Tame Geometry in Henselian Valued Fields
Mittwoch, 4.12.19, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
In real algebaric geometry, the objects of study are semi-algebraic sets, i.e., subsets of R^n defined using polynomial inequalities. In the 80s, Pillay and Steinhorn introduced o-minimality, a simple axiomatic description of classes of sets for which "geometry works as for semi-algebraic sets". More precisely, the sets in such a class are those which are first-order definable in a suitable language. This axiomatic approach had a huge impact on geometry in R, and many results known for semi-algebraic sets have then be proved in this much more general framework.\n\nSince the invention of o-minimality, various attempts have been made to come up with an analogous notion in (suitable) valued fields like the p-adics or fields of formal Laurent series. Understanding first-order definable sets in such fields has been crucial to obtain rationality of many kinds of Poincaré series, and in the late 90s, it also became the fundament of motivic integration. In this talk, I will present a new analogue of o-minimality for valued fields (a collaboration with Cluckers and Rideau) which is powerful enough so that all these applications (rationality, motivic integration) can be carried out within that framework.\n\nThe talk will only require some very basic knowledge about (some examples of) valued fields and some vague familiarity with o-minimality and/or model theory.\n
Graphs of bounded shrub-depth and first-order logic
Mittwoch, 11.12.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
We show that the expressive power of monadic second-order logic (MSO)\nand of first-order logic (FO) coincide on classes of graphs of bounded\nshrub-depth. Moreover we explain in what sense these classes are maximal\nclasses with MSO = FO.
Cohen reals und P-messbare Mengen
Mittwoch, 18.12.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Eine reelle Zahl heißt Cohen real, falls die Menge ihrer\nendlichen Anfangsstücke einen generischen Filter für das Cohen-Forcing\ndefiniert. Es folgt, dass Cohen reals keine Elemente des Grundmodells sein\nkönnen.\nFür eine Halbordnung P kann man die topologischen Eigenschaften "nirgends\ndicht" und "mager" sowie den Begriff der Messbarkeit verallgemeinern.\nIst die Halbordung P das Cohen Forcing, so entsprechen P-nirgends dicht\nund P-mager gerade ihren topologischen Definitionen und P-messbar der\nBaire-Eigenschaft.\nFür zwei Halbordnungen P und Q ergibt sich die interessante Fragestellung\nnach einem Zusammenhang der beiden Definitionen von Messbarkeit. Wenn Q das\nCohen Forcing ist, scheint es außerdem der Fall zu sein, dass es schon\ngenügt zu wissen, ob P Cohen reals addiert, um beantworten zu können, ob es\neinen Zusammhang zwischen P- und Q-messbar gibt.\n\nIn dem Vortrag stelle ich eine neue Forcinghalbordnung T vor. Ich werde\nexemplarisch an ihr zeigen, wie sich aus dem Nachweis von Cohen reals ein\nZusammenhang von T-messbar und der Baire Eigenschaft herstellen lässt. \nDer Vortrag beruht auf dem Paper "More on trees and Cohen reals", das in\nZusammenarbeit mit Giorgio Laguzzi entstanden ist.\n\n