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H2020Individual fellowship2019–2021

covtrans · Functional/Harmonic Analysis of Covariant Transforms

Horizon 2020 — Marie Skłodowska-Curie Actions

Duration
2019-03-01 → 2021-02-28
EU contribution
€183,455
Participants
1
Scheme
MSCA-IF-EF-ST

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Results in brief

Functional/Harmonic Analysis of Covariant Transforms

The covariant transforms/functions of characters (one-dimensional continuous irreducible unitary representations) of closed subgroups have applications in different mathematical areas such as number theory (automorphic forms), induced representations, homogeneous spaces, complex (hypercomplex) analysis, and coherent states. In general, classical harmonic analysis methods cannot be employed as a unified theory for covariant functions of a given closed subgroup. In this project, we developed harmonic/functional analysis foundations for covariant functions of characters. We developed some operator theoretic aspects related to Banach covariant function spaces of characters of normal or compact subgroups including a unified theory for structure of Banach convolution modules induced by the group algebras on Banach covariant function spaces of characters of subgroups (normal or compact) and covariant convolutions (convolution of covariant functions). The introduced structures and properties of covariant function spaces imply a better understanding of convolution type covariant transforms and presents a unified theory for harmonic/functional analysis of covariant functions/transforms. The presented theory can be applied in different directions including mathematical physics (coherent states), representation theory of groups, postmodern harmonic analysis. It also generalizes classical methods of abstract harmonic analysis on quotient groups and homogeneous spaces of compact subgroups.

Data: CORDIS, © European Union

Project objective

The topic of this proposal is a structured extension of abstract harmonic analysis and covariant/contravariant transforms over homogeneous spaces of locally compact groups. We are going to develop a systematic framework to study the structure and properties of convolution-type operators associated to locally compact groups, both from a theoretical perspective and in application to geometric analysis, operator theory and mathematical physics.The first objective is to present the abstract notion of relative dual space for homogeneous spaces. Then we present the theory of relative Fourier analysis over these homogeneous spaces by applying the the abstract theory of relative dual space. Finally, we explore a general model of covariant/contravariant analysis over homogeneous spaces. The entire project will be an important contribution to the field of abstract harmonic analysis, harmonic analysis, geometric analysis andtheoretical physics, presenting a unified perspective on the structure of these homogeneous spaces. Applications to symbolic calculus of operators and quantum information theory will be explored.The Fellow will gain new skills and experience in applications of his knowledge in mathematical physics and other areas. Progress in this ambitious project will reinforce Fellow's reputation and support him in obtaining a strong academic position. The host will gain from Fellow's expertise, on the representation theory/abstract harmonic analysis border, which Fellow has developed from his tree-year PostDoc in Vienna--the one of world's leading wavelets groups.

Original text from CORDIS.

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Data: CORDIS, © European Union