Mathematics Ph.D. students at Penn have the option to apply to the masters program in Computer Science. If they are accepted, then they will be candidates for both degrees. The Math Ph.D. requires 20 courses, and the Computer Science masters requires ten courses; but because of double-counting of courses, it is possible to fulfill the course requirements of both degrees with a total of 22 courses. As a result, students in this joint degree program take no longer than other Math Ph.D. students to finish (generally, five years).
Math Ph.D. students who wish to pursue a Computer Science masters will typically apply to the Computer Science Department at the end of their first year or in their second year. Students who wish to take Computer Science courses during their first two years, other than the summer, should first get permission from the Mathematics Graduate Group Chair. This is to insure that these courses do not interfere with taking the basic required Math courses. Math Ph.D. students who enter the Computer Science masters program can count any relevant courses they may have taken as grad students at Penn prior to entering the masters program, on the same basis as courses taken after entering that program.
There are currently two different Computer Science masters degrees offered. One of these, the MSE in Computer and Information Science, is available to students who already have a very strong background in computer science (equivalent to an undergraduate major in that subject). The other degree, the Masters in Computer and Information Technology, is for students with less background in computer science. Details of these programs are available on the Computer Science web page . Each of these two programs has certain required courses, and permits several electives. Under an arrangement between the Math and Computer Science Departments, up to four of these electives can be taken in the Math Department (subject to certain restrictions on the choice of those courses). Similarly, up to four of the 20 required courses for the Math Ph.D. can be taken in the Computer Science Department. Students pursuing both degrees should coordinate their programs with the two departments, to be sure that they will be able to double-count the desired courses.
Specifically, students in the MSE-CIS program will take the four required Computer Science courses for that degree, plus six electives, of which four can be taken in Mathematics. Of those four, at least one must be selected from a (long) list of Computer-related Math courses. Students in the MCIT program will take the six required courses for that degree (except that Math Ph.D. students can ordinarily be exempted from CIT 592) along with electives, for a total of ten courses. The electives may be selected from a list of pre-approved Computer-related Math course; see the Computer Science web page for the list and other info.
Additionally you may want to contact Mike Felker, the CIS Graduate Coordinator.