Goldbring proposes to continue the use of techniques from logic, specifically model theory, to answer questions about structures in analysis. More specifically, he plans on: (1) using the techniques of nonstandard analysis to investigate problems in infinite-dimensional Lie theory and geometric group theory; and (2) continuing the development of continuous logic and model theory for metric structures. In regards to item (1), he plans on investigating how one can use the technique of hyperfinite approximation in infinite-dimensional Lie theory by embedding infinite-dimensional objects inside of hyperfinite objects in the hopes that the finite-dimensional theory will apply to the hyperfinite objects and the infinite-dimensional objects embedded in them. In regards to (2), Goldbring plans on continuing the study of foundational aspects of continuous logic, including definability of sets and functions and the behavior of independence relations in various metric structures.
Model theory, a branch of mathematical logic, studies what properties of mathematical structures can be described in terms of first-order logic. This understanding can often lead to solutions to problems in various mathematical areas by taking general model-theoretic facts and interpreting them in specific contexts. This approach has been most successful in discrete situations, such as algebra and combinatorics. Recently, a model theory has been developed for structures arising in analysis, namely those for which a natural notion of distance exists. This model theory could then prove useful for solving problems in areas where classical model theory has yet to be able to help. That being said, model theory, in the form of nonstandard analysis, has found applications in analysis by providing a rigorous notion of infinitesimal and infinite numbers. Nonstandard analysis has found applications in such diverse areas as number theory, probability theory, and economics. This project proposes to use nonstandard analysis to answer questions in Lie theory, which is a very important mathematical area used in physics and engineering.
