ROe AlgebRas: between coarse geometry and operator algebras – ROAR

This project is about fundamental research in operator algebras and coarse geometry, two areas of pure mathematics. Its primary goal is to develop novel techniques to solve important problems on Roe-like C*-algebras. These algebras of operators were introduced by John Roe in the early 1990s to analyse coarse geometric notions with operator algebraic tools.

Coarse geometry studies metric spaces from far away (two spaces are the same if they look the same at a large scale). Coarse geometry underlies most of modern geometric group theory, and has applications to many areas of mathematics, including mathematical physics.

The study of algebras of operators is a branch of functional analysis, focusing on linear operators on complex Hilbert spaces. It is the noncommutative analog of topology and measure theory, and it has connections with virtually all other fields of mathematics, such as dynamics, ergodic theory, number theory and mathematical physics.

Roe type algebras are operator algebras associated to metric spaces introduced to approach problems of geometric nature. They found prominent applications in many fields of mathematics, such as geometric group theory and mathematical physics.
The goal of this project is to make significant and all around progress around the following broad question: Which coarse geometric properties are encoded by the algebraic behavior of Roe type algebras?

The project is divided into three highly connected work packages.
- WP1 focuses on the "Rigidity problem for Uniform Roe algebras", asking whether an isomorphism of two uniform Roe algebras gives a bijective coarse equivalence between the associated spaces.
- WP2 is centered around the Higson corona, a natural quotient algebra which encodes asymptotic properties of the metric spaces of interest. We plan to use the Higson corona to attack the dimension problem, an instance of the above question focusing on different notions of dimension in operator algebras and coarse geometry.
- WP3 is centered around the technical Finite Decomposition Complexity and amenability-like conditions, and consequently the existence of universal objects. A spin-off ambitious theme is the application of Finite Decomposition Complexity methods to the study of Thompson's groups.

The team is led by the PI Alessandro Vignati, and consists of young (K. Krutoy, a Ph.D. student) and experienced (F. Le Maître and R. Tessera) researchers in the areas of operator algebras and coarse geometry. The team will be completed by one postdoctoral fellow for three years.