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Professor Mahya Ghandehari has been awarded a National Science Foundation award for her project entitled as "Fourier and Fourier-type Algebras of Lie Groups" which lies within the field of non-commutative harmonic analysis.
Harmonic analysis is a fundamental branch of mathematics which addresses the following central problem: How can a function or signal be represented or approximated as a combination of basic waves? Such representations of signals are heavily used in today's technology for storage, transmission, and noise reduction of signals. Unfortunately, harmonic analytic methods perform poorly in various circumstances, as they are designed based on the assumption that the outcome of a series of actions is unchanged, regardless of the order in which the actions are performed. However, actions are often not interchangeable. For example, the result of performing consecutive rotations in 3D space is highly sensitive to the order of rotations. Indeed, the real world is a very "non-commutative" universe. Space of 3D rotations is a simple example of Lie groups, which arise naturally in physics. In this project, Prof. Ghandehari will use techniques of functional analysis, representation theory and Lie theory to advance the theory of non-commutative harmonic analysis for Lie groups. This project will be jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).
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