Omega-categorical structures and Macpherson-Steinhorn measurability. - PowerPoint PPT Presentation

Omega-categorical structures and Macpherson-Steinhorn measurability. David Evans Dept. of Mathematics, Imperial College London. FPS, Leeds, April 2018. 1 / 18

1. MS-measurability. For a (first-order) L -structure M we denote by Def ( M ) the collection of all non-empty parameter definable subsets of M n (for all n ≥ 1). D EF : A structure M is MS-measurable if there is a dimension - measure function h : Def ( M ) → N × R > 0 satisfying the following, where we write h ( X ) = ( dim ( X ) , ν ( X )) : (i) If X is finite (and non-empty) then h ( X ) = ( 0 , | X | ) ; y ) there is a finite set D φ ⊆ N × R > 0 of (ii) For every formula φ (¯ x , ¯ a ∈ M n ) and for each such possible vaules for h ( φ (¯ x , ¯ a )) (with ¯ value, the set of ¯ a giving this value is 0-definable; (iii) (Fubini property) Suppose X , Y ∈ Def ( M ) and f : X → Y is a definable surjection. By (ii), Y can be partitioned into disjoint definable sets Y 1 , . . . , Y r such that h ( f − 1 ( y )) is constant, equal to ( d i , m i ) , for y ∈ Y i . Let h ( Y i ) = ( e i , n i ) . Let c be the maximum of d i + e i and suppose this is attained for i = 1 , . . . , s . Then h ( X ) = ( c , m 1 n 1 + · · · + m s n s ) . 2 / 18

Measures Suppose h = ( dim , ν ) is a dimension-measure function for M . O BSERVATIONS : MS-measurability is a property of Th ( M ) . May assume ν ( M ) = 1, so h ( M n ) = ( n dim ( M ) , 1 ) . (R. Elwes, M. Ryten) M is supersimple of finite rank. M is (weak) unimodular: ◮ If f i : X → Y are k i -to-1 definable surjections (in M eq ), then k 1 = k 2 . Let B n ( M ) denote the definable subsets of M n . If D ∈ B n ( M ) , let � ν ( D ) if dim ( D ) = dim ( M n ) µ n ( D ) = . 0 otherwise This is a finitely-additive, definable probability measure on B n ( M ) . If L is countable and M is ℵ 1 -saturated, then µ n extends uniquely to a measure µ n on B σ n ( M ) , the σ -algebra generated by B n ( M ) . Fubini’s Theorem holds for the µ n . 3 / 18

Measures Suppose h = ( dim , ν ) is a dimension-measure function for M . O BSERVATIONS : MS-measurability is a property of Th ( M ) . May assume ν ( M ) = 1, so h ( M n ) = ( n dim ( M ) , 1 ) . (R. Elwes, M. Ryten) M is supersimple of finite rank. M is (weak) unimodular: ◮ If f i : X → Y are k i -to-1 definable surjections (in M eq ), then k 1 = k 2 . Let B n ( M ) denote the definable subsets of M n . If D ∈ B n ( M ) , let � ν ( D ) if dim ( D ) = dim ( M n ) µ n ( D ) = . 0 otherwise This is a finitely-additive, definable probability measure on B n ( M ) . If L is countable and M is ℵ 1 -saturated, then µ n extends uniquely to a measure µ n on B σ n ( M ) , the σ -algebra generated by B n ( M ) . Fubini’s Theorem holds for the µ n . 3 / 18

Measures Suppose h = ( dim , ν ) is a dimension-measure function for M . O BSERVATIONS : MS-measurability is a property of Th ( M ) . May assume ν ( M ) = 1, so h ( M n ) = ( n dim ( M ) , 1 ) . (R. Elwes, M. Ryten) M is supersimple of finite rank. M is (weak) unimodular: ◮ If f i : X → Y are k i -to-1 definable surjections (in M eq ), then k 1 = k 2 . Let B n ( M ) denote the definable subsets of M n . If D ∈ B n ( M ) , let � ν ( D ) if dim ( D ) = dim ( M n ) µ n ( D ) = . 0 otherwise This is a finitely-additive, definable probability measure on B n ( M ) . If L is countable and M is ℵ 1 -saturated, then µ n extends uniquely to a measure µ n on B σ n ( M ) , the σ -algebra generated by B n ( M ) . Fubini’s Theorem holds for the µ n . 3 / 18

Examples (Z. Chatzidakis, L. van den Dries, A. Macintyre) Pseudofinite fields. (H. D. Macpherson, C. Steinhorn) Ultraproducts of asymptotic classes of finite structures: ◮ Random graph ◮ Smoothly approximated structures ◮ .... 4 / 18

Examples (Z. Chatzidakis, L. van den Dries, A. Macintyre) Pseudofinite fields. (H. D. Macpherson, C. Steinhorn) Ultraproducts of asymptotic classes of finite structures: ◮ Random graph ◮ Smoothly approximated structures ◮ .... 4 / 18

Q UESTIONS : (R. Elwes, H. D. Macpherson) 1. Is an ω -categorical, MS-measurable structure necessarily one-based? 2. Is there an example of a supersimple, finite rank, unimodular structure which is not MS-measurable? R EMARKS : ω -categoricity implies unimodularity. Examples of non-one-based, ω -categorical, supersimple structures: Hrushovski constructions (1988, 1997). Q UESTION ′ : Are any of the ω -categorical Hrushovski constructions MS-measurable? 5 / 18

