Topics in Model Theory

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The focus of this project is model theory and its interactions with and applications to other areas of mathematics. Model theory is a part of mathematical logic, and studies the manner in which mathematical objects or classes of objects are defined linguistically. Consequently, model theory has developed various ways of measuring the simplicity and complexity of classes of functions or sets. Among the other areas of mathematics that the PI will study using model-theoretic methods, is combinatorics, in particular graphs, also informally called networks. Important results in combinatorics are "regularity" results, showing how networks can be decomposed into a small number of pieces, each such piece behaving "almost randomly". The PI will use model theoretic methods to extend and improve such regularity theorems, sometimes under a simplicity assumption. The core part of the proposal has three aspects: The first is to use methods from topological dynamics to obtain new invariants for first order theories and definable groups. The second concerns applications of model theory and nonstandard methods to arithmetic regularity theorems as well as the formulation of new regularity statements in a "tame" environment such as stability theory in continuous logic. The third involves the formulation of the notion of an approximate subgroup in a general model-theoretic context and giving a classification in the absence of invariant measures. Other parts of the proposal include describing p-adic semialgebraic groups and developing a model theory of affine group schemes (or pro-linear algebraic 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.