Final Report Summary - NCGSF (Noncommutative Geometry for Singular Foliations)
- The Baum-Connes conjecture and its importance
The Baum-Connes conjecture (BC) is the latest of a long sequence of important achievements linking Topology with Geometry and Analysis. It can be traced back to 1848, when the first form of the Gauss-Bonnet theorem appeared. This was followed by the Riemann-Roch theorem, Chern-Weil theory in the 1940s, and finally the much celebrated Atiyah-Singer theorem in the 1960s.
Actually BC is a far-reaching generalization of the latter. Its most simplified expression states that all the Analytic represeantations actually come from Geometry. The statement holds in a great deal of situations where internal symmetries (groups) are considered. On the other hand, Higson, V. Lafforgue and Skandalis came up with a counterexample. This raised the question of the range of applicability of BC. Well beyond this question though, the mere understanding of the properties of BC contains the key to understanding the local-to-global nature of the geometric situation under consideration. Namely, properties of BC imply other important results (Novikov and Kadison-Kaplansky conjectures), which provide the local-to-global information. Such questions have been in the spotlight among high-dimensional topologists since the 1970s, and can be traced back to the work of Poincare in the 19th century.
The applications of BC can be found in various different fields, for instance mathematical physics: For instance, Alain Connes showed that it can be used to make certain predictions which allow to calculate the spectrum of the Schroedinger operator. In fact, this approach used heavily the technology of foliations that this project is also concerned with. On the other hand, it is well-known that BC plays a major role in understanding the passage from classical to quantum mechanics (quantization).
- Singular foliations
Very roughly, foliations are geometric structures which come up in the process of solving differential equations. Actually, more than often, they determine completely the differential equations under consideration. From this point of view, foliations bring together various different fields of mathematics, such as dynamical systems, control theory, geometry, mathematical physics, etc. The working model of a foliation is the so-called "holonomy groupoid", an object which keeps track of the external symmetries of a foliation.
The counterexample of BC mentioned before appears for a well-behaved "regular" foliation. Such foliations are studied extensively, nevertheless the majority of foliations that appear in applications are very pathological, or "singular". It is exactly the topological properties of singular foliations that we would like to understand in order to extend the existing apparatus to the fields mentioned above, in particular mathematical physics (Poisson geometry).
- Objectives of the project
NCGSF is designed to formulate BC for any singular foliation. This will allow the deployment of the previous methods in order to understand better geometric structures as such. Much more importantly, we are interested in the applications of BC in mathematical physics: Working with singular foliations is really the key to studying the problem of quantization free from obstructions, and with the minimal possible information.
The methodology we follow builds on recent results by Androulidakis and Skandalis, who managed to construct all the necessary ingredients for the development of the Analytic part of BC (foliation C*-algebra, longitudinal pseudodifferential calculus, index theory). They key to their work is a novel construction of the (very pathological) holonomy groupoid, which is stable enough to allow for the development of the previous tools. What is missing in the formulation of BC is its Geometric part, namely a good model for a certain classifying space (of proper actions). In well-behaved situations (regular foliations), a model for this space was given by LeGall and Tu. In fact, we are looking for ways to generalize this model to the singular case.
The webpage of the project is: http://users.uoa.gr/~iandroul/NCGSF.html(opens in new window)
- Results
Unfortunately, the pathology of the holonomy groupoid (in the singular case) does not allow the application of the LeGall-Tu model to singular foliations. This is why, within the duration of the project we looked at the following:
1. Conditions ensuring the longitudinal smoothness of the holonomy groupoid.
2. To obtain model of the singular foliation around a compact leaf. In other words, to generalize the (local) Reeb stability theorem.
Understanding both of these will provide ways to deploy the LeGall-Tu model to singular foliations. In fact, both of these objectives were achieved in a series of 3 publications. Moreover, these publications explain how Connes' method for the calculation of the spectrum of the longitudinal Laplacian can be generalized to singular foliations using the Androulidakis-Skandalis construction.
- Impact
The main socio-economic impact of the final results will be to establish Noncommutative Geometry in Greece, making the University of Athens a reference point in the study of singular foliations. This is expected to create new opportunities for young mathematicians in Greece. The socio-economic importance of this is evident, given the current rates of youth unemployment in countries of the european south.
