Geometry behind the Baum-Connes Conjecture

This project concerns an unsolved problem in pure mathematics called the Baum-Connes conjec- ture. This article describes in absolutely non-technical terms what this problem is and how we propose to solve it. The problem is significant because it lies at the intersection of many branches of pure math- ematics. Hence resolving the Baum-Connes conjecture, or BCC for short, has implications for many unsolved problems such as Novikovs conjecture in topology or Kadison-Kaplanskys conjecture in op- erator theory. Let start with one version of BCC. There are two characters in this story, symmetries, and Dirac operators. Lets first get to know them better. The idea of symmetry is very intuitive to our minds. Any symmetry is a process that can be applied to a shape without changing it. Mathematicians abstract the concept of symmetries to what is called a group. Two shapes have the same kind of symmetries if the group abstracting them is the same. Groups allow us to say when two different shapes have the same kind of symmetries. Not surprisingly, we also seek symmetries in equations and their solutions. For many problems, it is important to know the symmetries of various equations or operators. A particularly relevant operator is Paul Diracs operator that describes an electrons motion in quantum mechanics. In mathematics, we refer to all operators with a structural similarity to Diracs operator with his name. Here is a question that has surprisingly many implications. How many essentially different Dirac operators are there with the same fixed group of symmetries? Mathematicians Paul Baum and Alain Connes con- jectured an answer to this question. This conjectured answer involves a simple "universal" shape for the given group of symmetries. In particular, their answer is easy to compute. However, their conjectured answer has not been verified for all groups. In the current project, we will concentrate on some important examples of groups. The main idea is to attach a boundary or a periphery to the universal shape given by Baum and Connes. The points on the boundary are located at infinity if we view from within the shape. Hence adding a boundary to a shape adds a scale to its geometry. Our goal is to utilize this scale in geometry to understand BCC. 1

The Baum-Connes conjecture is a profound statement in operator theory with far-reaching implications if proven true. This project considered a geometric approach to the validity of this conjecture. To this end, we study parabolic geometries, which are geometries associated with certain symmetries, called semi-simple Lie groups (in fact, lattices in this group). As part of this project, we construct and analyze modified hypoelliptic operators on parabolic geometries. These operators are expected to play the role of"Dirac" operators in this setup. In particular, we achieved the goal of constructing the required type of operators and also assigned to them a formula for their index. In the arguments a crucial role is palyed by deformations and groupoids that we device, depend upon a suitable choice of a submanifold.