Martin Hils

University of Münster | WWU · Institute of Mathematical Logic

We investigate when an ordered abelian group $G$ is stably embedded in a given elementary extension $H$. We focus on a large class of ordered groups which includes maximal ordered groups with interpretable archimedean valuation. We give a complete answer for groups in this class which takes the form of a transfer principle for valued groups. It fol...

We exhibit a theory where definable types lack the amalgamation property.

We study metric valued fields in continuous logic, following Ben Yaacov's approach, thus working in the metric space given by the projective line. As our main result, we obtain an approximate Ax-Kochen-Ershov principle in this framework, completely describing elementary equivalence in equicharacteristic 0 in terms of the residue field and value gro...

We exhibit a theory where definable types lack the amalgamation property.

We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called beautiful pairs, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties...

We study interpretable sets in henselian and sigma-henselian valued fields with value group elementarily equivalent to Q or Z. Our first result is an Ax-Kochen-Ershov type principle for weak elimination of imaginaries in finitely ramified characteristic zero henselian fields -- relative to value group imaginaries and residual linear imaginaries. We...

We study the domination monoid in various classes of structures arising from the model theory of henselian valuations, including RV-expansions of henselian valued fields of residue characteristic 0 (and, more generally, of benign valued fields), p-adically closed fields, monotone D-henselian differential valued fields with many constants, regular o...

For $G$ an algebraic group definable over a model of $\operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G}$ of $G$, as introduced by Loeser and the second author. For $G$ connected and stably dominated, assuming $G$ commutative or that the valued field is of equicharacterist...

Das Kolloquium über Geschichte und Didaktik der Mathematik feiert im Jahr 2021 bereits sein 70-jähriges Bestehen. Auch in den Jahren 2011, 2001, 1991 und 1971 gab es bereits Zusammen-fassungen mit Vortragsthemen aus den vergangenen Jahren mit einem Blick zurück auf die Themen der gehaltenen Vorträge der Frage, welche inhaltlichen Leitstränge das Pr...

We show that separably closed valued fields of finite imperfection degree (either with lambda-functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of interpretable sets to study stably dominated types in those structures. We show that separably closed valued fields of finite impe...

KimByunghan. Simplicity Theory. Oxford Logic Guides, 53. Oxford University Press, Oxford, 2014, x+224 pp. - Volume 22 Issue 2 - Martin Hils

Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory CCMA of compact complex manifolds with a generic automorphism. It is shown that while CCMA does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexampl...

We extend the construction of bad fields of characteristic zero to the case of arbitrary prescribed divisible green torsion.

Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tracta...

We show that the theory of the non-standard Frobenius automorphism, acting on an algebraically closed valued field of equal characteristic 0, is NTP2. More generally, in the contractive as well as in the isometric case, we prove that a sigma-henselian valued difference field of equicharacteristic 0 is NTP2, provided both the residue difference fiel...

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purel...

Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tracta...

Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tracta...

We show that the generic automorphism is axiomatizable in the green field of Poizat (once Morleyized) as well as in the bad fields that are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain ‘bad pseudofinite fields’ in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is p...

The universal-algebraic approach has proved a powerful tool in the study of the complexity of CSPs. This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates, and relies on two facts. The first is that in finite or omega-categorical structures A, a relation is primitive positive definable i...

Zusammenfassung Wir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen Untergruppe der Dimension Eins, so daβ der Körper selbst Dimension Zwei hat. D...

We construct the free fusion of two geometric thories over a common ω-categorical strongly minimal reduct. If the two theories are supersimple of rank 1 (and satisfy an additional hypothesis true in particular for stable theories or trivial reduct), the completions of the free fusion are supersimple of rank at most ω.

We study countable universes similar to a free action of a group G. It turns out that this is equivalent to the study of free semi-actions of G, with two universes being transformable iff one corresponding free semi-action can be obtained from the other by a finite alteration. In the case of a free group G (in finitely many or countably many genera...

The present thesis deals with constructions by Hrushovski's amalgamation method in the relative context. First, the free fusion of two simple rank 1 theories T(1) and T(2) is constructed, over a common reduct T(0) which is supposed to be strongly minimal and omega-categorical. It is shown that, in many situations, its competions are simple. If the...

Generalising Hrushovski's fusion technique we construct the free fusion of two strongly minimal theories T 1. T 2 intersecting in a totally categorical sub-theory T 0. We show that if. e.g., T 0 is the theory of infinite vector spaces over a finite field then the fusion theory T ω, exists, is complete and ω-stable of rank ω. We give a detailed geom...

We construct a bad field in characteristic zero. That is, we con- struct an algebraically closed field which carries a notion of dimension analo- gous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber. Resume....

L'objet de cette thèse est l'étude des amalgames de Hrushovski dans le contexte relatif. D'abord, la fusion libre de deux théories simples de rang 1 T(1) et T(2) est construite, au-dessus d'un réduit commun T(0) qui est supposé fortement minimal et omega-catégorique. Dans bien des cas, il est montré que ses complétions sont simples. Si les T(i) son...