Award: DMS 1105670, Principal Investigator: Markus J. Pflaum
The proposed work will advance the study of singularities by means of noncommutative geometry. Spaces with singularities appear abundantly and naturally in various areas of mathematics. Standard methods developed to study smooth manifolds or smooth varieties can in general not be extended to the singular setting, so one has to develop new approaches. Among the most promising new and original proposals which will provide progress for singularity theory is the idea to determine the cyclic homology theory of function algebras over spaces with singularities. This is the viewpoint from noncommutative geometry which goes back to the work of A. Connes and which has turned out to provide deeper mathematical insight not only into the structure theory of noncommutative but also of commutative algebras. In addition to the computation of cyclic homologies of function algebras over singular spaces, the PI plans to combine recent results from the stratification theory of singular spaces with noncommutative geometry to open up new paths to examine singularities. The construction of new topological invariants of singularities by this approach also promises to provide progress for index theory over spaces with singularities. In particular, it is intended to define inertia spaces associated to proper Lie groupoids and study their singularity structure with the goal of constructing a mathematical device which keeps track of the contribution of singularities to the cyclic homology of convolution algebras over proper Lie groupoids. Finally, relative cyclic cohomology theory will be used to construct and describe secondary invariants of geometric operators in singular situations.
Singularity theory is the mathematical discipline in which one describes and studies geometrical objects containing so-called singularities such as corners, edges or vertices. Besides these rather elementary singularities, considerably more complicated ones appear not only in mathematics itself but also in many physical or technical applications like for example hydro dynamics, string theory, robotics or catastrophe theory, which plays a fundamental role in the theoretical understanding of "catastrophic" phenomena in laser physics or population dynamics. A better mathematical understanding of singularities therefore will not only lead to progress within mathematics but also will have its impact for theoretical physics or engineering in situations where singular phenomena appear. The proposed project aims at improving the foundational knowledge on singularities by connecting singularity theory to another modern mathematical theory, namely noncommutative geometry. It is to be expected that this way new mathematical invariants for singularities can be constructed. This will provide further crucial steps towards a classification of singularities as they appear in mathematics, the sciences or engineering. To strengthen the broader impact of the project, the PI plans to present visualizations of singularities via a website specifically designed to disseminate mathematical knowledge.
Singularities appear abundantly in mathematics and its applications within the sciences and engineering. One usually understands by a singularity of a space a point whose neighborhood is not smooth. Elementary singularities are corners, edges or vertices. A significant class of spaces with a singular structure possess a so-called stratification which means that they have a locally finite decomposition into smooth parts.It was one of the main goals of the project to examine such stratified spaces from the viewpoint of noncommutative geometry, a far reaching mathematical theory invented by Alain Connes in the 1980ies which led to a better mathematical understanding of the geometry of quantized systems among other. In addition, noncommutative methods led to a better and more conceptual understanding of analytic problems and results, in particular of the index theory of elliptic operators. It was therefore a further goal of the project to derive new index theorems in the singular setting out of the results obtained by examining singularities through noncommutative geometry. In the first phase of the project the PI and collaborators Posthuma and Tang studied the singularities of so-called Lie groupoids. Groupoids represent a significant category of singular spaces and also serve to describe the generalized symmetries of the spaces they represent. One of the outcomes of the joint work with Posthuma and Tang has been the construction of an explicit stratification of the orbit space of a proper Lie groupoid using a metric linearization theorem for groupoids which the PI and collaborators derived from Zung's linearization theorem. Moreover, further topological and geometric properties of orbit spaces of groupoids have been determined. Altogether this led to a publication in the renown Crelle Journal. Following this work, the PI and collaborators Farsi and Seaton examined the stratification theory of inertia spaces of groupoids. Since these spaces serve as "book-keeping" devices for the contribution of singularities to the index theory over a groupoid, the work by Farsi, Seaton and the PI also served as progress in regrad to the second goal of the proposed project. A corresponding paper has been accepted for publication in Journal of Singularities. A major outcome of the project, mainly obtained during the main phase of the project, was the derivation of two new index theorems for groupoids, one for so-called longitudinally elliptic operators on Lie groupoids, the other one for appropriate operators along the action of a Lie groupoid on another space. The resulting index theorems are among the most general ones known so far and comprise several other deep index theorems such as those by Atiyah-Singer and Connes-Skandalis. Methodically, noncommutative geometry plaid a crucial role in the proof of these theorems thus confirming the strength of the chosen approach. The work on the index theorems on groupoids has been in collaboration with Posthuma and Tang, and led to two papers which have been accepted for publication in Journal of Differential Geometry and Advances In Mathematics, respectively. Finally, the PI has worked with a former PhD student of his on the real homotopy theory of semianalytic sets. It could be shown in this project that the Whitney-deRham complex over a simply connected seminalytic set determines the real homotopy type of the set. This generalizes a famous result by Sullivan to a wide class of singular spaces. A corresponding paper has been submitted.