This ANR project gathers 20 people from 10 different universities to solve problems in modern analysis using groupoids and methods from operator algebras.
Many problems in Mathematical Physics, Number Theory, Geometry, Partial Diﬀerential Equations and other areas of science lead to advanced questions in Functional Analysis. An example of such a problem is to understand analysis on non-compact and singular spaces. Classical analysis on euclidean spaces and bounded domains can be modeled by either commutative \(C^*\)-algebras or by the algebra of compact operators, and both of these are examples of operator algebras that correspond to groupoids. We plan to use larger classes of groupoids to model many of the singular spaces that arise in applications. In addition to groupoids, we shall also consider other tools of noncommutative geometry, such as \(C^*\)-algebras, \(K\)-theory, and cyclic homology. Natural questions to study are those related to the index and the eta-invariants of the corresponding elliptic operators, the Hadamard well-posedness of the resulting equations, and applications of such operators.
Not all problems in analysis on singular spaces involve partial diﬀerential equations, however. Problems in Number Theory, Representation Theory, and Geometry also lead to analytical questions on non-compact or singular spaces that do not involve diﬀerential operators. The methods that we will develop in this project will be useful also for these problems.
Victor Nistor (email@example.com)