In formal writing (in this case an abstract in a scientific paper), is there a difference between the sentences:
We report the applications of an uncommon interpretation of x to the field of y.
and
We report on the applications of an uncommon interpretation of x to the field of y.
I'm tempted to say that perhaps only the second sentence is acceptable, but I'm unsure.
Yes, there is a slight difference.
"Report" by itself, in this context, means that the rest of the text will list the the thing(s) being reported. To say
We report [X]
means basically the same thing as
We answer the question "what is/are [X]?"
So in your example "We report the applications...", you can expect that the paper will contain an answer to the question "What are the applications of an uncommon interpretation of x to the field of y?"
"Report on", in this context, means something more like "give information about", without being specific on what sort of information is being given. This is a broader meaning. In particular, it does not imply that the rest of the text will answer the previously mentioned question (although it does not rule it out either). It's possible that the paper will simply discuss some progress made in attempting to determine the applications, or give some speculation about the applications, or some such thing.
There is no difference. If anything, the first is preferable, since it avoids using an unnecessary word, which might make it a little clearer.
