BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Leonardo Coregliano (University of Chicago) DTSTART:20240926T180000Z DTEND:20240926T190000Z DTSTAMP:20260423T035720Z UID:OLS/152 DESCRIPTION:Title: Ex changeable random structures and quasirandomness\nby Leonardo Coreglia no (University of Chicago) as part of Online logic seminar\n\n\nAbstract\n A random structure on a vertex set $V$ (in a fixed finite relational langu age) is exchangeable if\nits distribution is invariant under permutations of $V$. The Aldous--Hoover Theorem says all such\ndistributions are genera ted from a collection of i.i.d. variables on $[0\,1]$\, one for each subse t\nof $V$\, using a simple rule that was later called "Euclidean structure " by combinatorialists. As the\nname suggests\, an Euclidean structure res embles a relational structure over $[0\,1]$\, except for the\npresence of "higher-order variables".\n\nOne of the original questions of Hoover was t o determine which such distributions admit simpler\ndescriptions\, that do not depend on certain variables. Very little progress was obtained in thi s\nproblem until it got revisited under the light of the theories of limit s of combinatorial objects\nand quasirandomness. It turns out that asking for a representation of an exchangeable hypergraph in\nwhich the Euclidean structure is a usual (measurable) relational structure over $[0\,1]$ (i.e .\, which\ndoes not need any higher-order variables) is equivalent to aski ng for "tamer" Szemeré\;di regularity\nlemmas and was solved using t he theory of hypergraphons.\n\nThe dual problem of determining when there is a representation that does not need any low-order\nvariable is more clo sely related to quasirandomness\, which informally is the property of "lac k of\ncorrelation with simple structures".\n\nIn this talk\, I will introd uce exchangeability and quasirandomness theory and talk about recent\nprog ress on the aforementioned dual problem. I will assume familiarity with ba sic logic/model\ntheory\, but no prior knowledge in extremal combinatorics \, limit theory or quasirandomness will be\nnecessary.\n\nThis talk is bas ed on joint works with Alexander Razborov and Henry Towsner.\n LOCATION:https://researchseminars.org/talk/OLS/152/ END:VEVENT END:VCALENDAR