I do math, and currently do not have an REU opportunity for the coming summer. Part of this is due to me being a US citizen studying in the UK (meaning many opportunities start before my classes end), and part of this is due to my own organizational shortcomings (causing me to miss a number of application deadlines).
Also, I will soon start sending introductory emails to prospective PhD supervisors.
I am considering adding a post-script to these introductory emails asking if they would want me as a research intern for the summer (and staggering the emails to avoid the possibility of two people saying yes).
My gut says that adding this post-script has a high risk of seeming unprofessional. Additionally, I am worried what would be the correct way to phrase the post-script. (So my question is: “Is this a bad idea? If not, how do I (roughly) phrase the post-script?”)
I already have solid research experience, having done two research internships in the past and publishing several solo papers (one of which has been accepted to a very good journal in my field).
I have already corresponded with most of my prospective advisors, asking about preprints of theirs on the arXiv. So I don’t know these professors too well, but have in the past shown a solid understanding of some of their work.