According to a Chinese scholar of English,"An infinitive without to as the completement of subject can be in the front of a sentence, for example, Die is the last thing I shall do, doctor." (NG1493)
Hence comes the one by me , "Swim is I want to learn this summer."
What do you think of the two?
But his conclusion (by which, I presume, you mean his analysis) is quite correct. Provided you begin with a grammaticallly acceptable sentence in which a bare infinitive stands as subject complement (such as The last thing I shall do is die), that infinitive can indeed be fronted for emphasis.
The problem with your example is that, as other contributors have indicated, it originated from a non-sentence (*I want to learn is swim).
The problem lies therefore, not in his analysis, but in your own attempt to apply it.
I would suggest that the root of your confusion is simply lack of awareness of the rules that permit the formation of a sentence such as
The last thing I shall do is die.
in the first place, where omission of 'to' is either optional, or (as here) obligatory. The possibility of fronting is, by comparison, a minor issue.