However, I get a large number of emails about this, and it would be impossible to fairly read and respond to all of them. If you want to do a PhD here with me (or one of my talented colleagues), the best way to do this is to submit a formal application.

If you've emailed me directly and I haven't responded, that doesn't mean there isn't a good match! It's just that a CV or a link to a paper is rarely enough information to make a judgment, which is why we have a process that asks for lots of information. If you mention wanting to work with me in your application, then (subject to decisions of the admissions committee) your application will find me, and I will give it careful consideration.

For my research area, a strong mathematical background (and a desire to further strengthen it) is very important. It is also a good idea to discuss what your interests are: Do you want to prove theorems? Build large-scale learning systems? Do you have a particular application area you are interested in? Are you interested in building new general tools, or in working with combinations of existing tools to solve real problems?

There are no "right" answers these questions, because there are an infinite number of important problems to work on! But it's important to show that you have thought about how you would actually potentially be spending a significant fraction of your waking hours for several years.

On a related note, I'm often asked by overseas undergraduate students about summer research internships, which unfortunately I cannot support. This includes self-funded internships.