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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Moment methods in energy minimization: New bounds for Riesz minimal energy problems
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by David de Laat PDF
Trans. Amer. Math. Soc. 373 (2020), 1407-1453 Request permission


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:;
  • 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}
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 1407-1453
  • MSC (2010): Primary 52C17, 90C22
  • DOI:
  • MathSciNet review: 4068268