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 is likely the best way to draw my (and others') attention to your file. Note that there are several faculty members at Columbia working on machine learning, theory of computation, and foundations of data science. Unfortunately, I do not generally have time to reply to inquiries from prospective students.
I do not have summer internship positions available in 2017.
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 and statistics.
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 visit me during office hours.