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<div dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);">Dear all, </div>
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<div dir="ltr" style="text-decoration: none; color: rgb(33, 33, 33);">Leonardo Coregliano's talk in the logic seminar tomorrow, "<span style="text-decoration: none; display: inline !important; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);">Exchangeable
random structures and quasirandomness</span><span style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">" (see below) </span>may
be of interest to people on this list. </div>
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<span dir="ltr">MM</span></div>
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<div dir="ltr"><span dir="ltr" style="text-decoration:none">---------</span></div>
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<div dir="ltr"><span dir="ltr" style="text-decoration:none">Logic Seminar </span></div>
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Speaker: </span><span style="font-size: inherit;">Leonardo Coregliano</span>
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<div dir="ltr" style="text-decoration:none">Time: <span style="font-size: inherit;">Eckhart 206, 5-6pm, Thursday 10/17</span></div>
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<div dir="ltr" style="text-decoration:none"><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">Title: </span><span style="font-size: inherit;">Exchangeable random structures and quasirandomness</span></div>
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<span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">Abstract:</span><br style="text-decoration:none">
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<span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">A random structure on a vertex set $V$ (in a fixed finite relational language) is exchangeable if </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">its
distribution is invariant under permutations of $V$. The Aldous--Hoover Theorem says all such </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">distributions are generated from a collection of i.i.d.
variables on $[0,1]$, one for each subset </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">of $V$, using a simple rule that was later called "Euclidean structure" by combinatorialists. As the </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">name
suggests, an Euclidean structure resembles a relational structure over $[0,1]$, except for the </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">presence of "higher-order variables".</span><br style="text-decoration:none">
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<span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">One of the original questions of Hoover was to determine which such distributions admit simpler </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">descriptions
that do not depend on certain variables. Very little progress was obtained in this </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">problem until it got revisited under the light of the theories
of limits of combinatorial objects </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">and quasirandomness. It turns out that asking for a representation of an exchangeable hypergraph in </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">which
the Euclidean structure is a usual (measurable) relational structure over $[0,1]$ (i.e., which </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">does not need any higher-order variables) is equivalent
to asking for "tamer" Szemerédi regularity </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">lemmas and was solved using the theory of hypergraphons.</span><br style="text-decoration:none">
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<span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">The dual problem of determining when there is a representation that does not need any low-order </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">variable
is more closely related to quasirandomness, which informally is the property of "lack of </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">correlation with simple structures".</span><br style="text-decoration:none">
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<span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">In this talk, I will introduce exchangeability and quasirandomness theory and talk about recent </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">progress
on the aforementioned dual problem. I will assume familiarity with basic logic/model </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">theory, but no prior knowledge in extremal combinatorics, limit
theory or quasirandomness will be </span><span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">necessary.</span><br style="text-decoration:none">
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<span style="text-decoration: none; display: inline !important; background-color: rgb(255, 255, 255);">This talk is based on joint works with Alexander Razborov and Henry Towsner. </span></div>
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