Group actions on 1-manifolds: a list of very concrete open questions

This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.Comment: 21 pages, 2 figure

A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets

We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal order type of the term ordering on the finite multisets over a wpo. Moreover we discuss an approach to compute maximal order types of well-partial orders which are related to tree embeddings

Classes of structures with no intermediate isomorphism problems

We say that a theory $T$ is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence $T$ is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle on a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle on a cone

Social Choice with Analytic Preferences

A social welfare function is a mapping from a set of profiles of individual preference orderings to the set of social orderings of a universal set of alternatives. A social choice correspondence specifies a nonempty subset of the agenda for each admissible preference profile and each admissible agenda. We provide examples of economic and political preference domains for which the Arrow social welfare function axioms are inconsistent, but whose choice-theoretic counterparts (with nondictatorship strengthened to anonymity) yield a social choice correspondence possibility theorem when combined with a natural agenda domain. In both examples, agendas are compact subsets of the nonnegative orthant of a multidimensional Euclidean space. In our first possibility theorem, we consider the standard Euclidean spatial model used in many political models. An agenda can be interpreted as being the feasible vectors of public goods given the resource constraints faced by a legislature. Preferences are restricted to be Euclidean spatial preferences. Our second possibility theorem is for economic domains. Alternatives are interpreted as being vectors of public goods. Preferences are monotone and representable by an analytic utility function with no critical points. Convexity of preferences can also be assumed. Many of the utility functions used in economic models, such as Cobb-Douglas and CES, are analytic. Further, the set of monotone, convex, and analytic preference orderings is dense in the set of continuous, monotone, convex preference orderings. Thus, our preference domain is a large subset of the classical domain of economic preferences. An agenda can be interpreted as the set of feasible allocations given an initial resource endowment and the firms' production technologies. To establish this theorem, an ordinal version of the Analytic Continuation Principle is developed.