{"id":121,"date":"2020-11-20T13:54:41","date_gmt":"2020-11-20T13:54:41","guid":{"rendered":"https:\/\/beaadamday.com\/?page_id=121"},"modified":"2022-09-14T14:09:12","modified_gmt":"2022-09-14T14:09:12","slug":"models-and-sets","status":"publish","type":"page","link":"https:\/\/beaadamday.com\/index.php\/models-and-sets\/","title":{"rendered":"Models and Sets"},"content":{"rendered":"\n

The Leeds Models and Sets seminar series is now run by Ibrahim Mohammed and Carla Simons, and its webpage can be found here<\/a>.<\/p>\n\n\n\n

Here are a list of some previous seminars – for upcoming talks please see the Leeds webpage.<\/p>\n\n\n\n

14th<\/sup> June 2022<\/strong>
Speaker: Daisuke Ikegami<\/a>, Shibaura Institute of Technology
Title: On preserving AD via forcings
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Abstract: It is well-known that forcings preserve \"\mathsf{ZFC}\", i.e., any set generic extension of any model of \"\mathsf{ZFC}\" is again a model of \"\mathsf{ZFC}\". How about the Axiom of Determinacy (\"\mathsf{AD}\") under \"\mathsf{ZF}\"? It is not difficult to see that Cohen forcing always destroys \"\mathsf{AD}\", i.e., any set generic extension of a model of \"\mathsf{ZF}+ \mathsf{AD}\" via Cohen forcing is not a model of \"\mathsf{AD}\". Actually it is open whether there is a forcing which adds a new real while preserving \"\mathsf{AD}\". In this talk, we present some results on preservation & non-preservation of \"\mathsf{AD}\" via forcings, whose details are as follows:
1. Starting with a model of \"\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R))\", any forcing increasing \"\Theta\" destroys \"\mathsf{AD}\".
2. It is consistent relative to \"\mathsf{ZF} + \mathsf{AD}_R\" that \"\mathsf{ZF} + \mathsf{AD}^{+} +\" There is a forcing which increases \"\Theta\" while preserving \"\mathsf{AD}\".
3. In \"\mathsf{ZF}\", no forcings on the reals preserve \"\mathsf{AD}\". (This is an improvement of the result of Chan and Jackson where they additionally assumed \"\Theta\" is regular.)
4. In \"\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R)) + \Theta\" is regular, there is a forcing on \"\Theta\" which adds a new subset of \"\Theta\" while preserving AD.
This is joint work with Nam Trang.<\/p>\n\n\n\n

7th<\/sup> June 2022<\/strong>
Speaker: Lorna Gregory<\/a>, Universit\u00e0 degli Studi della Campania Luigi Vanvitelli
Title: Decidability of Theories of Modules of Pr\u00fcfer domains<\/p>\n\n\n\n


Abstract: An integral domain is Pr\u00fcfer if its localisation at each maximal ideal is a valuation domain. Many classically important rings are Pr\u00fcfer domains. For instance, they include Dedekind domains and hence rings of integers of number fields; B\u00e9zout domains and hence the ring of complex entire functions and the ring of algebraic integers; the ring of integer valued polynomials with rational coefficients and the real holomorphy rings of formally real fields.
Over the last 15 years, efforts have been made to characterise when the theory of modules of (particular types of) Pr\u00fcfer domains are decidable. I will give an overview of such decidability results culminating in recently obtained elementary conditions completely characterising when the theory of modules of an arbitrary Pr\u00fcfer domain is decidable.<\/p>\n\n\n\n

31st<\/sup> May 2022<\/strong>
Speaker: Omer Ben-Neria<\/a>, Hebrew University of Jerusalem
Title: Diamonds, Compactness, and Global Scales<\/p>\n\n\n\n


Abstract:  In pursuit of an understanding of the relations between compactness and approximation principles we address the question:  To what extent do compactness principles assert the existence of a diamond sequence? It is well known that a cardinal \"\kappa\" that satisfies a sufficiently strong compactness assumption must also carry a diamond sequence. However, other results have shown that certain weak large cardinal assumptions are consistent with the failure of the full diamond principle. We will discuss this gap and describe recent results with Jing Zhang which connect this problem to the existence of a certain global notion of cardinal arithmetic scales.<\/p>\n\n\n\n

