The boundaries of index theory

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In recent years there has been an influx of new ideas from other disciplines into mathematics and this has led to major advances in many areas, notably geometry and topology. Classical problems have been solved and new perspectives exposed. In this spirit this project will use the methods of noncommutative analysis and noncommutative geometry to extend the mathematical area of spectral geometry. A primary objective is to determine how the geometric and differential structure of certain spaces interacts with the new spectral invariants that will be introduced. The project aims to obtain more subtle and refined information about these spaces. In this fashion it expects to resolve several long standing questions in mathematics.