Noncommutative Duality

Poincare duality is one of the central results in mathematics which is of great importance in algebraic topology. Noncommutative extensions of Poincare duality and various other duality results based on KK-theory have appeared since the 1980's and have played an important role in recent work on the Baum Connes conjecture, a central theme of research in operator algebras and noncommutative geometry.

Despite its long history and significant importance there are only very few explicit Poincare duality results in the current literature. The goal of this PhD research project is to establish and understand new Poincare duality results, in particular in connection with dynamical systems. The new innovative approaches to this problem include the systematic use of longer extensions to describe KK-theory classes and Kasparov products.

There are potential significant applications of the results of this project to the Baum Connes conjecture and other research problems.

This project being at the interface of analysis, dynamics and topology is in line with the EPSRC strategy to strengthen research in mathematical analysis in the UK.