
## 1. Discrete geometry and automorphic forms

1. #### Problem 1.02.

[Saff] Smale’s problem: Rapidly generate $N$ points on $S^2$ whose logarithmic energy differs from the optimal energy by $O(log N)$.
• #### Problem 1.04.

[Petrache] Can we extend the definition of universal optimality to potentials with slow decay (or functions that are not completely monotonic)?
• #### Problem 1.06.

[Saff] Which lattices $\Lambda$ in $\mathbb R^d$ of determinant $1$ maximize $\min_{y \in \mathbb R^d \setminus \Lambda} \sum_{x \in \Lambda} \frac{1}{|x-y|^s}?$ For $s \to \infty$ this is the covering problem. On $S^{n-1}$ this is called polarization.
• #### Problem 1.08.

[Musin] Are the Voronoi cells of the optimal $8$ or $24$ dimensional sphere packings as small as possible by volume?
• #### Problem 1.1.

[Musin/Cohn] Prove $A_2$ is universally optimal. Or find magic function for the sphere packing poblem in $\mathbb R^2$.
• #### Problem 1.12.

[Petrache] Choose $\Lambda \subseteq \mathbb R^d$ to minimize the Wasserstein (optimal transport) distance form $\sum_{x \in \Lambda} \delta_x$ to the Lebesque measure.
1. Remark. [Petrache] For the Wasserstein 2-distance in $\mathbb R^2$ the unique minimizer is the triangular lattice (proved by Bourne-Peletier-Theil in https://arxiv.org/abs/1212.6973)
• #### Problem 1.14.

[Kumar] Use Siegel modular forms to improve bounds for the maximal dimension of an extremal even lattice.
• #### Problem 1.16.

[Radchenko] To obtain LP bound in $\mathbb R^n$ numerically, use linear combinations of different Gaussian (with complex parameters).
• #### Problem 1.18.

[Cohn] Numerical LP bounds in high dimensions.
• #### Problem 1.2.

[Cohn] Let $A_-(d)$ be the minimum or $r$ for which there exists a Schwartz function $f \colon \mathbb R^d \to \mathbb R$ such that $f(0) = \hat f(0) = 1$, $f(x) \leq 0$ for $|x| \geq r$, and $\hat f(y) \geq 0$ for all $y$. For $d$ a non-integral positive number this bound is still defined by using radial functions and the Bessel transform. We know $A_-(1) = 1$, $A_-(8) = \sqrt{2}$, and $A_-(24) = 2$. Numerically it seems to be the case that $A_-(2) = (4/3)^{1/4}$, and that $A_-'(8) = \sqrt{2}/30 = (\sqrt(2))^7/240$ and $A_-'(24) = 368/12285 = 23 * 2^8/196560$.
• #### Problem 1.22.

[Kumar] It is know that $A_+(12) = \sqrt{2}$ from the paper by Kumar and Gonçalves. Numerically it seems $A_+'(12) = \sqrt{8}/63 = (\sqrt{2})^9/504$. Can we prove this?
• #### Problem 1.24.

[Kumar] Is it true that if you remove one point of the $24$-cell, it is still optimal? Same question for removing up to $7$ points of $120$ cell. What about $E_8$ and $\Lambda_{24}$?
• #### Problem 1.26.

[Kumar] Uniqueness of $4$-dimensional kissing configuration.
• #### Problem 1.28.

[Cohn] Let $f$ be the magic function in $8$-dimension and $M_f(s) = \int_0^\infty f(x) x^{s-1} dx$. Conjecture of Cohn and Miller: $M_f(4) = \frac{1}{15} = M_{\hat{f}}(4)$. Can we prove this? The same question for the magic function in $24$-dimension, where $M_f(12) = M_{\hat{f}}(12) = 0.17786094729650\ldots$?
• #### Problem 1.3.

