Research in Model Theory: Generalized Amalgamation Properties

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This project is a systematic study of objects related to generalized amalgamation properties in the frameworks of first-order theories and non-elementary classes. The properties are at the heart of Shelah's fundamental work on classification of first-order theories, as well as the results on excellent classes. Preliminary results allow to conjecture that failure of generalized amalgamation is witnessed by certain category-theoretic objects. Study of such objects is the central topic of this research. Kolesnikov's approach allows to treat generalized amalgamation properties in first-order theories and non-elementary classes as particular cases of a common phenomenon. The starting point of this research is the development of a theory of generalized amalgamation for algebraically closed sets in stable first-order theories, extending the work of Hrushovski. Such a theory appears to be rich even for totally categorical theories and could be of independent interest.A novel feature of this research is the idea to phrase generalized amalgamation in terms of functorial embeddings of simplicial complexes into models (of a first-order theory or those of an atomic class). The projected characterization of generalized amalgamation in terms of homological algebra would lead to influx of new ideas into model theory of both first-order theories and non-elementary classes.Model theory is a branch of mathematical logic that aims to analyze classes of mathematical structures that are axiomatized in some way (the terms "first-order theory" and "non-elementary class" refer to certain kinds of axioms that are used). One of the subfields of model theory, called classification theory, attempts to identify conditions on a class of structures that provide substantial information about the overall behavior of the elements of the class. The finer analysis of geometric model theory attempts to connect the combinatorial geometry of structures, defined in model-theoretic terms, with geometries coming from classically known structures. Generalized amalgamation properties originated in classification theory. The expectation is that Kolesnikov's research will connect the failure of properties to the existence of certain objects studied in a different area of mathematics. This research will contribute to the broader efforts on solving a long-standing (classification-theoretic) conjecture of Shelah and, potentially, will help in developing geometric methods in model theory of non-elementary classes.