Fourier Interpolation and Extremal Problems

Project Information

FourIntExP

Grant agreement ID: 101078782

DOI

10.3030/101078782

EC signature date 8 February 2023

Start date 1 May 2023

End date 30 April 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 158 000,00

EU contribution

€ 1 158 000,00

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 1 - FourIntExP (Fourier Interpolation and Extremal Problems)

Reporting period: 2023-05-01 to 2025-10-31

Summary of the context and overall objectives of the project

In 2006 Cohn and Kumar have conjectured that the A2 lattice is universally optimal, meaning that it has the lowest potential energy among all configurations of the same density for a large class of potentials. This conjecture has several very important corollaries. Among other consequences, it it would imply a positive solution to the 2D crystallization problem, a major unsolved problem coming from materials science, and it also implies a conjecture on the emergence of the triangular lattice of Abrikosov vortices in the Landau-Ginzburg theory of superconductivity.
Recently, the 8 and 24-dimensional cases of the Cohn-Kumar conjecture have been positively resolved using novel interpolation formulas for radial Schwartz functions. This formula recovers a radial function from the data of it and its Fourier transform on a discrete set of radii, and its construction uses classical modular and quasimodular forms.
The main objective of this project is to investigate analogues and generalizations of these interpolation formulas with a view towards applications in extremal problems in Fourier analysis and geometry, including both sphere packing and energy minimization problems, and many others. One of the main motivations and a strategic goal for this investigation is to obtain interpolation formulas that would be of use to attack the Cohn-Kumar conjecture in dimension 2 as well as to perform asymptotic analysis of the Cohn-Kumar optimization problem in higher dimensions.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

1. Together with K. Fedosova and K. Klinger-Logan we have proved the conjecture of Chester, Green, Pufu, Wang, and Wen on sums of even powers of divisors. This conjecture was based on the Ads/CFT correspondence in string theory. The proof gives a large family of analogous identities involving coefficients of holomorphic cusp forms of level 1.
2. Together with A. Arman, A. Bondarenko, F. Nazarov, and A. Prymak we have constructed bodies of constant width 2 of volume exponentially smaller than the volume of a unit ball in high dimensions, resolving an old question asked by Oded Schramm.
3. Together with J. Ramos, we have constructed Heisenberg uniqueness pairs for a class of perturbed lattice crosses. This complements the results of Hedenmalm and Montes-Rodriguez on Heisenberg uniqueness pairs involving uniform lattice crosses, but uses a different set of ideas that are more flexible and give rise to a much wider class of uniqueness pairs.
4. Together with A. Bondarenko, J. Ortega-Cerdà, and K. Seip, we gave a very precise description of the Hormander-Bernhardsson extremal function, characterizing it via differential and functional equations, and giving a way to compute the optimal associated constant for point evaluation in the space PW^1 to any precision. As a byproduct of this investigation, we have constructed a large family of interpolation formulas for Paley-Wiener class functions, parameterized by eigenfunctions of a certain family of second order differential operators.
5. Together with Steven Charlton and Daniil Rudenko, we have proved a recent conjecture about multiple polylogarithms by establishing an unexpected link between them and cohomology of GL_n and Steinberg modules.
6. Together with Qihang Sun, we have generalized the Poincare sum construction of interpolation bases for square roots of integers for the space of radial Schwartz functions from dimensions >=5 to dimensions 3 and 4.
7. We have obtained a common generalization of the interpolation formula of Radchenko-Viazovska and the interpolation formula of Cohn-Kumar-Miller-Radchenko-Viazovska. The resulting three-parametric family of Fourier interpolation formulas clarifies the relationship between the two, is considerably simpler to prove and analyze, and allows a more direct proof of positivity of the modular kernel needed for the universal optimality in dimensions 8 and 24. A manuscript is in preparation.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

1. A proof of the conjecture of Chester, Green, Pufu, Wang, and Wen on sums of even powers of divisors, and a generalization thereof to an identity involving holomorphic cusp forms of level 1.
2. Construction of bodies of constant width of small volume, resolving a conjecture of Oded Schramm.
3. Extension of the work of Hedenmalm and Montes-Rordiguez on Heisenberg uniqueness pairs to a class of perturbed lattice crosses.
4. A description of the Hormander-Bernhardsson extremal function, the discovery of differential and functional equations that characterize it, as well as associated interpolation formulas for Paley-Wiener class functions.
5. A new general family of functional equations for multiple polylogarithms that significantly improves the understanding of relations between different multiple polylogarithm functions of the same depth.
6. A Poincare series construction of Radchenko-Viazovska interpolation formulas for radial functions in 3 and 4 dimensions.
7. A common generalization of Radchenko-Viazovska and Cohn-Kumar-Miller-Radchenko-Viazovska interpolation formulas.