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Open problems UP24

Maryna Manskova

TL;DR

Open Problems UP24 collects a set of open questions spanning Riesz and logarithmic energies, equilibrium measures, convex-body minimization, p-frame energies, geodesic/chordal energies, integration weights, snake polynomials, chromatic numbers of spheres, and approximately Hadamard matrices. It highlights sharp thresholds $\alpha_{s,d}$ for sphere supports, conjectured connectivity of minimizer sets $S(\Omega)$, and phase-transition phenomena as central themes, and provides numerical evidence and concrete targets for proving new extremal properties. The compilation aims to stimulate advances in potential theory, discrete geometry, approximation theory, and combinatorial matrix theory by mapping precise open questions to tractable research directions. The significance is in organizing cross-disciplinary open problems that connect continuous energies to discrete configurations and combinatorial constructions.

Abstract

The conference Unexpected Phenomena in Energy Minimization and Polarization, held in Sofia, Bulgaria in 2024, provided a platform for researchers to discuss and propose challenging open questions across various fields, such as potential theory, approximation, special functions, point configurations, lattices, and numerical analysis. The open problems sessions were productive, fruitful and led to a range of interesting questions. In this document, we present these open problems.

Open problems UP24

TL;DR

Open Problems UP24 collects a set of open questions spanning Riesz and logarithmic energies, equilibrium measures, convex-body minimization, p-frame energies, geodesic/chordal energies, integration weights, snake polynomials, chromatic numbers of spheres, and approximately Hadamard matrices. It highlights sharp thresholds for sphere supports, conjectured connectivity of minimizer sets , and phase-transition phenomena as central themes, and provides numerical evidence and concrete targets for proving new extremal properties. The compilation aims to stimulate advances in potential theory, discrete geometry, approximation theory, and combinatorial matrix theory by mapping precise open questions to tractable research directions. The significance is in organizing cross-disciplinary open problems that connect continuous energies to discrete configurations and combinatorial constructions.

Abstract

The conference Unexpected Phenomena in Energy Minimization and Polarization, held in Sofia, Bulgaria in 2024, provided a platform for researchers to discuss and propose challenging open questions across various fields, such as potential theory, approximation, special functions, point configurations, lattices, and numerical analysis. The open problems sessions were productive, fruitful and led to a range of interesting questions. In this document, we present these open problems.

Paper Structure

This paper contains 9 sections, 8 theorems, 37 equations, 4 figures.

Table of Contents

  1. When is the equilibrium support a sphere? Robert Womersley and Edward Saff
  2. Inner product expansions for logarithmic and Riesz $\mathbf{s}$-energy Johann Brauchart
  3. Energy minimization on convex bodies Ryan Matzke
  4. Minimizers of the $\mathbf{p}$-frame energy. Distances on the torus. Dmitriy Bilyk
  5. Riesz energy for geodesic and chordal distances Peter Grabner
  6. Integration weights Jordi Marzo
  7. Snake Polynomials Geno Nikolov
  8. Chromatic numbers of spheres Danila Cherkashin
  9. Approximately Hadamard Matrices Stefan Steinerberger

Key Result

Theorem 1.1

Suppose that $-2<s<d-3$ and $V(x)=\frac{\gamma}{\alpha} \|x\|^\alpha$, where $\gamma>0$ and $\alpha>\max\{-s,0\}$. Define If $\alpha\geq \alpha_{s,d}$, then $\mu_{\text{eq}}=\sigma_{R_*}$, where Furthermore, the threshold $\alpha_{s,d}$ is a sharp bound for $\alpha$, meaning that if $\max\{-s,0\} < \alpha < \alpha_{s,d}$, then for all $R>0$, $\sigma_R$ is not a minimizer of $I_{s,V}$.

Figures (4)

  • Figure 1: Plot for $d=10$ and Riesz parameter $-2<s<d-3$, where the colour gives the value of $R_*$ when the equilibrium support is $\mathbb{S}^{d-1}_{R_*}$. The external field power $\alpha\geq \alpha_{s,d}$ as in Theorem \ref{['Theorem_sphere']} with $\gamma=1$. Outside the coloured region the support is not a sphere.
  • Figure 2: Numerical results for $d=5$, $s=d-6=-1$, and $\alpha=\frac{3}{2}<\alpha_{s,d}$.
  • Figure 3: Maximizers of $I_\alpha$.
  • Figure 4: Examples of snake polynomials $\omega_\mu$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1: Johann Brauchart
  • Conjecture 2.1
  • Theorem 2.1: Johann Brauchart
  • Remark 2.1
  • Conjecture 4.1: D. Bilyk, A. Glazyrin, R. Matzke, J. Park, O. Vlasiuk, Bilyk2021Bilyk2022
  • Conjecture 7.1: Geno Nikolov
  • Theorem 7.1: V. A. Markov 1892
  • Theorem 7.2: R. J. Duffin and A. C. Schaeffer 1941
  • Theorem 7.3: A. Shadrin and G. Nikolov, Nikolov2012Nikolov2014
  • ...and 2 more