See the GSAS Dissertation Rules which are available in 107 Low Memorial Library, or consult http://www.columbia.edu/cu/gsas/pages/cstudents/
diss-office/rules/index.html for more details.

The student must prepare a dissertation, embodying original research, under the supervision of a sponsor approved for that Ph.D. program by the Executive Committee of the Faculty of the Graduate School and the Dean. Graduate School policy requires that all dissertations be written in English. Rare exceptions may be granted only with prior approval of the Executive Committee of the Graduate School of Arts and Sciences and the Dean (after departmental approval of the topic, but well before research has begun). Guidelines governing exceptions are available from the Dissertation Office, 107 Low Memorial Library.