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We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We…
It was realized early on that topologies can model constructive systems, as the open sets form a Heyting algebra. After the development of forcing, in the form of Boolean-valued models, it became clear that, just as over ZF any…
We outline a portfolio of novel iterable properties of c.c.c. and proper forcing notions and study its most important instantiations, Y-c.c. and Y-properness. These properties have interesting consequences for partition-type forcings and…
We introduce and study in detail the notion of compatibility between valuations and orderings in real hyperfields. We investigate their relation with valuations and orderings induced on factor and residue hyperfields. Much of the theory…
We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and $p$-adically closed valued fields. We give an effectiveness condition, related to Hensel's lemma, on a valued…
Monadic stability and the more general monadic dependence (or NIP) are tameness conditions for classes of logical structures, studied in the 80's in Shelah's classification program in model theory. They recently emerged in algorithmic and…
A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce-Pillay, is formulated in terms of a relative notion of prolongation for Kolchin-closed…
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…
In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the…
We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…
We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general…
We define forcing orders which add witnesses to the failure of various forms of Friedman's Property. These posets behave similarly to the forcing order adding a nonreflecting stationary set but have the advantage of allowing the…
We study algebraic, combinatorial and topological properties of the set of preorders on a group, and the set of valuations on a field. We show strong analogies between these two kinds of sets and develop a dictionary for these ones. Among…
In this paper, we study several definitions of generalized rank weights for arbitrary finite extensions of fields. We prove that all these definitions coincide, generalizing known results for extensions of finite fields.
We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can…
Whenever P is a proper definable forcing for adding a real, the countable support iteration of P has all the preservation properties it can possibly have, within a wide syntactically identified class of properties.
We classify and construct irreducible completely splittable representations of affine and finite Hecke-Clifford algebras over an algebraically closed field of characteristic not equal to 2.
There exists an absolute constant $C$ with the following property. Let $A \subseteq \mathbb{F}_p$ be a set in the prime order finite field with $p$ elements. Suppose that $|A| > C p^{5/8}$. The set \[ (A \pm A)(A \pm A) = \{(a_1 \pm…
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…
Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term…