Math 490 is a course number reserved for undergraduate students to earn credit while doing research under the direction of a Math Department faculty member. See this page for suggestions on locating a faculty advisor.
Most Math 490 courses earn 1-2 units and do not count toward a major or minor program. A Math faculty member only needs to register the course with the Math Department and get D-clearance for the student to add. This faculty advisor is responsible for the course content, assessment, and grading decisions.
If a student would like an instructor outside the Math department to supervise a Math 490, the student should ask the outside instructor to write a short summary of the proposed course content, how the student will be evaluated, and the number of units. The instructor can send this proposal to the Math DUS (Paul Tokorcheck) for review. If Math approves the proposal one of our faculty can (formally) act as the instructor of record, with the two faculty communicating about the student's performance so a grade can be assigned at the end of the semester.
If the Math 490 is intended to be used as an elective toward a MATH or AMCM major (ostensibly then worth 4 units), this substitution of an elective requires department approval and the involvement of the student's academic advisor. The supervising faculty should write up a course plan as described above and send it to the Math DUS for review.