Ai| =
SLIDE 40
- 3. The cardinality of a compact metric space
Idea
Given a compact metric space A, choose a sequence A0 ⊆ A1 ⊆ · · ·
- f finite subsets of A, with
i Ai dense in A. Try to define
|A| = lim
i→∞ |Ai|.
Theorem
Let A = [0, ℓ] and take any sequence (Ai) as above. Then lim
i→∞ |Ai| = ℓ + 1.
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- 3. The cardinality of a compact metric space
Idea
Given a compact metric space A, choose a sequence A0 ⊆ A1 ⊆ · · ·
- f finite subsets of A, with
i Ai dense in A. Try to define
|A| = lim
i→∞ |Ai|.
Theorem
Let A = [0, ℓ] and take any sequence (Ai) as above. Then lim
i→∞ |Ai| = ℓ + 1.
Remark
This is the reason for the choice of the constant e−2.
SLIDE 42
- 3. The cardinality of a compact metric space
Idea
Given a compact metric space A, choose a sequence A0 ⊆ A1 ⊆ · · ·
- f finite subsets of A, with
i Ai dense in A. Try to define
|A| = lim
i→∞ |Ai|.
Theorem
Let A = [0, ℓ] and take any sequence (Ai) as above. Then lim
i→∞ |Ai| = ℓ + 1.
Remark
[0, ℓ] has cardinality function t → | [0, tℓ] | = ℓt + 1: so ‘t = cm’.
SLIDE 43
- 3. The cardinality of a compact metric space
Assume now that all spaces mentioned have well-defined cardinality.
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- 3. The cardinality of a compact metric space
Assume now that all spaces mentioned have well-defined cardinality.
Products
Let A and B be compact metric spaces. Then |A × B| = |A| · |B|
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- 3. The cardinality of a compact metric space
Assume now that all spaces mentioned have well-defined cardinality.
Products
Let A and B be compact metric spaces. Then |A × B| = |A| · |B| as long as we give A × B the ‘d1 metric’: d((a, b), (a′, b′)) = d(a, a′) + d(b, b′).
SLIDE 46
- 3. The cardinality of a compact metric space
Assume now that all spaces mentioned have well-defined cardinality.
Products
Let A and B be compact metric spaces. Then |A × B| = |A| · |B| as long as we give A × B the ‘d1 metric’: d((a, b), (a′, b′)) = d(a, a′) + d(b, b′).
Example (rectangle)
With this metric, [0, k] × [0, ℓ] = ℓ k has cardinality function t → (kt + 1)(ℓt + 1) = kℓ t2 + (k + ℓ)t + 1.
SLIDE 47
- 3. The cardinality of a compact metric space
Example (circle)
Let Cℓ be the circle of circumference ℓ, with metric given by length of shortest arc. a b d(a, b)
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- 3. The cardinality of a compact metric space
Example (circle)
Let Cℓ be the circle of circumference ℓ, with metric given by length of shortest arc. a b d(a, b) Then |Cℓ| = ℓ 1 − e−ℓ =
∞
- n=0
Bn (−ℓ)n n! where Bn is the nth Bernoulli number.
SLIDE 49
- 3. The cardinality of a compact metric space
Example (circle)
Let Cℓ be the circle of circumference ℓ, with metric given by length of shortest arc. a b d(a, b) Then |Cℓ| = ℓ 1 − e−ℓ =
∞
- n=0
Bn (−ℓ)n n! where Bn is the nth Bernoulli number.
Asymptotics
- |Cℓ| → 1 as ℓ → 0.
SLIDE 50
- 3. The cardinality of a compact metric space
Example (circle)
Let Cℓ be the circle of circumference ℓ, with metric given by length of shortest arc. a b d(a, b) Then |Cℓ| = ℓ 1 − e−ℓ =
∞
- n=0
Bn (−ℓ)n n! where Bn is the nth Bernoulli number.
Asymptotics
- |Cℓ| → 1 as ℓ → 0.
- |Cℓ| − ℓ → 0 as ℓ → ∞
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- 3. The cardinality of a compact metric space
Example (circle)
Let Cℓ be the circle of circumference ℓ, with metric given by length of shortest arc. a b d(a, b) Then |Cℓ| = ℓ 1 − e−ℓ =
∞
- n=0
Bn (−ℓ)n n! where Bn is the nth Bernoulli number.
Asymptotics
- |Cℓ| → 1 as ℓ → 0.
- |Cℓ| − ℓ → 0 as ℓ → ∞: so when ℓ is large, |Cℓ| ≈
- perimeter
ℓ +
- Euler char
0.
SLIDE 52
- 3. The cardinality of a compact metric space
Hypothesis
All the important invariants of compact metric spaces can be derived from cardinality
SLIDE 53
- 3. The cardinality of a compact metric space
Hypothesis
All the important invariants of compact metric spaces can be derived from cardinality
Examples
- Euler characteristic
SLIDE 54
- 3. The cardinality of a compact metric space
Hypothesis
All the important invariants of compact metric spaces can be derived from cardinality
Examples
- Euler characteristic
- Intrinsic volumes µ0, µ1, . . .
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- 3. The cardinality of a compact metric space
Hypothesis
All the important invariants of compact metric spaces can be derived from cardinality
Examples
- Euler characteristic
- Intrinsic volumes µ0, µ1, . . .
- Hausdorff dimension
SLIDE 56
Review
metric spaces as enriched categories cardinality (Euler char)
- f finite categories
SLIDE 57
Review
metric spaces as enriched categories cardinality (Euler char)
- f finite categories
❅ ❅ ❘
- ✠
cardinality of finite metric spaces
SLIDE 58
Review
metric spaces as enriched categories cardinality (Euler char)
- f finite categories
❅ ❅ ❘
- ✠
cardinality of finite metric spaces
❄
cardinality of compact metric spaces
SLIDE 59
Review
metric spaces as enriched categories cardinality (Euler char)
- f finite categories
❅ ❅ ❘
- ✠
cardinality of finite metric spaces
❄
cardinality of compact metric spaces
❄
invariants from geometric measure theory
SLIDE 60
How do you quantify the diversity of an ecosystem?
SLIDE 61
How do you quantify the diversity of an ecosystem?
Hundreds of numerical measures of biodiversity have been proposed.
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How do you quantify the diversity of an ecosystem?
Hundreds of numerical measures of biodiversity have been proposed. diversity: low
SLIDE 63
How do you quantify the diversity of an ecosystem?
Hundreds of numerical measures of biodiversity have been proposed. diversity: higher
SLIDE 64
How do you quantify the diversity of an ecosystem?
Hundreds of numerical measures of biodiversity have been proposed.
Example of a diversity measure
‘Effective number of species’
SLIDE 65
How do you quantify the diversity of an ecosystem?
Hundreds of numerical measures of biodiversity have been proposed.
Example of a diversity measure
‘Effective number of species’ ‘Measuring biological diversity’, Andrew Solow, Stephen Polasky, Environmental and Ecological Statistics 1 (1994), 95–107.
SLIDE 66
How do you quantify the diversity of an ecosystem?
Hundreds of numerical measures of biodiversity have been proposed.
Example of a diversity measure
‘Effective number of species’ ‘Measuring biological diversity’, Andrew Solow (Marine Policy Center, Woods Hole), Stephen Polasky (Agricultural and Resource Economics, Oregon State), Environmental and Ecological Statistics 1 (1994), 95–107.
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