Construction and Representation of Generalised Equitable Preference Relations (Based on a Joint Paper with Ram Sewak Dubey)
Dienstag, 9.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In recent years a strict connection between set theory and social welfare relations has been studied in theoretical economics. In this talk I present some generalised versions of redistributional equity principles for infinite populations. More specifically we focus on the representation and construction of social welfare relations satisfying these generalised principles and we also combine them with other known efficiency and intergenerational equity principles in economic theory; in particular, important roles from set theory are played by ultrafilters and non-Ramsey sets.\n\n
The number of models of the theory of existentially closed differential fields revisited.
Dienstag, 16.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In 1973, Shelah showed that the theory of existentially closed differential fields of characteristic 0 although complete and totally transcendental admits the maximal number of models in any uncountable cardinality. This was extended by Hrushovski-Sokolovic and independently by Pillay to countable models in the 90’s. \n\nIn my talk, I will discuss how to recover (local and global versions of) Shelah’s result from the study of a specific family of differential equations: the differential equations of the form y’’/y’ = f(y) where f(y) is an arbitrary rational function introduced by Poizat in the 90's. \n\nThis is joint work with J. Freitag, D. Marker and R. Nagloo.
On abelian corners and squares
Dienstag, 23.11.21, 14:45-15:45, Raum 404, Ernst-Zermelo-Str. 1
Given an abelian group G, a corner is a a subset of pairs of the form \(\b{(x,y), (x+g, y), (x, y+g)\b}\) with \(g\) non trivial. Ajtai and Szemerédi proved that, asymptotically for finite abelian groups, every dense subset \(S\) of \(G\btimes G\) contains an corner. Shkredov gave a quantitative lower bound on the density of the subset \(S\). In this talk, we will explain how model-theoretic conditions on the subset \(S\), such as local stability, will imply the existence of corners and of cubes for (pseudo-)finite abelian groups. This is joint work with D. Palacin (Madrid/Freiburg) and J. Wolf (Cambridge).\n
On the Distributivity of Perfect Tree Forcings for Singulars of Uncountable Cofinality
Dienstag, 30.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Forcing with perfect trees is a major topic of research in set theory. One example is Namba forcing, which was originally developed as an example of a forcing that is \((\baleph_0,\baleph_1)\)-distributive but not \((\baleph_0,\baleph_2)\)-distributive. A recent paper of Dobrinen, Hathaway, and Prikry shows that a classical singular Namba forcing \(P_\bkappa\) is \((\bomega,\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) is a singular strong limit cardinal of countable cofinality. The authors then ask whether this result generalizes, i.e\(.\) if \(P_\bkappa\) is (cf\((\bkappa),\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) has uncountable cofinality. In joint work with Heike Mildenberger, we answer this question in the negative by showing that in this case \({\bmathbb P}_\bkappa\) is not (cf\((\bkappa),2)\)-distributive.
On the Distributivity of Perfect Tree Forcings for Singulars of Uncountable Cofinality
Dienstag, 30.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Forcing with perfect trees is a major topic of research in set theory. One example is Namba forcing, which was originally developed as an example of a forcing that is \((\baleph_0,\baleph_1)\)-distributive but not \n\((\baleph_0,\baleph_2)\)-distributive. A recent paper of Dobrinen, Hathaway, and Prikry shows that a classical singular Namba forcing \n\(P_\bkappa\) is \((\bomega,\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) is a singular strong limit cardinal of countable cofinality. The authors then ask whether this result generalizes, i.e. if \n\(P_\bkappa\) is (cf\((\bkappa),\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) has uncountable cofinality. In joint work with Heike Mildenberger, we answer this question in the negative by showing that in this case \n\(P_\bkappa\) is not (cf\((\bkappa),2)\)-distributive.