This article addresses the question which structures occur as fixed structures of stable structures with a generic automorphism. In particular we give a Galois theoretic characterization. Furthermore, we prove that any pseudofinite field is the fixed field of some model of ACFA, any one-free pseudo-differentially closed field of characteristic zero is the fixed field of some model of DCFA, and that any one-free PAC field of finite degree of imperfection is the fixed field of some model of SCFA.

Many classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.

This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality.

In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimension coming from the order.

We investigate Talagrand’s construction of an exhaustive pathological submeasure. We consider the forcing associated with this submeasure and we also begin an effort to explicitly describe this construction.

In this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

We show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called type shortness, and show that it follows similarly from large cardinals.

We investigate some common points between stable structures and weakly small structures and define a structure M to be fine if the Cantor-Bendixson rank of the topological space ${S_\varphi }\left( {dc{l^{eq}}\left( A \right)} \right)$ is an ordinal for every finite subset A of M and every formula $\varphi \left( {x,y} \right)$ where x is of arity 1. By definition, a theory is fine if all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subset A of a fine group G, the traces on the algebraic closure $acl\left( A \right)$ of A of definable subgroups of G over $acl\left( A \right)$ which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions.

We prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions f : K → K are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.

Given a class ${\cal C}$ of subgroups of a topological group G, we say that a subgroup $H \in {\cal C}$ is a universal${\cal C}$subgroup of G if every subgroup $K \in {\cal C}$ is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of ${\cal C}$ with respect to a natural preorder on the set of subgroups of G. We show that for any locally compact Polish group G, the countable power Gω has a universal Kσ subgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universal Kσ and compactly generated subgroups. As an aside, we explore the relationship between the classes of Kσ and compactly generated subgroups and give conditions under which the two coincide.

Assume ZF + AD + V = L(ℝ) and let κ < Θ be an uncountable cardinal. We show that κ is Jónsson, and that if cof (κ) = ω then κ is Rowbottom. We also establish some other partition properties.

Assume G is a group definable in a model M of a stable theory T. We prove that the semigroup SG (M) of complete G-types over M is an inverse limit of some semigroups type-definable in Meq. We prove that the maximal subgroups of SG (M) are inverse limits of some definable quotients of subgroups of G. We consider the powers of types in the semigroup SG (M) and prove that in a way every type in SG (M) is profinitely many steps away from a type in a subgroup of SG (M).

We study the expressive power of independence-friendly quantifier prefixes composed of universal $\left( {\forall x/X} \right)$, existential $\left( {\exists x/X} \right)$, and majority quantifiers $\left( {Mx/X} \right)$. We provide four quantifier prefixes that can express NP hard properties and show that all quantifier prefixes capable of expressing NP-hard properties embed at least one of these four quantifier prefixes. As for the quantifier prefixes that do not embed any of these four quantifier prefixes, we show that they are equivalent to a first-order quantifier prefix composed of $\forall x$, $\exists x$, and Mx. In unison, our results imply a dichotomy result: every independence-friendly quantifier prefix is either decidable in LOGSPACE or NP hard.

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:

1. (1) If ${\cal I}$ is a normal ideal on $\omega _2 $ which satisfies stationary antichain catching, then there is an inner model with a Woodin cardinal;

2. (2) For any $n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal ${\cal I}$ on $\omega _n $ which satisfies projective antichain catching, yet ${\cal I}$ is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

The main result of this paper is a combinatorial characterization of Magidor-generic sequences. Using this characterization, I show that the critical sequences of certain iterations are Magidor-generic over the target model. I then employ these results in order to analyze which other Magidor sequences exist in a Magidor extension. One result in this direction is that if we temporarily identify Magidor sequences with their ranges, then Magidor sequences are maximal, in the sense that they contain any other Magidor sequence that is present in their forcing extension, even if the other sequence is generic for a different Magidor forcing. A stronger result holds if both sequences come from the same forcing: I show that a Magidor sequence is almost unique in its forcing extension, in the sense that any other sequence generic for the same forcing which is present in the same forcing extension coincides with the original sequence at all but finitely many coordinates, and at all limit coordinates. Further, I ask the question: If d ε V[c], where c and d are Magidor-generic over V, then which Magidor forcing can d be generic for? It turns out that it must essentially be a collapsed version of the Magidor forcing for which c was generic. I treat several related questions as well. Finally, I introduce a special case of Magidor forcing which I call minimal Magidor forcing. This approach simplifies the forcing, and I prove that it doesn’t restrict the class of possible Magidor sequences. That is, if c is generic for a Magidor forcing over V, then it is generic for a minimal Magidor forcing over V.

We describe a variation of Ash’s η-system and give a new proof of Ash’s metatheorem. As an application, we prove a generalization of Ash and Knight’s theorem on pairs of structures.

We show that the valuation ring $F_q [[t]]$ in the local field $F_q \left( t \right)$ is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an ∃-$F_q $-definable bounded neighbouhood of 0. Then we “tweak” this set by subtracting, taking roots, and applying Hensel’s Lemma in order to find an ∃-$F_q $-definable subset of $F_q [[t]]$ which contains $tF_q [[t]]$. Finally, we use the fact that $F_q $ is defined by the formula $x^q - x = 0$ to extend the definition to the whole of $F_q [[t]]$ and to rid the definition of parameters.

Several extensions of the theorem are obtained, notably an ∃-∅-definition of the valuation ring of a nontrivial valuation with divisible value group.