# Logic Seminar - Dugald Macpherson (Leeds)

Dates: | 11 May 2022 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Title: Uniform families of definable sets in finite structures

Abstract: A theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if \phi(x,y) is a formula in the language of rings (where x,y are tuples) then the size of the solution set of \phi(x,a) in any finite field F_q (where a is a parameter tuple from F_q) takes one of finitely many dimension-measure pairs as F_q and a vary: for a finite set E of pairs (\mu,d) (\mu rational, d integer) dependent on \phi, any set \phi(F_q,a) has size roughly \mu.q^d for some (\mu,d) \in E. This led in work of Elwes, Steinhorn and myself to the notion of `asymptotic class’ of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. There is a corresponding notion for infinite structures of `measurable structure’ (e.g. a pseudofinite field, by the Chatzidakis- van den Dries-Macintyre theorem, or certain pseudofinite difference fields). I will discuss a body of work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a richer range of examples with fewer model-theoretic constraints; for example, the corresponding infinite `generalised measurable’ structures, for which the definable sets are assigned values in some ordered semiring, need no longer have simple theory. I will also discuss a variant in which sizes of definable sets in finite structures are given exactly rather than asymptotically.