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SUMMARY:Michael Pinsker (Technical University Vienna\, Austria)
DTSTART:20211026T190000Z
DTEND:20211026T200000Z
DTSTAMP:20260423T024557Z
UID:PALS/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PALS/15/">Un
 iqueness of Polish topologies on endomorphism monoids of countable structu
 res</a>\nby Michael Pinsker (Technical University Vienna\, Austria) as par
 t of PALS Panglobal Algebra and Logic Seminar\n\n\nAbstract\nMany mathemat
 ical objects are naturally equipped with both an algebraic and a\ntopologi
 cal structure. For example\, the automorphism group of any\nfirst-order st
 ructure is\, of course\,\na group\, and in fact a topological group when e
 quipped with the\ntopology of pointwise convergence.\n\nWhile in some case
 s\, e.g. the additive group of the reals\, the\nalgebraic structure\nof th
 e object alone carries strictly less information than together with the\nt
 opological structure\, in other cases its algebraic structure is so\nrich 
 that it actually determines\nthe topology (under some requirements for the
  topology): by a result\nof Kechris and Solecki\,\nthe pointwise convergen
 ce topology is the only compatible separable\ntopology on the full symmetr
 ic group on a\ncountable set. Which topologies are compatible with a given
  algebraic object has\nintrigued mathematicians for decades: for example\,
  Ulam asked whether\nthere exists a compatible\nlocally compact Polish top
 ology on the full symmetric group on a\ncountable set (by the above\, the 
 answer is negative).\n\nIn the case of automorphism groups of first-order 
 structures\, the\nquestion of the relationship between the algebraic and\n
 the topological structure  has been pursued actively over the past 40\nyea
 rs\, and numerous results have been obtained:\nmany of the most popular au
 tomorphism groups\, including that of the\norder of the rationals and of t
 he random graph\,\ndo have unique Polish topologies.\n\nThe endomorphism m
 onoid of a first-order structure is algebraically\nnot as rich as its auto
 morphism group\, and\noften allows many different compatible topologies. W
 e show\, however\,\nthat there is a unique compatible Polish topology on t
 he endomorphism\nmonoids of the random graph\, the weak linear order\nof t
 he rational numbers\, the random poset\, and many more.\n\nThis is joint w
 ork with L. Elliott\, J. Jonušas\, J. D. Mitchell\, Y.\nPéresse\, and C.
  Schindler.\n
LOCATION:https://researchseminars.org/talk/PALS/15/
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