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Transactions of the American Mathematical Society

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Moment methods in energy minimization: New bounds for Riesz minimal energy problems


Author: David de Laat
Journal: Trans. Amer. Math. Soc. 373 (2020), 1407-1453
MSC (2010): Primary 52C17, 90C22
DOI: https://doi.org/10.1090/tran/7976
Published electronically: October 2, 2019
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Abstract: We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite-dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop symmetric sum-of-squares techniques. We numerically compute the second step of our hierarchy for Riesz $ s$-energy problems with five particles on the two-dimensional unit sphere, where the $ s=1$ case is the Thomson problem. This yields new numerically sharp bounds (up to high precision) and suggests that the second step of our hierarchy may be sharp throughout a phase transition and may be universally sharp for five particles on the unit sphere. This is the first time a four-point bound has been computed for a problem in discrete geometry.


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Additional Information

David de Laat
Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, 2628 XE Delft, The Netherlands
Email: d.delaat@tudelft.nl; mail@daviddelaat.nl

DOI: https://doi.org/10.1090/tran/7976
Keywords: Thomson problem, Riesz $s$-energy, four-point bounds, semidefinite programming, Lasserre hierarchy, harmonic analysis on spaces of subsets, invariant polynomials
Received by editor(s): July 31, 2017
Received by editor(s) in revised form: August 15, 2019
Published electronically: October 2, 2019
Additional Notes: The author was funded by Vidi grant 639.032.917 and TOP grant 617.001.351 from the Netherlands Organization for Scientific Research (NWO). A preliminary version of this paper appeared in the author’s Ph.D. thesis \cite{Laat2016}
Article copyright: © Copyright 2019 American Mathematical Society