# PhD Requirements

The major requirement for the PhD in mathematics at Caltech is the presentation and acceptance by the faculty of a thesis containing results of original research.

For admission to candidacy for the PhD, students are required to:

Demonstrate a good working knowledge in the three core areas: Algebra, Analysis, and Topology/Geometry by

earning grades of B or better in the core courses, Ma 110, 120, 151 (unless excused)

taking and passing a written qualifying exam in two of the three areas.

Complete at least 9 quarter-courses in advanced mathematics in addition to the core courses. At least two of these must be in discrete mathematics (combinatorics, logic, complexity and computability).

Pass a candidacy examination consisting of an oral presentation to a committee of mathematics faculty describing the proposed thesis research and the area of research it belongs to. At least a week prior to the examination, candidates submit a written summary of their presentation.

Qualifying examinations are usually taken in June. Some entering students take a qualifying examination in October in order to demonstrate knowledge of one of the core areas and be excused from taking the corresponding course. It is expected that the core courses will be completed in the first year, unless the student needs to take a preparatory course, such as Ma 109 (Introduction to Geometry and Topology). The candidacy examination is expected to be completed before the end of the student's third year.

In addition to the core courses, students are required to complete nine quarters of other advanced mathematics courses, of which at least two quarters must be in the area of discrete mathematics: combinatorics, logic, complexity, and computability. Under special circumstances (e.g., finishing the degree in three years), exceptions to these requirements may be granted by the graduate option representative.

Details of the candidacy and course requirements are given in theĀ catalog.