Logic seminar: Pablo Andujar Guerrero (University of Leeds)
| Dates: | 21 February 2024 |
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| Times: | 15:15 - 16:30 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |
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Title: Defining definable compactness
Abstract: The first notion of definable topological compactness was introduced within o-minimality in 1999 by K. Peterzil and C. Steinhorn for definable manifold spaces (e.g. definable groups), with the general motivation of extending the role of compactness in topological algebra to the o-minimal setting. Their definition was the property that every definable curve converges (onwards curve-compactness). Its study was later motivated by the fact that it played a crucial role in the formulation of Pillay's conjecture about o-minimal definable groups.
In the 2000s the investigation of o-minimal forking led to an equivalent characterization of definable compactness: every definable family of closed sets has a finite transversal. The equivalence with a third notion, that of having fsg (finitely satisfiable generics), was also observed. In the 2010s E. Hrushovski and F. Loeser adapted curve-compactness to the valued field setting by considering the property that every definable type converges. Furthermore, authors such as A. Fornasiero and W. Johnson started exploring yet another notion of definable compactness: every definable filter of closed sets has non-empty intersection. The latter began extending the topological algebraic work of Peterzil and Steinhorn in o-minimality to the the field of p-adics. In this talk we discuss the relationship between all the aforementioned definitions of definable compactness in various NIP settings (including o-minimal, p-adic and distal dp-minimal). We present the current literature, open questions, and the model theory behind characterizing definable compactness.
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