The Baum-Connes conjecture (BC) is the latest of a long sequence of important achievements linking Topology with Geometry and Analysis. It can be traced back to 1848, when the first form of the Gauss-Bonnet theorem appeared. This was followed by the Riemann-Roch theorem, Chern-Weil theory in the 1940s, and finally the much celebrated Atiyah-Singer theorem in the 1960s.
Actually BC is a far-reaching generalization of the latter. Its most simplified expression states that all the Analytic represeantations actually come from Geometry. The statement holds in a great deal of situations where internal symmetries (groups) are considered. On the other hand, Higson, V. Lafforgue and Skandalis came up with a counterexample. This raised the question of the range of applicability of BC. Well beyond this question though, the mere understanding of the properties of BC contains the key to understanding the local-to-global nature of the geometric situation under consideration. Namely, properties of BC imply other important results (Novikov and Kadison-Kaplansky conjectures), which provide the local-to-global information. Such questions have been in the spotlight among high-dimensional topologists since the 1970s, and can be traced back to the work of Poincare in the 19th century.
The applications of BC can be found in various different fields, for instance mathematical physics: For instance, Alain Connes showed that it can be used to make certain predictions which allow to calculate the spectrum of the Schroedinger operator. In fact, this approach used heavily the technology of foliations that this project is also concerned with. On the other hand, it is well-known that BC plays a major role in understanding the passage from classical to quantum mechanics (quantization).
- Singular foliations
Very roughly, foliations are geometric structures which come up in the process of solving differential equations. Actually, more than often, they determine completely the differential equations under consideration. From this point of view, foliations bring together various different fields of mathematics, such as dynamical systems, control theory, geometry, mathematical physics, etc. The working model of a foliation is the so-called "holonomy groupoid", an object which keeps track of the external symmetries of a foliation.
The counterexample of BC mentioned before appears for a well-behaved "regular" foliation. Such foliations are studied extensively, nevertheless the majority of foliations that appear in applications are very pathological, or "singular". It is exactly the topological properties of singular foliations that we would like to understand in order to extend the existing apparatus to the fields mentioned above, in particular mathematical physics (Poisson geometry).
- Objectives of the project
NCGSF is designed to formulate BC for any singular foliation. This will allow the deployment of the previous methods in order to understand better geometric structures as such. Much more importantly, we are interested in the applications of BC in mathematical physics: Working with singular foliations is really the key to studying the problem of quantization free from obstructions, and with the minimal possible information.
The methodology we follow builds on recent results by Androulidakis and Skandalis, who managed to construct all the necessary ingredients for the development of the Analytic part of BC (foliation C*-algebra, longitudinal pseudodifferential calculus, index theory). They key to their work is a novel construction of the (very pathological) holonomy groupoid, which is stable enough to allow for the development of the previous tools. What is missing in the formulation of BC is its Geometric part, namely a good model for a certain classifying space (of proper actions). In well-behaved situations (regular foliations), a model for this space was given by LeGall and Tu. In fact, we are looking for ways to generalize this model to the singular case.
The webpage of the project is: http://users.uoa.gr/~iandroul/NCGSF.html(opens in new window)
- Results
Unfortunately, the pathology of the holonomy groupoid (in the singular case) does not allow the application of the LeGall-Tu model to singular foliations. This is why, within the duration of the project we looked at the following:
1. Conditions ensuring the longitudinal smoothness of the holonomy groupoid.
2. To obtain model of the singular foliation around a compact leaf. In other words, to generalize the (local) Reeb stability theorem.
Understanding both of these will provide ways to deploy the LeGall-Tu model to singular foliations. In fact, both of these objectives were achieved in a series of 3 publications. Moreover, these publications explain how Connes' method for the calculation of the spectrum of the longitudinal Laplacian can be generalized to singular foliations using the Androulidakis-Skandalis construction.
- Impact
The main socio-economic impact of the final results will be to establish Noncommutative Geometry in Greece, making the University of Athens a reference point in the study of singular foliations. This is expected to create new opportunities for young mathematicians in Greece. The socio-economic importance of this is evident, given the current rates of youth unemployment in countries of the european south.