Non-Commutative Spaces, Their Symmetries, and Geometric Quantum Group Theory

The project fits within the scope of the branch of mathematics known as noncommutative geometry, originating primarily in the discovery early in the 20th century that the then-newly-discovered phenomena of quantum mechanics require a novel mathematical formalism. The main point of departure from classical (or non-quantum) mathematics is the fact that by the very small-scale nature of our ambient world, certain pairs of measurable physical quantities cannot be measured simultaneously (with the momentum and position of a particle serving as the preeminent example). Mathematically, this manifests as the non-commutativity of a pair of transformations on a physical system, justifying the name "noncommutative" for the relevant field of study. The formal objects that model the symmetries (that is, structure-preserving transformations) of a physical system modeled according to this paradigm are known as a "quantum groups", and they are the central theme of the present research proposal. The training of graduate students is an important part of this project. A number of broader themes inform the problems under consideration. As one example, discrete quantum groups, like their classical counterparts, fall into a constellation of taxonomic classes based on the approximation properties enjoyed by their group algebras. The quantum versions of the classical results are typically more technically demanding, only partially settled, and good test beds for the strengths of the geometric-group-theoretic and operator-algebraic techniques that jointly make classical discrete groups such rich geometric and analytical objects. In another direction, much light can be shed on the structure and above-mentioned approximation properties of quantum groups (discrete or more generally, locally compact) by category and representation-theoretic means. For that reason, results to the effect that group-theoretic data (e.g. the center of a locally compact quantum group) can be reconstructed from categories of unitary representations with their underlying structure are of some interest in the field and the project. As a third example, randomness features heavily in the study of groups and other discrete objects (probabilistic methods are very important in graph theory, for instance); the general phenomenon whereby a "generic" object, constructed randomly (with the technical meaning of that phrase depending on the specifics of the problem) tends to be highly asymmetric appears to replicate in the quantum setting, with "most" finite graphs, finite metric spaces, etc. admitting no quantum symmetries. Such generic rigidity results always recover their classical counterparts (given that quantum symmetries always encompass ordinary ones), but usually require more involved and often more enlightening proof techniques. It is hoped the requisite eclectic mix of approaches to the problems (combinatorial, representation-theoretic, probabilistic, operator-algebraic, etc.) will offer insight not only into the nature of the quantum-mathematical objects ostensibly being studied, but also into the classical versions thereof. 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.