This paper shows that in general, difference fields do not have a difference closure. However, we introduce a stronger notion of closure (kappa-closure), and show that every algebraically closed difference field K of characteristic 0, with fixed field satisfying a certain natural condition, has a closure, and this closure is unique up to isomorphism over K.
{"title":"Remarks around the nonexistence of difference closure","authors":"Zoé Chatzidakis","doi":"10.2140/mt.2023.2.405","DOIUrl":"https://doi.org/10.2140/mt.2023.2.405","url":null,"abstract":"This paper shows that in general, difference fields do not have a difference closure. However, we introduce a stronger notion of closure (kappa-closure), and show that every algebraically closed difference field K of characteristic 0, with fixed field satisfying a certain natural condition, has a closure, and this closure is unique up to isomorphism over K.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135463810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that K is an infinite field which is large (in the sense of Pop) and whose first-order theory is simple. We show that K is bounded , namely has only finitely many separable extensions of any given finite degree. We also show that any genus 0 curve over K has a K -point and if K is additionally perfect then K has trivial Brauer group. These results give evidence towards the conjecture that large simple fields are bounded PAC. Combining our results with a theorem of Lubotzky and van den Dries we show that there is a bounded PAC field L with the same absolute Galois group as K . In the appendix we show that if K is large and NSOP ∞ and v is a nontrivial valuation on K then ( K , v) has separably closed Henselization, so in particular the residue field of ( K , v) is algebraically closed and the value group is divisible. The appendix also shows that formally real and formally p -adic fields are SOP ∞ (without assuming largeness).
{"title":"Galois groups of large simple fields","authors":"Anand Pillay, Erik Walsberg","doi":"10.2140/mt.2023.2.357","DOIUrl":"https://doi.org/10.2140/mt.2023.2.357","url":null,"abstract":"Suppose that K is an infinite field which is large (in the sense of Pop) and whose first-order theory is simple. We show that K is bounded , namely has only finitely many separable extensions of any given finite degree. We also show that any genus 0 curve over K has a K -point and if K is additionally perfect then K has trivial Brauer group. These results give evidence towards the conjecture that large simple fields are bounded PAC. Combining our results with a theorem of Lubotzky and van den Dries we show that there is a bounded PAC field L with the same absolute Galois group as K . In the appendix we show that if K is large and NSOP ∞ and v is a nontrivial valuation on K then ( K , v) has separably closed Henselization, so in particular the residue field of ( K , v) is algebraically closed and the value group is divisible. The appendix also shows that formally real and formally p -adic fields are SOP ∞ (without assuming largeness).","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"93 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using ideas from geometric stability theory we construct differentially closed fields with no non-trivial automorphisms.
{"title":"Rigid differentially closed fields","authors":"David Marker","doi":"10.2140/mt.2023.2.177","DOIUrl":"https://doi.org/10.2140/mt.2023.2.177","url":null,"abstract":"Using ideas from geometric stability theory we construct differentially closed fields with no non-trivial automorphisms.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"13 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.
{"title":"Residue field domination in some henselian valued fields","authors":"Clifton Ealy, Deirdre Haskell, Pierre Simon","doi":"10.2140/mt.2023.2.255","DOIUrl":"https://doi.org/10.2140/mt.2023.2.255","url":null,"abstract":"We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"86 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a reconstruction theorem valid for arbitrary theories in continuous (or classical) logic in a countable language, that is to say that we provide a complete bi-interpretation invariant for such theories, taking the form of an open Polish topological groupoid. More explicitly, for every such theory $T$ we construct a groupoid $mathbf{G}^*(T)$ that only depends on the bi-interpretation class of $T$, and conversely, we reconstruct from $mathbf{G}^*(T)$ a theory that is bi-interpretable with $T$. The basis of $mathbf{G}^*(T)$ (namely, the set of objects, when viewed as a category) is always homeomorphic to the Lelek fan. We break the construction of the invariant into two steps. In the second step we construct a groupoid from any emph{reconstruction sort}, while in the first step such a sort is constructed. This allows us to place our result in a common framework with previously established ones, which only differ by their different choice of a reconstruction sort.
