Mathematics

We initiate a systematic study of the convolution operation on Keisler measures, generalizing the work of Newelski in the case of types. Adapting results of Glicksberg, we show that the supports of generically stable (or just definable, assuming NIP) measures are nice semigroups, and classify idempotent measures in stable groups as invariant measures on type-definable subgroups. We establish left-continuity of the convolution map in NIP theories, and use it to show that the convolution semigroup on finitely satisfiable measures is isomorphic to a particular Ellis semigroup in this context.

We continue the study of a class of topological L -fields endowed with a generic derivation δ , focussing on describing definable groups. We show that one can associate to an L δ definable group a type L -definable topological group. We use the group configuration tool in o-minimal structures as developed by K. Peterzil.

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of Polish spaces. We show that there exists a 0 ′ -computable low 3 Polish space which is not homeomorphic to a computable one, and that, for any natural number n , there exists a Polish space X n such that exactly the high 2n+3 -degrees are required to present the homeomorphism type of X n . We also show that no compact Polish space has an easiest presentation with respect to Turing reducibility.

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal K n -free graph for all n≥3 . Furthermore, we show that finite group actions on finite metric spaces or finite relational structures form a Fraïssé class, where Hall's group appears as the acting group of the Fraïssé limit. We also embed continuum many non-isomorphic countable universal locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn Δ -metric spaces in terms of the distance value set Δ .

We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include "Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?" and "Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?" We also provide a new bound for Borel chromatic numbers of graphs whose connected components all have two ends.

A Kaufmann model is an ω 1 -like, recursively saturated, rather classless model of PA . Such models were constructed by Kaufmann under ♢ and Shelah showed they exist in ZFC by an absoluteness argument. Kaufmann models are an important witness to the incompactness of ω 1 similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly naïve question of whether such a model can be ``killed" by forcing without collapsing ω 1 . We show that the answer to this question is independent of ZFC and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of ZFC whether or not Kaufmann models can be axiomatized in the logic L ω 1 ,ω (Q) where Q is the quantifier ``there exists uncountably many".

We show that the derivative of a log-analytic function is log-analytic. We prove that log-analytic functions exhibit strong quasianalytic properties. We establish the parametric version of Tamm's theorem for log-analytic functions.

We prove the Existential Closedness conjecture for the differential equation of the j -function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the j -function have solutions. Its consequences include a complete axiomatisation of j -reducts of differentially closed fields, a dichotomy result for strongly minimal sets in those reducts, and a functional analogue of the Modular Zilber-Pink with Derivatives conjecture.

We axiomatize a class of existentially closed exponential fields equipped with an E -derivation. We apply our results to the field of real numbers endowed with exp(x) the classical exponential function defined by its power series expansion and to the field of p-adic numbers endowed with the function exp(px) defined on the p -adic integers where p is a prime number strictly bigger than 2 (or with exp(4x) when p=2 ).

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let f:X→ R n be a definable map, where X is a definable set and R is the universe of the structure. We demonstrate the inequality dim(f(X))≤dim(X) in this paper. As a corollary, we get that the set of the points at which f is discontinuous is of dimension smaller than dim(X) . We also show that the structure is defiably Baire in the course of the proof of the inequality.