Nicholas Ramsey
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View article: A TWO-SORTED THEORY OF NILPOTENT LIE ALGEBRAS
A TWO-SORTED THEORY OF NILPOTENT LIE ALGEBRAS Open
We prove the existence of a model companion of the two-sorted theory of c -nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field s…
View article: Primitive pseudo-finite permutation groups of finite SU-rank
Primitive pseudo-finite permutation groups of finite SU-rank Open
We study definably primitive pseudo-finite permutation groups of finite $SU$-rank. We show that if $(G,X)$ is such a permutation group, then the rank of $G$ can be bounded in terms of the rank of $X$, providing an analogue of a theorem of …
View article: Some model theory of quadratic geometries
Some model theory of quadratic geometries Open
Orthogonal spaces are vector spaces together with a quadratic form whose associated bilinear form is non-degenerate. Over fields of characteristic two, there are many quadratic forms associated to a given bilinear form and quadratic geomet…
View article: A two-sorted theory of nilpotent Lie algebras
A two-sorted theory of nilpotent Lie algebras Open
We prove the existence of a model companion of the two-sorted theory of $c$-nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field …
View article: Generic Stability Independence and Treeless Theories
Generic Stability Independence and Treeless Theories Open
We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both bina…
View article: Model-theoretic properties of nilpotent groups and Lie algebras
Model-theoretic properties of nilpotent groups and Lie algebras Open
We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent $p$ studied by Baudisch is 2-dependent and NSOP$_{1}$. We prove that the class of…
View article: A New Kim's Lemma
A New Kim's Lemma Open
Kim's Lemma is a key ingredient in the theory of forking independence in simple theories. It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters. Variants of Kim's Lemma have formed the …
TRANSITIVITY, LOWNESS, AND RANKS IN NSOP THEORIES Open
We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As application…
View article: Generic Stability Independence and Treeless theories
Generic Stability Independence and Treeless theories Open
We initiate a systematic study of \emph{generic stability independence} and introduce the class of \emph{treeless theories} in which this notion of independence is particularly well-behaved. We show that the class of treeless theories cont…
View article: Measures on perfect e-free PAC fields
Measures on perfect e-free PAC fields Open
We construct measures on definable sets in $e$-free perfect PAC fields, as well as on perfect PAC fields whose absolute Galois groups are free pro-$p$ of finite rank. We deduce the definable amenability of all groups definable in such fiel…
View article: Invariant measures in simple and in small theories
Invariant measures in simple and in small theories Open
We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such …
View article: Exact saturation in pseudo-elementary classes for simple and stable theories
Exact saturation in pseudo-elementary classes for simple and stable theories Open
We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple…
View article: Transitivity, lowness, and ranks in NSOP$_1$ theories
Transitivity, lowness, and ranks in NSOP$_1$ theories Open
We develop the theory of Kim-independence in the context of NSOP$_{1}$ theories satsifying the existence axiom. We show that, in such theories, Kim-independence is transitive and that $\ind^{K}$-Morley sequences witness Kim-dividing. As ap…
View article: On Kim-independence
On Kim-independence Open
We study NSOP _1 theories. We define Kim-independence , which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim’s lemma, loca…
View article: Local character of Kim-independence
Local character of Kim-independence Open
We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of ele…
View article: A note on NSOP$_{1}$ in one variable
A note on NSOP$_{1}$ in one variable Open
We prove that, in order to establish that a theory is NSOP$_{1}$, it suffices to show that no formula in a single free variable has SOP$_{1}$.
View article: On model-theoretic tree properties
On model-theoretic tree properties Open
We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see text]) for c…