I spent time reading a professor's papers, and I developed an idea about how to expand the professor's work. I told the professor this and got the professor's approval. Now the next job is to persuade the admission committee (adcom).
Should I tell the adcom about how I found the professor while writing my SOP (statement of purpose)? I don't see a problem with it. I would be happy if a stranger told me how they found me. So I wrote about it the first time I contacted the professor. However, the adcom is not the professor, and I'm not sure if they care or not. I think that they only care about the benefit that I will bring to the department, as well as the approval of the professor. How I reach them is not important.
But I don't have any reason to remove that part, right? At best, it will show how hard I worked to reach them. At worst, there will be nothing to lose.
My draft is as follows:
I have read some of the most recent and commonly cited papers of Professor X, starting with Paper 1 written in 2014. Among the introduced methods, I found the Method the most interesting prompting further study. I then focused on Paper 2, written with Y. Its introduction on Quanta Magazine gave me a great insight into the Method. However, there are known limitations that Prof X is keen to discern. For example, when the system undergoes rapid changes the method breaks down, the reason for this remains unclear.
My hypothesis is that the changing of the system might suggest that... (this paragraph is for showing my idea to expand the work)
A minor question: should I include this in the email to professors too? Why or why not?