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DTSTART:20240310T070000
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DTSTART:20231105T060000
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UID:69fa0bbdbb338
DTSTART;TZID=America/Toronto:20241007T143000
SEQUENCE:0
TRANSP:TRANSPARENT
DTEND;TZID=America/Toronto:20241007T153000
URL:https://uwaterloo.ca/pure-mathematics/events/pure-math-department-collo
 quium-0
SUMMARY:Pure Math Department Colloquium
CLASS:PUBLIC
DESCRIPTION:MATTHEW HARRISON-TRAINOR\, UNIVERSITY OF ILLINOIS AT CHICAGO\n\
 nBack-and-forth games to characterize countable structures\n\nGiven two co
 untable structures A and B of the same type\, such as\ngraphs\, linear ord
 ers\, or groups\, two players Spoiler and Copier can\nplay a back-and-fort
 h games as follows. Spoiler begins by playing a\ntuple from A\, to which C
 opier responds by playing a tuple of the same\nsize from B. Spoiler then p
 lays a tuple from B (adding it to the tuple\nfrom B already played by Copi
 er)\, and Copier responds by playing a\ntuple from B (adding it to the tup
 le already played by Spoiler). They\ncontinue in this way\, alternating be
 tween the two structures. Copier\nloses if at any point the tuples from A 
 and B look different\, e.g.\, if\nA and B are linear orders then the two t
 uples must be ordered in the\nsame way. If Copier can keep copying forever
 \, they win. A and B are\nisomorphic if and only if Copier has a winning s
 trategy for this game.\n  Even if Copier does not have a winning strategy
 \, they may be able\nto avoid losing for some (ordinal) amount of time. Th
 is gives a\nmeasure of similarity between A and B. A classical theorem of 
 Scott\nsays that for every structure A\, there is an α such that if B is 
 any\ncountable structure\, A is isomorphic to B if and only if Copier can\
 navoid losing for α steps of the back-and-forth game\, that is\, when A\n
 is involved we only need to play the back-and-forth game for α many\nstep
 s rather than the full infinite game. This gives a measure of\ncomplexity 
 for A\, called the Scott rank. I will introduce these ideas\nand talk abou
 t some recent results.\n\nMC 5501
DTSTAMP:20260505T152445Z
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