| Private Homepage | http://www.wwu.de/Logik/hils/ |
| Research Interests | Model Theory: Geometric Stability and Simplicity Theory, Hrushovski Amalgamation Valued Fields and their Model Theory, in particular Valued Fields with Automorphism |
| Selected Publications | • Hasson A, Hils M Fusion over sublanguages. Journal of Symbolic Logic Vol. 71 (2), 2006, pp 361-398 online • Hils M Generic automorphisms and green fields. J. London Math. Soc. (2) Vol. 85 (2), 2012, pp 223-244 online • Hils M La fusion libre: le cas simple. Journal of the Institute of Mathematics of Jussieu Vol. 7 (4), 2008, pp 825-868 online • Chernikov A, Hils M Valued difference fields and NTP2. Israel Journal of Mathematics Vol. 204 (1), 2014, pp 299-327 online • Baudisch A, Hils M Martin Pizarro A, Wagner F-O Die böse Farbe. Journal of the Institute of Mathematics of Jussieu Vol. 8 (3), 2009 online • Bays M, Hils M, Moosa R Model Theory of Compact Complex Manifolds with an Automorphism. Transactions of the American Mathematical Society Vol. 369 (6), 2017, pp 4485-4516 online • Bays M, Gavrilovich M, Hils M Some Definability Results in Abstract Kummer Theory. International Mathematics Research Notices Vol. 2014 (14), 2014, pp 3975-4000 online |
| Project membership Mathematics Münster | A: Arithmetic and Groups A2: Groups, model theory and sets |
| Current Talks | • Die Amalgamierungseigenschaft für definierbare Typen. Model Theory Conference , University of Wroclaw Link to event • Lang-Weil Abschätzungen in endlichen Differenzenkörpern. Model theory of valued fields, CIRM (Marseille) Link to event • Räume definierbarer Typen und schöne Paare in unstabilen Theorien. Géométrie et théorie des modèles, Institut Henri Poincaré, Paris Link to event • Schöne Paare in unstabilen Theorien und Räume definierbarer Typen. Research Seminar in Model Theory, Universität Wien Link to event • Lang-Weil Abschätzungen in endlichen Differenzenkörpern. Donau-Rhein-Modelltheorie-Seminar, Universität Passau Link to event • Lang-Weil Abschätzungen in endlichen Differenzenkörpern. Logic Colloquium, Universität Wien Link to event • Lang-Weil Abschätzungen in endlichen Differenzenkörpern. Model Theory: Combinatorics, Groups, Valued Fields and Neostability, Mathematisches Forschungsinstitut Oberwolfach Link to event • Getwistete Lang-Weil Abschätzungen und definierbare Quotienten. Zahlentheoretisches Kolloquium, TU Graz Link to event • Definierbare Quotienten. Mathematics Colloquium, Bosporus University, Istanbul Link to event |
| Current Publications | • Hils, Martin; Mennuni, Rosario Some definable types that cannot be amalgamated. Mathematical Logic Quarterly Vol. 69 (1), 2023 online • Hils, Martin; Liccardo, Martina; Touchard, Pierre Stably Embedded Pairs of Ordered Abelian Groups. , 2023 online • Hils, Martin; Ludwig, Stefan Marian An Approximate AKE Principle for Metric Valued Fields. , 2022 online • Hils, Martin; Hrushovski, Ehud; Simon, Pierre Definable Equivariant Retractions in Non-Archimedean Geometry. arXiv Vol. 2021, 2021 online • Cubides Kovacsics Pablo, Hils Martin, Ye Jinhe Beautiful pairs. arXiv Vol. 2021, 2021 online • Hils Martin, Mennuni Rosario The domination monoid in henselian valued fields. arXiv Vol. 2021, 2021 online • Hils Martin, Loeser François A First Journey through Logic. Student Mathematical Library, 2019 online • Hils M, Kamensky M, Rideau S Imaginaries in Separably Closed Valued Fields. Proceedings of the London Mathematical Society Vol. 2018, 2018 online • Hils Martin Model theory of valued fields. Lectures in Model TheoryMünster Lectures in Mathematics, 2018, pp 151-180 online |
| Current Projects | • Model Theory of Valued Fields with Endomorphism We propose a model-theoretic investigation of valued fields with non-surjective endomorphism. The model theory of valued fields with automorphism, in particular the Witt Frobenius case treated by Bélair, Macintyre and Scanlon, was extensively developed over the last 15 years, e.g., obtaining Ax-Kochen-Ershov principles for various classes of σ-henselian valued difference fields. We plan to generalize these results to the non-surjective context. The most natural example of a non-surjective endomorphism is the Frobenius map on an imperfect field. In analogy to the Witt Frobenius, the main examples for our study are Cohen fields over imperfect residue fields endowed with a lift of the Frobenius. The model theory of Cohen fields was recently developed in the work of Anscombe and Jahnke. In order to obtain Ax-Kochen-Ershov type results in our setting, it will be necessary to firstunderstand the equicharacteristic 0 case. This case is interesting in its own right, as it encompasses the asymptotic theory of Cohen fields with Frobenius lift. We are particularly interested in obtaining relative completeness and transfer of model theoretic tameness notions from value group and residue field to the valued difference field, as well as in identifying model companions for various subclasses. In all these cases, the endomorphism is an isometry of the valued field.
Another natural example of a σ-henselian valued difference fields is given by ultraproducts ofseparably closed valued fields with Frobenius. Here, the endomorphism is no longer an isometry, and the induced automorphism of the value group is ω-increasing. By work of Chatzidakis and Hrushovski the residue difference field of such an ultraproduct is existentially closed as a field with distinguished endomorphism. We aim to show the analogous result for the valued difference field, namely that it is existentially closed in a natural language, and infer existence and an axiomatization of the model-companion from this. online• Geometric and Combinatorial Configurations in Model Theory Model theory studies structures from the point of view of first-order logic. It isolates combinatorial properties of definable sets and uses these to obtain algebraic consequences. A key example is the group configuration theorem, a powerful tool in geometric stability used, e.g., to prove the trichotomy for Zariski geometries and in recent applications to combinatorics. Valued fields are an example of the confluence of stability theory and algebraic model theory. While Robinson studied algebraically closed valued fields already in 1959, the tools from geometric stability were only made available in this context in work of Haskell-Hrushovski-Macpherson, brought to bear in Hrushovski-Loeser's approach to non-archimedean geometry. In the project, we aim to strengthen the recent relations between model theory and combinatorics, develop the model theory of valued fields using tools from geometric stability and carry out an abstract study of the configurations which are a fundamental tool in these two areas. online • EXC 2044 - A2: Groups, model theory and sets Model theory and, more generally, mathematical logic as a whole has seen striking applicationsto arithmetic geometry, topological dynamics and group theory. Such applications as well asfundamental questions in geometric group theory, on the foundations of model theory and in settheory are at the focus of our work. The research in our group reaches from group theoretic questions to the model theory of groups andvalued fields as well as set theory. The study of automorphism groups of first order structures astopological groups, for examples, uses tools from descriptive set theory leading to pure set theoretic questions. online | hils at uni-muenster dot de |
| Phone | +49 251 83-32685 |
| FAX | +49 251 83-33078 |
| Room | 815a |
| Secretary | Sekretariat Weischer Frau Paulina Weischer Telefon +49 251 83-33790 Fax +49 251 83-33078 Zimmer 811 |
| Address | Prof. Dr. Martin Hils Institut für Mathematische Logik und Grundlagenforschung Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |
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