In general, the admissions office at Ecole Polytechnique takes care of all admissions related issues.
This is especially true if any kind of credit transfer is to take place to a foreign university, either for research internships or courses.
In any event, if you’re interested in studying with me, the appropriate procedure would be to contact me first, to ensure that there’s a match of interests, and then to contact the admissions office for formal procedures to commence. Yes, the admissions office are the good guys, they can even help with accommodation and such.
For ideas for a research project, you may examine at my research page, the page of my research team or the internships proposed by myself and my colleagues in my team.
It’s of great help if an initial contact (by email) is followed by a grade transcript, and a letter of recommendation and/or a number of references who may be contacted.