[Kumar] Does every lattice have a $\mathbb{Z}$-basis of Voronoi vectors? (I.e., vectors defining facets of Voronoi cells)
• #### Problem 1.32.

[de Laat] Consider the following $3$-point bound generalization of the Cohn-Elkies bound: $\inf \Big\{f(0,0) : f \in S(\mathbb R^{2n}), \, \hat f(0, 0) = 1, \, \hat f \geq 0, \,f \leq 0 \text{ on } C_2 \Big\}^{1/2},$ where $C_2 = \big\{(x,y) \in \mathbb R^{2n} : \|x\|, \|y\|, \|x-y\| \in \{0\} \cup [1, \infty)\big\} \setminus \big\{(0,0)\big\}.$ This gives an upper bound on the optimal Lattice sphere packing center density. Computational results suggest this is equal to the Cohn-Elkies bound, and the computed solutions are invariant under the action of the group $O(n) \times O(n)$ on $\mathbb R^{2n}$. Why are the solutions invariant under $O(n) \times O(n)$. Can we use a solution of the Cohn-Elkies bound to formulate a solution for the above bound and vice versa?
• #### Problem 1.34.

[Saff] Minimize energy for $f(r) = e^{-\alpha r^2}$ in $\mathbb{R}^n$ as $\alpha \rightarrow 0$
• #### Problem 1.36.

[Cohn] Behavior of energy as $n \rightarrow \infty$ with $\alpha$ fixed. (Gaussian core model). Known for small $\alpha$ by the paper of Cohn and de Courcy-Ireland.
• #### Problem 1.38.

[Saff] Minimize energy for $5$ points on $S^2$ under $f(r) = r^{-s}$, for the values of $s$ where this problem is still open.
• #### Problem 1.4.

[Cohn] Numerical computation of LP bound in $\mathbb{H}^n$.
• #### Problem 1.42.

[Musin] Is it true that for $1 \leq k \leq d$, the only critical points for $d+k$ particles in $S^{d-1} \subset \mathbb{R}^d$ under some potential functions (logarthmic + others) are orthogonal unions of $k$ regular simplices?
• #### Problem 1.44.

[Kumar] Prove there is no lattice whose shells are spherical $12$-designs
• #### Problem 1.46.

[Kumar] Prove there exist spherical $t$-designs on $S^2$ with $(\frac{1}{2} + o(1))t^2$ points.
• #### Conjecture 1.48.

[Kumar] Is it true that no shell of $E_8$ is a $8$-design? (This is equivalent to Lehman’s conjecture)
• #### Problem 1.5.

[Cohn] Suppose we have a radial function $f$ on $\mathbb{R}^n$ ($0 < n < 10$) such that $f$ has a single root at $\sqrt{2}$, double roots at $\sqrt{2k}$ for $k > 1$, and $\hat f$ has double roots at $\sqrt{2k}$ for $k \geq 1$. Then, $$\frac{f(0)}{\hat f(0)} = - \frac{(n^4 -56n^3 + 1184n^2 - 11200 n+ 40320}{16(n-10)(n-14)(n-18)}$$
• #### Problem 1.52.

[Saff] Prove/Disprove that for optimal $N$-point codes on $S^2$ non-hexagonal Voronoi cells are dense in $S^2$ as $N \rightarrow \infty$.
• #### Problem 1.54.

[Petrache] Pick $n$ points in $\Lambda$ to maximize the number of adjacent pairs. What is the limit shape as $n \rightarrow \infty$?
1. Remark. We should specify that \Lambda is a lattice?
• #### Problem 1.56.

[Cohn] (G. Kuperberg - Schramm) How large can the average kissing number be for packing with spheres of varying radii? It is known that in $\mathbb{R}^3$ it is larger than $12$ and at most $24$. Is it true that it is strictly larger than the kissing number for dimension $4, 8,$ and $24$?

Cite this as: AimPL: Discrete geometry and automorphic forms, available at http://aimpl.org/discreteaf.