Analysis on Singular Spaces

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AbstractNistorThe project is devoted to application of certain areas of Analysis, especially Index theory, K-theory, and Hochschild and cyclic homology, to a broad range of problems that have implications in several other areas of mathematics and physics. A common issue in these problems is to identify the relevant algebras that model specific situation of interest. Sometimes these algebras already exist, sometimes they have to be constructed. These algebras will then be studied with the indicated homological tools, and the results will be interpreted as providing information on the specific situations that are studied. Some of the applications include Index theory on singular spaces, the spectral and Index theory of Dirac operators coupled with vector potentials, and determinants of elliptic operators. A different but closely related type of application is a cohomological study of the representation space of a p-adic group.The index of an operator in its simplest form is a number. Many quantities from mathematics and even from physics and chemistry can be identified with the index of a suitable operator - for example, the number of electrons occupying a certain energy level in an atom is the index of a suitable operator. The Dirac equation (or operator) is one of the fundamental equations in physics; the number of solution of the Dirac equation is very closely related and can often be identified with the index of that operator.