CAREER: Model theory, independence, and approximation

This is a project in model theory, a branch of mathematical logic that studies the features of mathematical structures that can be expressed by formal languages. Model theory usually studies infinite structures, though, within model theory, there are techniques for taking a collection of finite structures--graphs, orders, groups, etc.--and producing an infinite limit structure that encodes useful information about the finite structures. For example, there are Fraïssé limits, like the Rado graph or the rational order, which allow for the study of all finite graphs and orders via their rich collection of symmetries, and ultraproducts, which are logical limits that reflect the asymptotic features of a sequence of structures detected by first order logic. Some of the most striking developments in model theory concern the 'smoothly approximable' structures, a special class of structures that can be seen simultaneously as both kinds of limit, which are built out of classical geometries over finite fields and have deep connections to combinatorics and group theory. The geometries that form their basic building blocks also can be considered over infinite fields and are important objects of study in several areas of mathematics. As with the infinite limits of finite structures, model-theoretic techniques allow for the construction of infinite-dimensional limits of these (infinite but) finite-dimensional geometries and new tools are needed to understand them. This project is focused on the development of those tools. The educational component involves organization of a summer school at University of Notre Dame and working with graduate students. In addition, the PI plans to work with students in several countries where access to high level math education is more limited. The PI's research agenda will develop a structure theory for NSOP_1 theories, pushing beyond the reaches of simplicity theory and enabling the treatment of smoothly approximable structures over infinite fields. The name NSOP_1 is an unfortunate label for a very natural class of theories, serving as a kind of 'compactification' of the simple theories and encompassing many important examples. The PI has proposed a two-part program. The first part would develop the 'geometric' aspects of the theory, by analogy with geometric stability theory, with a focus on definable groups and higher amalgamation. The second part proposes a far-reaching program to expand the reach of the existing structure theory for smoothly approximable structures developed, in its ultimate form, by Cherlin and Hrushovski. In that work, group theory, via work of Kantor, Liebeck, and Macpherson, provided a catalogue of the basic building blocks and classification theory, via the then-nascent theory of simple theories, explained how a smoothly approximable structure is assembled out of these basic pieces. For the analogue over infinite fields, there are significant questions both on the group-theoretic side, concerning the basic geometries, and the classification-theoretic side, concerning a 'geometric' theory of NSOP_1 theories. The PI aims to develop the analysis of geometries in two distinct regimes: the algebraic and the pseudo-finite. This will enable meaningful applications to algebra, representation theory, and combinatorics, particularly to groups of finite Morley rank and to the asymptotic behavior of finite permutation groups. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.