We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (ove
r a sort $Gamma$) if every type over a sufficiently rich base structure can be viewed as part of a $Gamma$-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from $Gamma$, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF.
In 2011, Rideau and Winskel introduced concurrent games and strategies as event structures, generalizing prior work on causal formulations of games. In this paper we give a detailed, self-contained and slightly-updated account of the results of Ridea
u and Winskel: a notion of pre-strategy based on event structures; a characterisation of those pre-strategies (deemed strategies) which are preserved by composition with a copycat strategy; and the construction of a bicategory of these strategies. Furthermore, we prove that the corresponding category has a compact closed structure, and hence forms the basis for the semantics of concurrent higher-order computation.
We answer two open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invaria
nt extension property. These two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
We prove field quantifier elimination for valued fields endowed with both an analytic structure and an automorphism that are $sigma$-Henselian. From this result we can deduce various Ax-Kochen-Ersov type results with respect to completeness and the N
IP property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first order theory and that this theory is NIP.
We prove that the theory of the $p$-adics ${mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${rm GL}_n({mathbb Q}_p)/{rm GL}_n({mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$.
Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.
