# Logic seminar: Pietro Freni

Dates: | 5 June 2024 |
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Times: | 15:15 - 16:30 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Title: T-convexity and T-lambda-spherical completions of o-minimal structures.

Abstract: After introducing the basic notions relevant to the talk (valued field, o-minimal theory, T-convexity), I will discuss generalizations of the Kaplanski embedding theorem for RCVFs to T-convexly valued o-minimal fields 4.

If T is power bounded, there is a known straightforward generalization: every T-convexly valued (E, O) has a unique-up-to-non-unique-isomorphism spherically complete immediate elementary extension (follows from the residue-valuation property 3,5). This cannot happen if T is exponential 2.

However, the following generalization still holds: for every uncountable cardinal lambda, every T-convexly valued (E, O) has a unique-up-to-non-unique lambda-spherically complete el. extension which is weakly initial among the lambda-spherically complete el. extensions and weakly terminal among the "lambda-wim-constructible" extensions, moreover such extension has the same residue field as (E,O) 1.

1 Freni, P. (2024). T-convexity, Weakly Immediate Types and T-lambda-Spherical Completions of o-minimal Structures. arXiv preprint arXiv:2404.07646. 2 Franz-Viktor Kuhlmann, Salma Kuhlmann, and Saharon Shelah. Exponentiation in power series fields. Proceedings of the American Mathematical Society, 125(11):3177–3183, 1997 3 James Michael Tyne. T-levels and T-convexity. University of Illinois at Urbana-Champaign, 2003 4 Lou Van Den Dries and Adam H Lewenberg. T-convexity and tame extensions. The Journal of Symbolic Logic, 60(1):74–102, 1995. 5 Lou Van Den Dries and Patrick Speissegger. The field of reals with multisummable series and the exponential function. Proceedings of the London Mathematical Society, 81(3):513–565, 2000.