They came up with a map that looks like this, and this map has been a source of contention, for reasons we may get to later. 
[CLICK] 
Now for the really interesting, or awful, depending on how you view it part of the paper, and arguably the reason why this paper was published. Using the geographic data, and the phylogenetic data, they attempt to 'trace back' the root (or possible roots) for the tree. To do this, they adapt a Bayesian phylogeographic method originally developed in this top paper, for tracing the origins of virus outbreaks, from their lineages.
Similar idea, but with different data.  The method models spatial diffusion of languages as a Brownian ‘ random walk’ in two dimensions (latitude and longitude) along the branches of the tree.
This means that for a set period of time, the geographic distribution of languages expanding from a point of origin is assumed to be approximated by Brownian motion – some languages will have moved loads, some won't have moved at all but most will have moved a bit.
They also 'relaxed' the random walk, which essentially means different branches can have different rates 
Like they did with cognates, by allowing different words to evolve at different rates, the average rate of movement was allowed to vary across the tree (ie, some populations/families will diffuse and disperse quicker than others)
But enough about that, what were the results?
[CLICK]
As you can see, it's pretty conclusive from their data. The results provide evidence that maybe Anatolia is the homeland of PIA.