Large Scale Geometry in Functional Analysis

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Functional analysis is a branch of mathematics that investigates vector spaces endowed with some notion of convergence and structure-preserving maps between them. Since many objects in nature can be modeled by such spaces, functional analysis is frequently used by scientists from different backgrounds. Topics in functional analysis under the optics of large-scale geometry will be of particular interest. In a nutshell, large-scale geometry is the study of the global structure of certain mathematical objects (think of geometric behavior measured by an observer far away from the object of interest). This subject is motivated by computer science. Indeed, when working with large data sets, such methods aid the understanding of global behavior. Moreover, this framework provides the appropriate tools to study the relation between different data sets through large-scale geometric embeddings and equivalences between those objects. The PI will continue his engagement with undergraduate and graduate students, and seminar/conference organization. This project will improve our understanding of certain linear objects (for instance, operator algebras, Banach spaces, operator spaces, etc.), given some nonlinear information about them. It is divided into three main parts: (1) Roe algebras. The goal is to understand how much of the large-scale geometry of a uniformly locally finite metric space (or more generally, of a uniformly locally finite coarse space) is encoded in their uniform Roe algebras and Roe algebras. The questions in this area are often referred to as ‘rigidity problems’ for Roe algebras. Embeddings and isomorphisms between those algebras and how their existence affects the geometry of the metric spaces will be studied. (2) Quantization. The quantization of classic mathematical objects allows one to interpret structures connected to Hilbert spaces as "noncommutative" or "quantum" versions of their classical counterparts. In collaborative work, the PI has recently proposed a quantization of coarse spaces and uniform Roe algebras. Further developments in the quantization of large-scale geometric properties and their relation with uniform Roe algebras will be investigated. (3) Operator spaces. Although the nonlinear geometry of Banach spaces has been receiving attention, especially in the last two decades, its natural noncommutative counterpart (that is, the nonlinear theory of operator spaces) has been waiting to be developed. The plan is to understand how much of the commutative theory holds for operator spaces and apply those results to the strictly noncommutative scenario.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.