Q UESTIONS : (R. Elwes, H. D. Macpherson) 1. Is an ω -categorical, MS-measurable structure necessarily one-based? 2. Is there an example of a supersimple, finite rank, unimodular structure which is not MS-measurable? R EMARKS : ω -categoricity implies unimodularity. Examples of non-one-based, ω -categorical, supersimple structures: Hrushovski constructions (1988, 1997). Q UESTION ′ : Are any of the ω -categorical Hrushovski constructions MS-measurable? 5 / 18

Q UESTIONS : (R. Elwes, H. D. Macpherson) 1. Is an ω -categorical, MS-measurable structure necessarily one-based? 2. Is there an example of a supersimple, finite rank, unimodular structure which is not MS-measurable? R EMARKS : ω -categoricity implies unimodularity. Examples of non-one-based, ω -categorical, supersimple structures: Hrushovski constructions (1988, 1997). Q UESTION ′ : Are any of the ω -categorical Hrushovski constructions MS-measurable? 5 / 18

Q UESTIONS : (R. Elwes, H. D. Macpherson) 1. Is an ω -categorical, MS-measurable structure necessarily one-based? 2. Is there an example of a supersimple, finite rank, unimodular structure which is not MS-measurable? R EMARKS : ω -categoricity implies unimodularity. Examples of non-one-based, ω -categorical, supersimple structures: Hrushovski constructions (1988, 1997). Q UESTION ′ : Are any of the ω -categorical Hrushovski constructions MS-measurable? 5 / 18

Q UESTIONS : (R. Elwes, H. D. Macpherson) 1. Is an ω -categorical, MS-measurable structure necessarily one-based? 2. Is there an example of a supersimple, finite rank, unimodular structure which is not MS-measurable? R EMARKS : ω -categoricity implies unimodularity. Examples of non-one-based, ω -categorical, supersimple structures: Hrushovski constructions (1988, 1997). Q UESTION ′ : Are any of the ω -categorical Hrushovski constructions MS-measurable? 5 / 18

A modest result Theorem A There is an ω -categorical supersimple structure of SU -rank 1 which is not MS-measurable. – The example is a Hrushovski construction. – Possibly the proof is over-elaborate. 6 / 18

A modest result Theorem A There is an ω -categorical supersimple structure of SU -rank 1 which is not MS-measurable. – The example is a Hrushovski construction. – Possibly the proof is over-elaborate. 6 / 18

2. A higher amalgamation result N OTATION : L countable; M an L -structure. If k ≤ n ∈ N , denote by [ n ] k the k -subsets of [ n ] = { 1 , . . . , n } . If I ∈ [ n ] k , then π I : M n → M k denotes the projection to coordinates in I . Theorem B Suppose M is a MS-measurable structure and E ⊆ M n is a definable subset. Suppose that: (a) dim ( π J ( E )) = dim ( M n − 1 ) , where J = { 1 , . . . , n − 1 } , and (b) if ( a 1 , . . . , a n ) ∈ E , then a i ∈ acl ( { a j : j � = i } ) (for all i ≤ n ). Then b ∈ M n : π I (¯ dim ( { ¯ b ) ∈ π I ( E ) for all I ∈ [ n ] n − 1 } ) = dim ( M n ) . R EMARKS : 1. Is there a relation between Theorem B and independent n -amalgamation over a model? 2. Do MS-measurable structures satisfy independent n -amalgamation over a model? 7 / 18

2. A higher amalgamation result N OTATION : L countable; M an L -structure. If k ≤ n ∈ N , denote by [ n ] k the k -subsets of [ n ] = { 1 , . . . , n } . If I ∈ [ n ] k , then π I : M n → M k denotes the projection to coordinates in I . Theorem B Suppose M is a MS-measurable structure and E ⊆ M n is a definable subset. Suppose that: (a) dim ( π J ( E )) = dim ( M n − 1 ) , where J = { 1 , . . . , n − 1 } , and (b) if ( a 1 , . . . , a n ) ∈ E , then a i ∈ acl ( { a j : j � = i } ) (for all i ≤ n ). Then b ∈ M n : π I (¯ dim ( { ¯ b ) ∈ π I ( E ) for all I ∈ [ n ] n − 1 } ) = dim ( M n ) . R EMARKS : 1. Is there a relation between Theorem B and independent n -amalgamation over a model? 2. Do MS-measurable structures satisfy independent n -amalgamation over a model? 7 / 18

2. A higher amalgamation result N OTATION : L countable; M an L -structure. If k ≤ n ∈ N , denote by [ n ] k the k -subsets of [ n ] = { 1 , . . . , n } . If I ∈ [ n ] k , then π I : M n → M k denotes the projection to coordinates in I . Theorem B Suppose M is a MS-measurable structure and E ⊆ M n is a definable subset. Suppose that: (a) dim ( π J ( E )) = dim ( M n − 1 ) , where J = { 1 , . . . , n − 1 } , and (b) if ( a 1 , . . . , a n ) ∈ E , then a i ∈ acl ( { a j : j � = i } ) (for all i ≤ n ). Then b ∈ M n : π I (¯ dim ( { ¯ b ) ∈ π I ( E ) for all I ∈ [ n ] n − 1 } ) = dim ( M n ) . R EMARKS : 1. Is there a relation between Theorem B and independent n -amalgamation over a model? 2. Do MS-measurable structures satisfy independent n -amalgamation over a model? 7 / 18

Deducing Thm A Produce an ω -categorical Hrushovski structure M f of SU-rank 1 1 which fails to have the amalgamation property in Theorem B for dim equal to SU-rank. Show that if ( dim , ν ) is a dimension-measure function on Def ( M f ) 2 then dim is equal to SU-rank (after normalising). 8 / 18