In particular, individual faculty do not make admission decisions and do not have internships or post-doctoral positions to award. Any summer funding or projects that might become available would be used to support our current students already here.
For graduate study, students are admitted to the mathematics department and not to research groups of faculty members. Selection of a research advisor occurs only after taking the basic courses and completing the basic examinations on graduate mathematics. There is no benefit to be gained by sending inquiries to professors requesting internships or PhD supervision before receiving admission to the PhD degree program.
I have received many such inquiries from students in Asia, India, and Iran. I have been told that some students in India are required to seek summer internships abroad. Such internships in mathematics generally do not exist, especially in the United States. Random solicitations constitute impolite spam with no positive outcome; please point this out to whoever requires it. I never had summer internships or research positions to award. International students who send me requests for internships, research assistantships, or post-docs will receive a pointer to this page in response.
Finally, note that my position at Illinois is Professor Emeritus. I retired in 2011 and have been spending much of my time since then in China. I do not have a research group. Also, examination of my publication list gives a good sense of my research interests; they generally do not include algorithms or algebraic combinatorics.
Please understand my purpose in referring such inquiries to this page. It contains exactly what my letter in response to an individual would say and is not meant to discourage anyone from applying via normal channels. I do not have time to read attached resumes that students send to me in email, since the information is of no use.
Note also that our department only admits strong students, and we encourage strong students to apply. There is no other information or assistance that I can provide. Furthermore, I cannot provide any estimate of the chances for acceptance on the basis of any information sent to me in email. Admission and financial aid decisions are made by departmental committees and the Director of Graduate Studies during February and March.
Best wishes and good luck with your application,
Douglas B. West