The affine Grassmannian and the Bogomolny equations In "Electric-Magnetic Duality and The Geometric Langlands Program", Sections 9 and 10, Kapustin and Witten describe certain convolution varieties in the affine Grassmannian (and more generally, in the Beilinson-Drinfeld) as moduli spaces of solutions to "the Bogomolny equations with 't Hooft operators added."  While I can roughly make sense of what they are doing, it is not such easy reading for a mathematician, and of course, the proofs are pretty loose in nature.  My (admittedly very vague) question is

Have any mathematicians followed up on this description i.e. written things in more mathematical language and done the proofs rigorously, or used it to understand the affine Grassmannian better?

 A: Well, I think that there is no problem making that part of the paper rigorous (basically it is rigorous, modulo
some well known results about moduli spaces of monopoles). In terms of how useful it is, the only thing that comes to my mind is this: it is a theorem of Jacob Lurie that the derived Satake category is an $E_3$-category, which
means that you can make it live over the configuration space of points in a 3-dimensional space (informally
$E_2$ is very close to just being symmetric monoidal and $E_3$ is some sort of higher commutativity; you can show
that $E_3$ is the best thing you can hope for as the derived Satake category is not $E_4$ even for a torus).
Now Lurie's argument is rather abstract, whereas probably you can give a purely geometric proof of this result
using Witten-Kapustin construction (since they define some space over the configuration space of points in
$\Sigma\times {\mathbb R}$ ($\Sigma$ is a Riemann surface) which simultaneously takes care of the "convolution" and "fusion" in the affine
Grassmannian). This is not done anywhere but this is a well defined mathematical problem (define an $E_3$-structure
on the derived Satake category using Witten-Kapustin space and show that it is equivalent to Lurie's).
A: For those who could be interested, I worked out a formal construction of the E3-structure on the derived Satake category here, following the arguments hinted at by Lurie.
A: As far as I know the only paper which gives a mathematical version of those ideas is the following work of Benoit Charbonneau and Jacques Hurtubise.
https://arxiv.org/abs/0812.0221
It doesn't quite use the language of the affine Grassmannian, but otherwise I think that it does what you want.
As far as understanding the affine Grassmannian better, the one application I know (which I already told you about) is to produce the symplectic structures on those slices that you (and Sasha, and I) like so much.
