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SUMMARY:Subgroup Tests and the Aldous-Lyons conjecture - Michael Chapman (
 NYU)
DTSTART:20241113T133000Z
DTEND:20241113T150000Z
UID:TALK220492@talks.cam.ac.uk
CONTACT:Francesco Fournier-Facio
DESCRIPTION:The Aldous-Lyons conjecture from probability theory states tha
 t every (unimodular) infinite graph can be (Benjamini-Schramm) approximate
 d by finite graphs. This conjecture is an analogue of other influential co
 njectures in mathematics concerning how well certain infinite objects can 
 be approximated by finite ones\; examples include Connes' embedding proble
 m (CEP) in functional analysis and the soficity problem of Gromov-Weiss in
  group theory. These became major open problems in their respective fields
 \, as many other long standing open problems\, that seem unrelated to any 
 approximation property\, were shown to be true for the class of finitely-a
 pproximated objects. For example\, Gottschalk's conjecture and Kaplansky's
  direct finiteness conjecture are known to be true for sofic groups\, but 
 are still wide open for general groups.\n\nIn 2019\, Ji\, Natarajan\, Vidi
 ck\, Wright and Yuen resolved CEP in the negative. Quite remarkably\, thei
 r result is deduced from complexity theory\, and specifically from undecid
 ability in certain quantum interactive proof systems. Inspired by their wo
 rk\, we suggest a novel interactive proof system which is related to the A
 ldous-Lyons conjecture in the following way: If the Aldous-Lyons conjectur
 e was true\, then every language in this interactive proof system is decid
 able. A key concept we introduce for this purpose is that of a Subgroup Te
 st\, which is our analogue of a Non-local Game. By providing a reduction f
 rom the Halting Problem to this new proof system\, we refute the Aldous-Ly
 ons conjecture.\n\nThis talk is based on joint work with Lewis Bowen\, Ale
 x Lubotzky\, and Thomas Vidick. No special background in probability theor
 y or complexity theory will be assumed.
LOCATION:MR4
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