{"title":"Star sorts, Lelek fans, and the reconstruction of non-ℵ0-categorical theories in continuous logic","authors":"Itaï Ben Yaacov","doi":"10.2140/mt.2023.2.285","DOIUrl":"https://doi.org/10.2140/mt.2023.2.285","url":null,"abstract":"We prove a reconstruction theorem valid for arbitrary theories in continuous (or classical) logic in a countable language, that is to say that we provide a complete bi-interpretation invariant for such theories, taking the form of an open Polish topological groupoid. More explicitly, for every such theory $T$ we construct a groupoid $mathbf{G}^*(T)$ that only depends on the bi-interpretation class of $T$, and conversely, we reconstruct from $mathbf{G}^*(T)$ a theory that is bi-interpretable with $T$. The basis of $mathbf{G}^*(T)$ (namely, the set of objects, when viewed as a category) is always homeomorphic to the Lelek fan. We break the construction of the invariant into two steps. In the second step we construct a groupoid from any emph{reconstruction sort}, while in the first step such a sort is constructed. This allows us to place our result in a common framework with previously established ones, which only differ by their different choice of a reconstruction sort.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"101 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study convolution semigroups of invariant/finitely satisfiable Keisler measures in NIP groups. We show that the ideal (Ellis) subgroups are always trivial and describe minimal left ideals in the definably amenable case, demonstrating that they always form a Bauer simplex. Under some assumptions, we give an explicit construction of a minimal left ideal in the semigroup of measures from a minimal left ideal in the corresponding semigroup of types (this includes the case of SL$_{2}(mathbb{R})$, which is not definably amenable). We also show that the canonical push-forward map is a homomorphism from definable convolution on $mathcal{G}$ to classical convolution on the compact group $mathcal{G}/mathcal{G}^{00}$, and use it to classify $mathcal{G}^{00}$-invariant idempotent measures.
{"title":"Definable convolution and idempotent Keisler measures, II","authors":"Artem Chernikov, Kyle Gannon","doi":"10.2140/mt.2023.2.185","DOIUrl":"https://doi.org/10.2140/mt.2023.2.185","url":null,"abstract":"We study convolution semigroups of invariant/finitely satisfiable Keisler measures in NIP groups. We show that the ideal (Ellis) subgroups are always trivial and describe minimal left ideals in the definably amenable case, demonstrating that they always form a Bauer simplex. Under some assumptions, we give an explicit construction of a minimal left ideal in the semigroup of measures from a minimal left ideal in the corresponding semigroup of types (this includes the case of SL$_{2}(mathbb{R})$, which is not definably amenable). We also show that the canonical push-forward map is a homomorphism from definable convolution on $mathcal{G}$ to classical convolution on the compact group $mathcal{G}/mathcal{G}^{00}$, and use it to classify $mathcal{G}^{00}$-invariant idempotent measures.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. A regular partition P for a 3-uniform hypergraph H = ( V, E ) consists of a partition V = V 1 ∪ . . . ∪ V t and for each ij ∈
{"title":"An improved bound for regular decompositions of 3-uniform hypergraphs of bounded VC2-dimension","authors":"Caroline Terry","doi":"10.2140/mt.2023.2.325","DOIUrl":"https://doi.org/10.2140/mt.2023.2.325","url":null,"abstract":". A regular partition P for a 3-uniform hypergraph H = ( V, E ) consists of a partition V = V 1 ∪ . . . ∪ V t and for each ij ∈","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"39 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss Jordan's theorem on finite subgroups of invertible matrices and give an account of his original proof.
{"title":"An exposition of Jordan’s original proof of his theorem on finite subgroups of GLn(ℂ)","authors":"Emmanuel Breuillard","doi":"10.2140/mt.2023.2.429","DOIUrl":"https://doi.org/10.2140/mt.2023.2.429","url":null,"abstract":"We discuss Jordan's theorem on finite subgroups of invertible matrices and give an account of his original proof.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"4 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define and study a higher-dimensional version of model theoretic internality, and relate it to higher-dimensional definable groupoids in the base theory.
{"title":"Higher internal covers","authors":"Moshe Kamensky","doi":"10.2140/mt.2023.2.449","DOIUrl":"https://doi.org/10.2140/mt.2023.2.449","url":null,"abstract":"We define and study a higher-dimensional version of model theoretic internality, and relate it to higher-dimensional definable groupoids in the base theory.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, of strongly bounded reducts of linearly ordered structures: A reduct $mathcal M$ of a linearly ordered structure $langle R;<,cdotsrangle $ is called emph{strongly bounded} if every $mathcal M$-definable subset of $R$ is either bounded or co-bounded in $R$. We investigate strongly bounded additive reducts of o-minimal structures and as a corollary prove the above theorem on additive reducts of real closed fields.
{"title":"Additive reducts of real closed fields and strongly bounded structures","authors":"Hind Abu Saleh, Ya’acov Peterzil","doi":"10.2140/mt.2023.2.381","DOIUrl":"https://doi.org/10.2140/mt.2023.2.381","url":null,"abstract":"Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, of strongly bounded reducts of linearly ordered structures: A reduct $mathcal M$ of a linearly ordered structure $langle R;<,cdotsrangle $ is called emph{strongly bounded} if every $mathcal M$-definable subset of $R$ is either bounded or co-bounded in $R$. We investigate strongly bounded additive reducts of o-minimal structures and as a corollary prove the above theorem on additive reducts of real closed fields.","PeriodicalId":21757,"journal":{"name":"Simul. Model. Pract. Theory","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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