I frequently get email from prospective Ph.D. students asking if I have an opening in my research group. However, individual professors at Cornell do not admit Ph.D. students. Admissions to the Center for Applied Mathematics (CAM), Computer Science (CS), and Operations Research (OR) Ph.D. programs at Cornell are handled by an admissions committee, and while I am sometimes asked for my input on particular candidates who are directly in my research area, I do not have any control over which students are admitted.
If a student is admitted into the Ph.D. program in OR or CS, the student then has about a year to take classes and discuss research opportunities with various potential Ph.D. advisors. Students and advisors make a mutual decision to work together. I suggest that students who are interested in working with me take a graduate-level class from me and do very well in it.