Project description
Bridging harmonic analysis and conformal field theory for breakthroughs in both
Two seemingly unrelated mathematical fields – abstract harmonic analysis and the conformal bootstrap method of mathematical physics – have recently revealed surprising deep connections. This has led to new results in both areas, including rigorous nearly-sharp bounds on the spectra of hyperbolic manifolds. The ERC-funded HARMONICON project aims to dramatically extend this emerging correspondence. It will apply conformal bootstrap methods to establish sharper spectral estimates for hyperbolic manifolds and L-functions, while simultaneously reformulating higher-dimensional conformal field theory within the framework of harmonic analysis. This unified approach could unlock the conformal bootstrap for a broader class of theories currently beyond its reach, delivering quantitative breakthroughs across pure mathematics and mathematical physics.
Objective
The study of spectral properties of locally symmetric spaces, particularly hyperbolic manifolds, stands as a central objective of abstract harmonic analysis, with far-reaching implications across mathematics. Despite the diverse range of tools employed to establish spectral estimates in this context, the conjecturally optimal results have remained out of reach.
In a seemingly unrelated realm, the conformal bootstrap has emerged as a powerful method of mathematical physics, used for elucidating conformal field theories (CFTs) in any number of dimensions. However, this method sheds little light on the problem of construction of higher-dimensional CFTs. Furthermore, it is ineffective in non-unitary CFTs, many of which are of great mathematical interest.
Recently, the principal investigator discovered surprising connections between harmonic analysis and the conformal bootstrap. This convergence has already yielded rigorous nearly sharp bounds on the spectra of hyperbolic manifolds as well as an independent solution of the sphere packing problem in dimensions 8 and 24.
The goal of the proposed program is to dramatically extend the emerging interplay between harmonic analysis and conformal field theory in general dimension and use it to achieve quantitative breakthroughs on both sides of the correspondence.
Firstly, I will apply conformal bootstrap methods to establish novel estimates on the spectral data of locally symmetric spaces. In arithmetic cases, this approach will improve upon the state-of-the-art bounds on the corresponding L-functions.
Secondly, I will formulate higher-dimensional conformal field theory rigorously in the framework of abstract harmonic analysis. A previously overlooked notion of positivity which emerges in the new formulation will allow me to extend the conformal bootstrap to nonunitary CFTs and thereby establish novel estimates on their scaling exponents.
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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(opens in new window) ERC-2025-COG
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75015 Paris
France
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