1. Discrete geometry and automorphic forms

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 \[ \min_{y \in \mathbb R^d \setminus \Lambda} \sum_{x \in \Lambda} \frac{1}{xy^s}? \] For $s \to \infty$ this is the covering problem. On $S^{n1}$ 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.
Remark. [Petrache] For the Wasserstein 2distance in $\mathbb R^2$ the unique minimizer is the triangular lattice (proved by BournePeletierTheil 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 1.2.
[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 nonintegral 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 1.28.
[Cohn] Let $f$ be the magic function in $8$dimension and $M_f(s) = \int_0^\infty f(x) x^{s1} dx$. 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 CohnElkies 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 \[ C_2 = \big\{(x,y) \in \mathbb R^{2n} : \x\, \y\, \xy\ \in \{0\} \cup [1, \infty)\big\} \setminus \big\{(0,0)\big\}. \] This gives an upper bound on the optimal Lattice sphere packing center density. Computational results suggest this is equal to the CohnElkies 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 CohnElkies 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 CourcyIreland. 
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 1.42.
[Musin] Is it true that for $1 \leq k \leq d$, the only critical points for $d+k$ particles in $S^{d1} \subset \mathbb{R}^d$ under some potential functions (logarthmic + others) are orthogonal unions of $k$ regular simplices? 
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, $$ \frac{f(0)}{\hat f(0)} =  \frac{(n^4 56n^3 + 1184n^2  11200 n+ 40320}{16(n10)(n14)(n18)}$$ 
Problem 1.52.
[Saff] Prove/Disprove that for optimal $N$point codes on $S^2$ nonhexagonal 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 $n \rightarrow \infty$?
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.