IA Scholar Query: The Lascar groups and the first homology groups in model theory.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 31 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Remarks on classification theory for abstract elementary classes with applications to abelian group theory and ring theory
https://scholar.archive.org/work/ezootjr2pffmxpwvzs2sy3qbdu
This thesis has two parts. The first part deals with the classification theory of abstract elementary classes and the second part deals with links and applications of this theory to algebra. Part I: Remarks on classification theory for abstract elementary classes This part of the thesis is made up of three chapters based on the corresponding papers: [Ch. 2], [Ch. 3] (a joint work with S. Vasey), and [Ch. 4] (a joint work with R. Grossberg). Chapter 2, Non-forking w-good frames. We introduce and study the notion of a w-good λ-frame which is a weakening of Shelah's notion of a good λ-frame. W-good λ-frames are useful as they imply the existence of larger models. We show that if K has a w-good λ-frame, then K has a model of size λ ++. This result extends [Sh:h, §II.4.13.3], [JaSh13, 3.1.9], and [Vas16a, 8.9]. Chapter 3, Universal classes near ℵ1 (a joint work with S. Vasey). Shelah has provided sufficient conditions for an Lω1,-ωsentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ℵn's. Using tools of Boney, Shelah, and Vasey, we give assumptions on ℵ0 and ℵ1 which suffice when ψ is restricted to be universal. Chapter 4, Simple-like independence relations in abstract elementary classes (a joint work with R. Grossberg). We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. We show that if K has a simple independence relation with the (< ℵ0)-witness property for singletons, then K does not have the tree property. We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation. Part II: Applications to abelian group theory and ring theory [...]Marcos Mazari Armidawork_ezootjr2pffmxpwvzs2sy3qbduMon, 31 Oct 2022 00:00:00 GMTThe model theory of the curve graph
https://scholar.archive.org/work/cmqwd52klzcwrjjq77ao5j3ywq
In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the general question: why does the curve graph of a surface play such a central role in the study of surfaces and mapping class groups? More specifically, we consider a surface Σ of finite type and its curve graph 𝒞(Σ), and we investigate its first-order theory in the language of graph theory. Crucially, 𝒞(Σ) is bi-interpretable with a certain object called the augmented Cayley graph of the mapping class group of the surface. We use this bi-interpretation to prove that the theory of the curve graph is ω–stable, to compute its Morley rank, and to show that it has quantifier elimination with respect to the class of ∀∃–formulae. We also show that many of the complexes which are naturally associated to a surface are interpretable in 𝒞(Σ). This shows that these complexes are all ω–stable and admit certain a priori bounds on their Morley ranks. We are able to use Morley ranks to prove that various complexes are not bi–interpretable with the curve graph. As a consequence of quantifier elimination, we show that algebraic intersection number is not definable in the first order theory of the curve graph. Finally, we prove that the curve graph of a surface enjoys a novel phenomenon that we call interpretation rigidity. That is, if surfaces Σ_1 and Σ_2 admits curve graphs that are mutually interpretable, then Σ_1 and Σ_2 are homeomorphic to each other. Along the way, numerous technical results are obtained.Valentina Disarlo, Thomas Koberda, J. de la Nuez Gonzálezwork_cmqwd52klzcwrjjq77ao5j3ywqMon, 31 Oct 2022 00:00:00 GMTNiche differentiation within a cryptic pathogen complex: climatic drivers and hyperparasitism at multiple spatial scales
https://scholar.archive.org/work/fipwvxvbnfgvblcf3ov7k6wrnm
Pathogens are embedded in multi-trophic food webs, which often include co-occurring cryptic species within the same pathogen complex. Nonetheless, we still lack an understanding of what dimensions of the ecological niche might allow these cryptic species to coexist. We explored the role of climate, host characteristics (tree autumn phenology) and attack by the fungal hyperparasite Ampelomyces (a group of fungi attacking plant pathogens) in defining the niches of three powdery mildew species (Erysiphe alphitoides, E. hypophylla and E. quercicola) within a cryptic pathogen complex on the pedunculate oak Quercus robur at the continental (Europe), national (Sweden and France) and landscape scales (a 5 km 2 island in southwestern Finland). Previous studies have shown that climate separated the niches of three powdery mildew species (E. alphitoides, E. hypophylla and E. quercicola) in Europe and two species (E. alphitoides and E. quercicola) in France. In our study, we did not detect a significant relationship between temperature or precipitation and the distribution of E. alphitoides and E. hypophylla present in Sweden, while at the landscape scale, temperature, but not relative humidity, negatively affected disease incidence of E. alphitoides in an exceptionally warm year. Tree variation in autumn phenology did not influence disease incidence of powdery mildew species, and hyperparasite presence did not differ among powdery mildew species at the continental, national and landscape scale. Climate did not affect the distribution of the hyperparasite at the continental scale and at the national scale in Sweden. However, climate affected the hyperparasite distribution in France, with a negative relationship between non-growing season temperature and presence of the hyperparasite. Overall, our findings, in combination with earlier evidence, suggest that climatic factors are more important than species interactions in defining the niches of cryptic species within a pathogen complex on oak.Maria Faticov, Marie‐Laure Desprez‐Loustau, Levente Kiss, Marie Massot, Julie Faivre d'Arcier, Jessie Mutz, Márk Z. Németh, Tomas Roslin, Ayco J. M. Tackwork_fipwvxvbnfgvblcf3ov7k6wrnmMon, 17 Jan 2022 00:00:00 GMT