Given the [march 2016 breakthrough concerning sphere packings](https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/) by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that the known cases $\Delta\_1=1$, $\Delta\_2=\frac{\pi}{\sqrt{12}}$, $\Delta\_3=\frac{\pi}{\sqrt{18}}$, $\Delta\_8=\frac{\pi^4}{384}$ and $\Delta\_{24}=\frac{\pi^{12}}{12!}$ all have a lattice solution with neat number-theoretic properties, yet the densities obtained do not seem to follow any obvious pattern, except for the presence of $\pi$. 



> 
> **Question A**: Is there a consensus among experts that one should not hope for a synthetic formula for densities, or are there still reasonable candidate formulas standing ? 
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> **Question B**: are there workshops planned on the recent papers soon (say, this fall, maybe even this summer) ? 
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Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?
Here's an attempt at answering your Question A:


Currently, one of the most powerful methods for proving upper bounds on sphere packing densities is the linear programming bound of [Cohn and Elkies](http://arxiv.org/abs/math/0110009) (which is what Viazovska used).


### Exact answers


According to the numerical computations of Cohn and Elkies, we do not know any dimensions other than 1, 2, 8, and 24 where their linear program has a chance of proving the exact bound (see their Figure 1). There are exceptionally good lattices in those four dimensions that match the bounds given by the Cohn-Elkies linear program. It could be the case (likely?) that these are the only dimensions where the bounds match, although we don't know how to rule out other possibilities. There could be some deep reason (perhaps related to finite simple groups) why these four dimensions are special.


[Hales' proof](http://arxiv.org/abs/math/9811071v2) of Kepler's conjecture (sphere packing in dimension 3) uses a completely different method.


### Asymptotic bounds


An important open question, for which there is still an exponential amount of room for improvement, is the problem of sphere packing bounds in very high dimensions.


The current best asymptotic upper bound to the packing density $\Delta\_n$ in $\mathbb{R}^n$ is by Kabatiansky and Levenshtein (1977)
$$
\Delta\_n \le 2^{-(0.5990\dots + o(1))n}
$$
(see, e.g., [my paper with Henry Cohn](http://arxiv.org/abs/1212.5966) on the matter; in particular, this asymptotic bound lies within the scope of the Cohn-Elkies linear program). The current best lower bound is due to [Venkatesh](http://math.stanford.edu/~akshay/research/sp.pdf). For all $n$,
$$
\Delta\_n \ge c n 2^{-n}
$$
for some explicit $c$ (that has been improved over time), and for a specific infinite set of $n$,
$$
\Delta\_n \ge c n (\log\log n) 2^{-n}.
$$


Closing the gap seems to be a difficult problem.


### Further reading


* [Henry Cohn's slides](http://research.microsoft.com/en-us/um/people/cohn/spherepacking.pdf)
* [Henry Cohn's PCMI 2014 lecture notes](https://arxiv.org/abs/1603.05202)