24th<\/sup> May 2022<\/strong>
Speaker: Juvenal Murwanashyaka<\/a>, University of Oslo
Title: Weak Essentially Undecidable Theories of Concatenation<\/p>\n\n\n\n


Abstract: We sketch a proof of mutual interpretability of Robinson arithmetic and a weak finitely axiomatized theory of concatenation.<\/p>\n\n\n\n

17th<\/sup> May 2022<\/strong>
Speaker: Julia Knight<\/a>, University of Notre Dame
Title: Freeness and typical behavior for algebraic structures<\/p>\n\n\n\n

Abstract: The talk is on joint work with Johanna Franklin and Turbo Ho.  Gromov asked “What is a typical group?”  He was thinking of finitely presented groups.  He proposed an approach involving limiting density.  In 2013, I conjectured that for elementary first order sentences \"\varphi\", and for group presentations with \"n\" generators (\"n\geq 2\") and a single relator, the limiting density for groups satisfying \"\varphi\" always exists, with value \"0\" or \"1\", and the value is 1 iff \"\varphi\" is true in the non-Abelian free groups.  The conjecture is still open, but there are positive partial results by Kharlampovich and Sklinos, and by Coulon, Ho, and Logan.  We ask Gromov’s question about structures in other equational classes, or algebraic varieties<\/em> in the sense of universal algebra.  We give examples illustrating different possible behaviors.  Focusing on languages with just finitely many unary function symbols, we prove a result with conditions sufficient to guarantee that the analogue of the conjecture holds.  The proof uses a version of Gaifman’s Locality Theorem, plus ideas from random group theory and probability. <\/p>\n\n\n\n

10th<\/sup> May 2022<\/strong>
Speaker: Diana Carolina Montoya<\/a>, Kurt G\u00f6del Research Center for Mathematical Logic
Title: Higher independence at regular cardinals.<\/p>\n\n\n\n


Abstract: In the first part of this talk I will introduce the classical concept of a maximal independent family and its main properties. The second part of the talk will be devoted to deal with the generalisation of independence for regular uncountable cardinals. I will show the differences and similarities with the classical setting, as well as new lines of research that appear when dealing with this generalisation.
Finally,  I will mention some recent results of Vera Fischer and myself regarding independence.<\/p>\n\n\n\n

3rd<\/sup> May 2022<\/strong>
Speaker: Noa Lavi, Hebrew University of Jerusalem
Title: New irreducible generalised power series<\/p>\n\n\n\n


Abstract: A classical tool in the study of real closed fields are the fields \"K((G))\" of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field \"K\" of characteristic \"0\" and exponents in an ordered abelian group \"G\". A fundamental result of Berarducci ensures the existence of irreducible series in the subring \"K((G\le0))\" of \"K((G))\" consisting of the generalised power series with non-positive exponents.  We generalize previous results and show that for certain order types almost all series are irreducible or irreducible up to a monomial.<\/p>\n\n\n\n

26th<\/sup> April 2022<\/strong>
Speaker: Tamara Servi<\/a>, IMJ-PRG & Fields Institute
Title: Interdefinability and compatibility in certain o-minimal expansions of the real field<\/p>\n\n\n\n

Abstract: The sets definable in an o-minimal expansion of the real field have a tame topological behaviour (uniform finiteness, good dimension theory, no pathological phenomena). Being able to tell if a certain real set or function is definable in a given o-minimal structure gives us information on how tame the geometry of that object is.
Let us say that a real function \"f\" is o-minimal if the expansion \"(R,f)\" of the real field \"R\" by \"f\" is o-minimal. A function \"g\" is definable from \"f\" if \"g\" is definable in \"(R,f)\". Two o-minimal functions \"f\" and \"g\" are compatible if \"(R,f,g)\" is o-minimal. I will discuss the o-minimality, the interdefinability and the compatibility of two special functions, Euler’s Gamma and Riemann’s Zeta, restricted to the reals. Joint work with J.-P. Rolin and P. Speissegger.<\/p>\n\n\n\n

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29th<\/sup> March 2022<\/strong> – will be at 15:45-16:55 BST
Speaker: Kameryn J Williams<\/a>, Sam Houston State University
Title: The potentialist multiverse of classes<\/p>\n\n\n

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