Invariant polynomials in harmonic analysis
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Full Text E-thesis
Date
2024
Authors
Murphy, Fergal
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Volume Title
Publisher
University College Cork
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Abstract
This thesis presents a methodology for analysing equiaffine invariant measures on surfaces, building on a modified version of a comparability lemma from a 2019 work of P. Gressman. The research focuses on simplifying the process of calculating the equiaffine invariant measure for 2-surfaces in R^n by avoiding the need for a complete set of generators for the algebra of invariant polynomials. Instead, we compute a sufficient set of invariant polynomials whose intersection characterises the set of unstable points under the SL_2(C) action.
Our approach is demonstrated on specific surfaces in R^6, R^10, and R^15, where we successfully identify the relevant invariant polynomials and use them, along with the modified lemma, to derive the associated density functions. These density functions can then be used to define the equiaffine invariant measure.
The results offer a practical framework for understanding affine invariants in geometric contexts and suggest possible extensions to higher-dimensional surfaces and different group actions. This work aims to provide a foundation for future studies in geometric invariant theory and harmonic analysis, exploring the interplay between algebraic geometry, invariant theory, and analysis.
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Keywords
Harmonic analysis , Algebraic geometry , Geometric invariant theory , Equiaffine invariant measure
Citation
Murphy, F. M. 2024. Invariant polynomials in harmonic analysis. MSc Thesis, University College Cork.