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SUMMARY:Exchangeable constructions of countable structures - Cameron Freer
  (Massachusetts Institute of Technology)
DTSTART:20160725T130000Z
DTEND:20160725T133000Z
UID:TALK66840@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<span>Co-authors: Nathanael Ackerman (Harvard  University)\, D
 iana Cai (University of Chicago)\, Alex Kruckman  (UC Berkeley)\, Aleksand
 ra Kwiatkowska (University of Bonn)\,  Jaroslav Nesetril (Charles Universi
 ty in Prague)\, Rehana Patel  (Franklin W. Olin College of Engineering)\, 
 Jan Reimann (Penn State  University) <br></span> <br>The Aldous-Hoover-Kal
 lenberg theorem and the theory of graph limits connect  three kinds of obj
 ects: sequences of finite graphs\, random countably infinite  graphs\, and
  certain continuum-sized measurable "limit" objects (graphons).  Graphons 
 induce exchangeable countably infinite graphs via sampling\, and all  exch
 angeable graphs arise from a mixture of such sampling procedures -- a  two
 -dimensional generalization of de Finetti&#39\;s theorem. <br> <br>This na
 turally leads to the question of which countably infinite graphs (or  othe
 r structures) can arise via an exchangeable construction. More formally\, 
  consider a random structure with a fixed countably infinite underlying se
 t. The  random structure is exchangeable when its joint distribution is in
 variant under  permutations of the underlying set. For example\, the count
 ably infinite  Erd&#x151\;s&ndash\;R&eacute\;nyi graph is exchangeable\; m
 oreover\, it is almost surely isomorphic to a  particular graph\, known as
  the Rado graph\, and so we say that the Rado graph  admits an exchangeabl
 e construction. On the other hand\, one can show that no  infinite tree ad
 mits an exchangeable construction. <br> <br>In joint work with Ackerman an
 d Patel\, we provide a necessary and sufficient  condition for a countably
  infinite structure to admit an exchangeable  construction. We also addres
 s related questions\, such as what structures admit a  unique exchangeable
  construction\, and give examples involving graphs\, directed  graphs\, an
 d partial orders. <br> <span><br>Joint work with Nathanael Ackerman\, Dian
 a Cai\, Alex Kruckman\, Aleksandra  Kwiatkowska\, Jaroslav Ne&scaron\;et&#
 x159\;il\, Rehana Patel\, and Jan Reimann.&nbsp\;</span>
LOCATION:Seminar Room 1\, Newton Institute
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