Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular forms. I understand some sphere packing and issues going with it but my understanding is most upperbounds come from linear programming and the bounds that are currently proven optimal (including at $n=8$ and $n=24$) already come from linear programming bounds.

1. How do modular forms become part of the story that provide the lower bound (do they arise naturally from some packing related structure)?

2. Is there a bigger story that this is just a chapter of that may apply to other upper bounds generated from linear programming? What makes modular forms click for this class of linear programming bounds?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular forms. I understand some sphere packing and issues going with it but my understanding is most upperbounds come from linear programming and the bounds that are currently proven optimal (including at $n=8$ and $n=24$) already come from linear programming bounds.

1. How do modular forms become part of the story that provide the lower bound (do they arise naturally from some packing related structure)?

2. Is there a bigger story that this is just a chapter of that may apply to other upper bounds generated from linear programming? What makes modular forms click for this class of linear programming bounds (perhaps Sphere packing and quantum gravity is of utility?)?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular forms. I understand some sphere packing and issues going with it but my understanding is most upperbounds come from linear programming and the bounds that are proven optimal already come from linear programming bounds.

1. How does modular forms become part of the story that provide the lower bound?

2. Is there a bigger story that this is just a chapter of that may apply to other upper bounds from linear programming? What makes modular forms click for this class of linear programming bounds?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular forms. I understand some sphere packing and issues going with it but my understanding is most upperbounds come from linear programming and the bounds that are currently proven optimal (including at $n=8$ and $n=24$) already come from linear programming bounds.

1. How do modular forms become part of the story that provide the lower bound (do they arise naturally from some packing related structure)?

2. Is there a bigger story that this is just a chapter of that may apply to other upper bounds generated from linear programming? What makes modular forms click for this class of linear programming bounds?