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BEGIN:VEVENT
DTSTAMP:20220506T140758Z
DTSTART:20220511T140000Z
DTEND:20220511T150000Z
SUMMARY:Logic Seminar - Dugald Macpherson (Leeds)
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}f1kd-l27q3d
vt-uq5mw3
DESCRIPTION:Title: Uniform families of definable sets in finite structure
s\n\nAbstract: A theorem of Chatzidakis\, van den Dries and Macintyre\,
stemming ultimately from the Lang-Weil estimates\, asserts\, roughly\, t
hat 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 s
et \\phi(F_q\,a) has size roughly \\mu.q^d for some (\\mu\,d) \\in E.\nT
his led in work of Elwes\, Steinhorn and myself to the notion of `asympt
otic class’ of finite structures (a class satisfying essentially the con
clusion of Chatzidakis-van den Dries-Macintyre). As an example\, by a th
eorem of Ryten\, any family of finite simple groups of fixed Lie type fo
rms an asymptotic class. There is a corresponding notion for infinite st
ructures of `measurable structure’ (e.g. a pseudofinite field\, by the
Chatzidakis- van den Dries-Macintyre theorem\, or certain pseudofinite d
ifference fields).\nI 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’ structure
s\, for which the definable sets are assigned values in some ordered sem
iring\, 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.
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:Frank Adams 1 (and zoom\, link in email)\, Alan Turing Building\
, Manchester
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