Related papers: Stable domination and independence in algebraicall…

We show that many nice properties of a theory $T$ follow from the corresponding properties of its reducts to finite subsignatures. If $\{ T_i \}_{i \in I}$ is a directed family of conservative expansions of first-order theories and each…

This the first of a series of articles dealing with abstract classification theory. The apparatus to assign systems of cardinal invariants to models of a first order theory (or determine its impossibility) is developed in [Sh:a]. It is…

A definable type of a first-order theory is the same as a section (retraction) of the simplicial path space (decalage) of its space of types viewed as a simplicial topological space; as is well-known, in the category of simplicial sets such…

Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its type setting counterpart. We prove that…

We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs $C$ is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs…

Let p=tp(a/A) be a stationary type in an arbitrary finite rank stable theory, and P an A-invariant family of partial types. The following property is introduced and characterised: whenever c is definable over (A,a) and a is not algebraic…

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm{ACFG}$. This theory was introduced recently as a new example of $\mathrm{NSOP}_1$ non…

The v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in algebraic coding theory. In this paper we study the case of binomial edge ideals and we establish a significant…

A prevailing assumption in machine learning is that model correctness must be enforced after the fact. We observe that the properties determining whether an AI model is numerically stable, computationally correct, or consistent with a…

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and…

We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of 1-types. We show this to hold in Real…

We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more…

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\bar\kappa$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\bar\kappa$ type definable…

Stable homotopy theory is governed by the principle that after inverting loop spaces, homotopy types become the representing objects for homology theories. We show that this principle extends to higher category theory: inverting…

We provide a treatment of isomorphism within a set-theoretic formulation of dependent type theory. Type expressions are assigned their natural set-theoretic compositional meaning. Types are divided into small and large types --- sets and…

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature…

We study differential forms invariant under a finite reflection group over a field of arbitrary characteristic. In particular, we prove an analogue of Saito's freeness criterion for invariant differential 1-forms. We also discuss how…

We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We…

We survey various approaches to axiomatic stable homotopy theory, with examples including derived categories, categories of (possibly equivariant or localized) spectra, and stable categories of modular representations of finite groups. We…

We study domination of quadratic forms in the abstract setting of ordered Hilbert spaces. Our main result gives a characterization in terms of the associated forms. This generalizes and unifies various earlier works. Along the way we…