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 categories ⊂ ⊂ finite categories metric spaces every finite category A has a cardinality (or Euler characteristic) | A |

Where does the idea come from? enriched categories ⊂ ⊂ finite categories metric spaces every finite category A But where does this idea come from? has a cardinality ✛ See two papers (or Euler characteristic) listed on my web page | A |

Where does the idea come from? enriched categories ⊂ ⊂ finite categories metric spaces every finite category A has a cardinality (or Euler characteristic) | A |

Where does the idea come from? finite enriched categories every finite enriched category A has a cardinality | A | GENERALIZE ✯ ⊂ ⊂ finite categories metric spaces every finite category A has a cardinality (or Euler characteristic) | A |

Where does the idea come from? finite enriched categories every finite enriched category A has a cardinality | A | GENERALIZE SPECIALIZE ✯ ⊂ ⊂ ❄ finite categories finite metric spaces every finite category A every finite metric space A has a cardinality has a cardinality | A | (or Euler characteristic) | A |

Where does the idea come from? finite enriched categories every finite enriched category A has a cardinality | A | GENERALIZE SPECIALIZE ✯ ⊂ ⊂ ❄ finite categories finite metric spaces every finite category A every finite metric space A has a cardinality has a cardinality | A | (or Euler characteristic) | A |

1. The cardinality of a finite metric space Definition Let A = { a 1 , . . . , a n } be a finite metric space. Write Z for the n × n matrix with Z ij = e − 2 d ( a i , a j ) .

1. The cardinality of a finite metric space Definition Let A = { a 1 , . . . , a n } be a finite metric space. Write Z for the n × n matrix with Z ij = e − 2 d ( a i , a j ) . The cardinality of A is � ( Z − 1 ) ij ∈ R . | A | = i , j

1. The cardinality of a finite metric space Definition Let A = { a 1 , . . . , a n } be a finite metric space. Write Z for the n × n matrix with Z ij = e − 2 d ( a i , a j ) . The cardinality of A is � ( Z − 1 ) ij ∈ R . | A | = i , j Remark In principle, Z is defined by Z ij = C d ( a i , a j ) for some constant C . We’ll see that taking C = e − 2 is most convenient.

1. The cardinality of a finite metric space Definition Let A = { a 1 , . . . , a n } be a finite metric space. Write Z for the n × n matrix with Z ij = e − 2 d ( a i , a j ) . The cardinality of A is � ( Z − 1 ) ij ∈ R . | A | = i , j Remark In principle, Z is defined by Z ij = C d ( a i , a j ) 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).

Reference ‘Metric spaces’, post at The n-Category Caf´ e , 9 February 2008

1. The cardinality of a finite metric space Definition Let A = { a 1 , . . . , a n } be a finite metric space. Write Z for the n × n matrix with Z ij = e − 2 d ( a i , a j ) . The cardinality of A is � ( Z − 1 ) ij ∈ R . | A | = i , j

1. The cardinality of a finite metric space Example (two-point spaces) ← d → Let A = ( • • ). Then � 1 � e − 2 · 0 � � e − 2 · d e − 2 d Z = = , e − 2 · d e − 2 · 0 e − 2 d 1

1. The cardinality of a finite metric space Example (two-point spaces) ← d → Let A = ( • • ). Then � 1 � e − 2 · 0 � � e − 2 · d e − 2 d Z = = , e − 2 · d e − 2 · 0 e − 2 d 1 � � − e − 2 d 1 1 Z − 1 = , − e − 2 d 1 − e − 4 d 1

1. The cardinality of a finite metric space Example (two-point spaces) ← d → Let A = ( • • ). Then � 1 � e − 2 · 0 � � e − 2 · d e − 2 d Z = = , e − 2 · d e − 2 · 0 e − 2 d 1 � � − e − 2 d 1 1 Z − 1 = , − e − 2 d 1 − e − 4 d 1 1 1 − e − 4 d (1 − e − 2 d − e − 2 d + 1) = 1 + tanh( d ) . | A | =

1. The cardinality of a finite metric space Example (two-point spaces) ← d → Let A = ( • • ). Then � 1 � e − 2 · 0 � � e − 2 · d e − 2 d Z = = , e − 2 · d e − 2 · 0 e − 2 d 1 � � − e − 2 d 1 1 Z − 1 = , − e − 2 d 1 − e − 4 d 1 1 1 − e − 4 d (1 − e − 2 d − e − 2 d + 1) = 1 + tanh( d ) . | A | = | A | 2 1 0 d

1. The cardinality of a finite metric space Cardinality assigns to each metric space not just a number , but a function .

