Dynamics Beyond Turbulence and Obstructions to Classification

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It is an empirical reality of our mathematical practice—and perhaps an inexorable artifact of any adequate mathematical formalism—that complex mathematical objects come with many different, yet equivalent, concrete descriptions. This multiverse of equivalent descriptions tends to grow in complexity as the associated objects of study become more complex, and it often becomes very difficult to solve the associated “classification problem”, i.e., to design an efficient method for telling whether two concrete descriptions correspond to the same object or to two different objects. It is a notoriously hard problem for example to decide whether two different concrete solutions of Einstein’s equations are descriptions of the exact same physical reality and an open problem to find measurable quantities which are invariant under the arbitrary change of coordinates. Invariant descriptive set-theory is an area of mathematical logic which provides a formal framework for measuring the intrinsic complexity of such classification problems and for deciding, in each case, which types of invariants are “too simple” to be used for a complete classification. It also provides an important link between topological dynamics and the meta-mathematics of classification as best exemplified by Hjorth’s theory of turbulence. Unfortunately, few things are known beyond the point of turbulence. Indeed, dynamical phenomena beyond the point of turbulence reside in groups of symmetries which are neither locally-compact nor non-archimedean, rendering classical methods of harmonic analysis and discrete model theory insufficient. However, a series of recent breakthroughs coming from the work of the PI and others suggest some new strategies for dealing with such dynamics. Through this program we initiate the systematic study of such dynamical phenomena which are “wilder” than turbulence and which can serve as obstructions to more general forms of classification.The proposed research program features four independent—yet mutually interacting—projects. The first project addresses the question of whether there exist obstructions to classification by actions of the unitary group of the infinite dimensional separable Hilbert space. In doing so it will examine the connections between the Borel reduction hierarchy and some recent developments in metric model theory regarding the correspondence between stability/NIP and reflexive/Rosenthal representability. The second project focuses on the dynamics of homeomorphism groups. Among others, it addresses the long-standing open problem of whether the homeomorphism group of the interval admits a turbulent action and it proposes some “higher dimensional” variants of turbulence which could be used as obstructions to classification by actions of homeomorphism groups of n-dimensional compacta. The third project proposes a unified framework for extracting the geometric content of various turbulent phenomena which are associated with dynamics of Banach spaces. The fourth project addresses several questions regarding the dynamics of Polish groups that do not admit two-sided invariant metrics which stem from the recent work of the PI and others.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.