BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Françoise Point (Université de Mons-Hainaut) DTSTART:20211007T180000Z DTEND:20211007T190000Z DTSTAMP:20260423T035736Z UID:OLS/75 DESCRIPTION:Title: Def inable groups in topological fields with a generic derivation\nby Fran çoise Point (Université de Mons-Hainaut) as part of Online logic seminar \n\n\nAbstract\nWe study a class of tame $\\mathcal L$-theories $T$ of top ological fields and their extensions by a generic derivation $\\delta$. Th e topological fields under consideration include henselian valued fields o f characteristic 0 and real closed fields. We axiomatize the class of the existentially closed $\\mathcal L_\\delta$-expansions.\nWe show that $T_\\ delta^*$ has $\\mathcal L$-open core (i.e.\, every $\\mathcal L_\\delta$-d efinable open set is $\\mathcal L$-definable) and derive both a cell decom position theorem and a transfer result of elimination of imaginaries. Othe r tame properties of $T$ such as relative elimination of field sort quanti fiers\, NIP and distality also transfer to $T_\\delta^*$. \n\\par Then let ting $\\mathcal K$ be a model of $T_\\delta^*$ and $\\mathcal M$ a $\\vert K\\vert^+$-saturated elementary extension of $\\mathcal K$\, we first ass ociate with an $\\mathcal L_\\delta(K)$-definable group $G$ in $\\mathcal M$\, a pro-$\\mathcal L$-definable set $G^{**}_{\\infty}$ in which the dif ferential prolongations $G^{\\nabla_\\infty}$ of elements of $G$ are dense \, using the $\\mathcal L$-open core property of $T_\\delta^*$. Following the same ideas as in the group configuration theorem in o-minimal structur es as developed by K. Peterzil\, we construct a type $\\mathcal L$-definab le topological group $H_\\infty\\subset G^{**}_{\\infty}$\, acting on a $K $-infinitesimal neighbourhood of a generic element of $G^{**}_\\infty$ in a faithful\, continuous and transitive way. Further $H_\\infty\\cap G^{\\n abla_\\infty}$ is dense in $H_\\infty$ and the action of $H_\\infty\\cap G ^{\\nabla_\\infty}$ coincides with the one induced by the initial $\\mathc al L_\\delta$-group action. \n\\par The first part of this work is joint w ith Pablo Cubid\\`es Kovacsics.\n LOCATION:https://researchseminars.org/talk/OLS/75/ END:VEVENT END:VCALENDAR