Please see the CS department's webpage on admissions. There, you'll find information about all aspects of the application process (e.g., deadlines, your chances of being admitted), as well as information about the academic programs themselves.
If you are interested in working with me, you can mention this in your application (and also see the section below); this will draw my attention (as well as others') to your file. Unfortunately, I do not generally have time to reply to inquiries from prospective students.
I work on algorithmic statistics and machine learning, with particular emphases on probabilistic models, interactive learning, and privacy. See my recent papers for examples of projects I've worked on. Most of my work involves algorithmic design and theoretical analysis, as motivated by problems in applied machine learning.
If you are a current (or already admitted) Columbia student and are interested in discussing a project with me, please feel free to contact me via e-mail or drop by my office.