Uniformity, Universality, and Computability Theory

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.Comment: 61 Page

Definability and Classification of Equivalence Relations and Logical Theories

This thesis consists of four independent papers. In the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth. In the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1. In the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone. In the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic L&#x03C9;1&#x03C9;: every countable L&#x03C9;1&#x03C9;-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (L,T) and (L',T'), every Borel functor Mod(L',T') &#x2192; Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'&#x03C9;1&#x03C9;-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are &#x2135;0-categorical.</p

Tree-like graphings, wallings, and median graphings of equivalence relations

We prove several results showing that every locally finite Borel graph whose large-scale geometry is "tree-like" induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call a proper walling. The proof is based on the Stone duality between proper wallings and median graphs, i.e., CAT(0) cube complexes.Comment: 38 page

Applications of nonstandard analysis in differentrial game theory

In this study we look at optimal control theory and differential game theory. In the control section, to illustrate some of the nonstandard methods which we will be using, we give existence and uniqueness proofs for standard and Loeb measurable controls. The standard existence is a well-known result, the proof we give is is due to Keisler; this proof was given by him in previously unpublished lecture notes at the University of Wisconsin ([27]). The uniqueness proof is a simple application of Gronwall's Lemma ([31]).We then show that there is always an optimal Loeb control even in situations where there is no optimal Lebesgue control. Using this result we are then able to show the well known result that there is always a standard optimal relaxed control.In the games section, by using nonstandard analysis we show that, under certain circumstances, we have the existence of value for two player, zero-sum differential games played over the unit time interval. We follow the work of Elliott and Kalton and, as they did, we show that if the Isaacs condition holds then the game has value in the sense of Friedman. Over the relaxed controls the Isaacs condition is always satisfied and so there is always value for relaxed controls. Like Elliott and Kalton, we do not need Friedman's hypothesis that the variables appear separated in the dynamics and payoff. By using nonstandard methods we are, unlike Elliott and Kalton, able to show these results without using the Isaacs-Bellman equation, other than to explain what the Isaacs condition is. We also find it unnecessary to impose as many restrictions on the functions as Elliott and Kalton

On dilation symmetries arising from scaling limits

Quantum field theories, at short scales, can be approximated by a scaling limit theory. In this approximation, an additional symmetry is gained, namely dilation covariance. To understand the structure of this dilation symmetry, we investigate it in a nonperturbative, model independent context. To that end, it turns out to be necessary to consider non-pure vacuum states in the limit. These can be decomposed into an integral of pure states; we investigate how the symmetries and observables of the theory behave under this decomposition. In particular, we consider several natural conditions of increasing strength that yield restrictions on the decomposed dilation symmetry.Comment: 40 pages, 1 figur

Measurable selection for purely atomic games

A general selection theorem is presented constructing a measurable mapping from a state space to a parameter space under the assumption that the state space can be decomposed as a collection of countable equivalence classes under a smooth equivalence relation. It is then shown how this selection theorem can be used as a general purpose tool for proving the existence of measurable equilibria in broad classes of several branches of games when an appropriate smoothness condition holds, including Bayesian games with atomic knowledge spaces, stochastic games with countable orbits, and graphical games of countable degree—examples of a subclass of games with uncountable state spaces that we term purely atomic games. Applications to repeated games with symmetric incomplete information and acceptable bets are also presented

Fourier theory and C*-algebras

We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included