Dear Colleagues, let me to send a reminder for 5.1.2023 - prof. Ehud Hrushovski - 121. Mathematical Colloquium
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121. kolokvium:
ELEMENTARY RAMSEY THEORY, APPROXIMATE SUBGROUPS AND A MODEL-THEORETIC GALOIS GROUP
E. Hrushovski
ctvrtek 5. ledna 2023 ve 14:00, aula (refektar), prvni patro
MFF UK, Malostranske nam. 25, Praha 1 ____________________________________________________________________
Abstract. A structure has the "elementary Ramsey property" if any colouring, restricted to a subset that approximates the full structure to a prescribed degree, becomes definable. The word elementary refers to the fact that it is really a property of the first-order theory, rather than the structure. This slight variation on structural Ramsey theory allows the following theorem: any theory T has a canonical minimal expansion T^{ram} to one with the elementary Ramsey property. It is a soft result in a hard field, but has the virtue of bringing out a hidden automorphism group, namely the automorphism group of T^{ram} over T. In the case of Ramsey's original theorem, the theory is that of a structureless set, the expansion consists of a linear ordering < (thus an ordering is not merely an artefact of the proof!), and the group is the two-element group exchanging < and its opposite ordering >. Evans, Hubička and Nešetřil have previously shown that such a theorem is not possible in a more usual (aleph_0-categorical) framework of structural Ramsey theory.
This construction is a special case of a canonical group construction with a long history in model theory, going back to many including Krupinski, Pillay, Lascar, Shelah, Galois. In another setting it provides a locally compact group associated with an approximate subgroup of any group, and leads to a general classification of approximate subgroups. I will define these various notions in the talk and try to explain their relationship. -------------------------------------------------------------------------------------
(pdf pozvanky - obsahuje laudatio a dalsi informace o kolokviich: https://kam.mff.cuni.cz/~klazar/hrush.pdf)