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 χ A : (0 , ∞ ) − → R , t �− → | tA | .

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 χ A : (0 , ∞ ) − → R , t �− → | tA | . Example ← 1 → A = ( • • ): χ A ( t ) 2 1 0 t

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 χ A : (0 , ∞ ) − → R , t �− → | tA | . Generic example χ A ( t ) no. points of A 0 t � �� � � �� � wild increasing

2. Some geometric measure theory Ref: Schanuel, ‘What is the length of a potato?’

2. Some geometric measure theory Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm : . 0 1 A half-open interval is good: 1 cm • ◦ 2 cm = • 1 cm ◦ ∪ • ◦

2. Some geometric measure theory Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm : . 0 1 A half-open interval is good: 1 cm • ◦ 2 cm = • 1 cm ◦ ∪ • ◦ A closed interval is not so good: 1 cm 2 cm • • = • • + • 1 cm ∪ • •

2. Some geometric measure theory Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm : . 0 1 A half-open interval is good: 1 cm • ◦ 2 cm = • 1 cm ◦ ∪ • ◦ A closed interval is not so good: 1 cm 2 cm • • = • • + • 1 cm ∪ • • So we declare: size([0 , 1]) = 1 cm + 1 point = 1 cm 1 + 1 cm 0 = 1 cm + 1 .

2. Some geometric measure theory Ref: Schanuel, ‘What is the length of a potato?’ Suppose we want a ruler of length 1 cm : . 0 1 A half-open interval is good: 1 cm • ◦ 2 cm = • 1 cm ◦ ∪ • ◦ A closed interval is not so good: 1 cm 2 cm • • = • • + • 1 cm ∪ • • So we declare: size([0 , 1]) = 1 cm + 1 point = 1 cm 1 + 1 cm 0 = 1 cm + 1 . In general, size([0 , ℓ ]) = ℓ cm + 1 .

2. Some geometric measure theory Examples ℓ • Size of rectangle is k ( k cm + 1)( ℓ cm + 1) = k ℓ cm 2 + ( k + ℓ ) cm + 1 .

2. Some geometric measure theory Examples area 1 2 × perimeter ℓ • Size of rectangle is Euler char k � � � ℓ cm 2 + ( k ( k cm + 1)( ℓ cm + 1) = k + ℓ ) cm + 1 .

2. Some geometric measure theory Examples area 1 2 × perimeter ℓ • Size of rectangle is Euler char k � � � ℓ cm 2 + ( k ( k cm + 1)( ℓ cm + 1) = k + ℓ ) cm + 1 . • Size of hollow triangle k ℓ is m ( k cm + 1) + ( ℓ cm + 1) + ( m cm + 1) − 3 = ( k + ℓ + m ) cm

2. Some geometric measure theory Examples area 1 2 × perimeter ℓ • Size of rectangle is Euler char k � � � ℓ cm 2 + ( k ( k cm + 1)( ℓ cm + 1) = k + ℓ ) cm + 1 . perimeter • Size of hollow triangle k ℓ is Euler char m � � ( k cm + 1) + ( ℓ cm + 1) + ( m cm + 1) − 3 = ( k + ℓ + m ) cm + 0 .

2. Some geometric measure theory Examples area 1 2 × perimeter ℓ • Size of rectangle is Euler char k � � � ℓ cm 2 + ( k ( k cm + 1)( ℓ cm + 1) = k + ℓ ) cm + 1 . perimeter • Size of hollow triangle k ℓ is Euler char m � � ( k cm + 1) + ( ℓ cm + 1) + ( m cm + 1) − 3 = ( k + ℓ + m ) cm + 0 . • Similarly, can compute sizes of , , , . . .

2. Some geometric measure theory Fix n ∈ N . What ‘measures’ can be defined on the ‘nice’ subsets of R n ?

2. Some geometric measure theory Fix n ∈ N . What ‘measures’ can be defined on the ‘nice’ subsets of R n ? Example When n = 2, have three measures: Euler characteristic perimeter area .

2. Some geometric measure theory Fix n ∈ N . What ‘measures’ can be defined on the ‘nice’ subsets of R n ? Hadwiger’s Theorem says that there are essentially n + 1 such measures. They are called the intrinsic volumes, µ 0 , µ 1 , . . . , µ n , and µ d is d -dimensional: µ d ( tA ) = t d µ d ( A ). Example When n = 2, have three measures: Euler characteristic perimeter area .