I see a lot of research on hypercomputation in the 1990's, but in more recent years there seems to be little work on the topic. Is it true that research in this area has died down? If so, what could be the reasons for it? Was this area convincingly shown to be unpromising?
It would be better if you specified what you mean exactly by hyper-computation and gave evidence for why you think it has "died down".
In any case, assuming that you are talking about computation of functions over natural numbers (and finite strings) (since I think it is clear that models for higher type computation is a very active area, e.g. CCA) and models of computation not equivalent to computability defined by Turing machines, I don't think the claim is correct, for example see CiE'05 and CiE'11. Also see the criticisms made against the claim that hyper-computation is something new:
- Martin Davis, "Why there is no such discipline as hypercomputation", 2006.
- Martin Davis, "The Myth of Hypercomputation", in "Alan Turing: Life and Legacy of a Great Thinker", 2004.
There have been recent several conferences on the topic of infinitary computability, which have treated many topics in hypercomputation.
- Bonn International Workshop on Ordinal Computability (BIWOC) 2007, see the program report there
- Conference on Effective Mathematics of the Uncountable (EMU) 2008
- Conference on Effective Mathematics of the Uncountable (EMU) 2009
In addition, there have been special sessions on infinitary computability in many of the CiE conferences.
I don't think this is true. Searching Arxiv for papers on hypercomputation gets a bunch of hits.