# Requirements for a Doctorate in Mathematics

The principal aim of the graduate program is to develop the student's ability to do original research in mathematics. Independent and critical thinking is fostered by direct contact with faculty members. 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.

The PhD requirements are below and are also available in the Caltech Catalog, Section 4: Information for Graduate Students.

MILESTONES | TIMEFRAME |

Submit Plan of Study for approval by Graduate Option Rep | By end of first term |

Complete Core Courses and Qualifying Exam | By beginning of second year |

Complete the Advanced Math Courses | Before candidacy exam |

Complete the Candidacy Exam | By end of third year |

Hold Annual Thesis Advisory Committee meetings (beginning in the 2018-2019 academic year) | 6 months to 1 year after the oral candidacy exam and every year thereafter |

Final PhD Defense | By the end of fifth year |

### Plan of Study

The plan of study is the set of courses that a student will take as part of their graduate curriculum. Students should consult with the Option Rep on their Plan of Study.

**What to do in REGIS?**

Log in to Regis and navigate to the Ph. D. Candidacy Tab of your Graduate Degree Progress page. Add you courses into the Plan of Study section. When complete, click the "Submit Plan of Study to Option Rep" button. This will generate a notice to the Option Rep to approve your plan of study.

**Core Courses and Qualifying Exams**

The three core courses—Ma 110 in analysis, Ma 120 in algebra, and Ma 151 in geometry and topology—are required of all graduate students. Students are expected to complete these core courses in the first year, unless the student needs to take a preparatory course, such as Ma 109 (Introduction to Geometry and Topology). Entering students are allowed to take a qualifying examination in September or October in order to demonstrate knowledge of one or more of the core areas. By passing the examination, they are excused from taking the corresponding course series. For current course offerings and schedule, please see the Math Course Schedule.

**Ma 110 abc Real and Complex Analysis:**First, second, third terms. Analytic functions, conformal mappings, Riemann surfaces, abstract measure theory, Fubini and Radon-Nikodyn theorems, Riesz representation theorem. Banach spaces, duality, L p spaces, Hilbert spaces. Application to Fourier series and integrals, elements of spectral theory.**Ma 120 abc Abstract Algebra:**First, second, third terms. Abstract development of the basic structure theorems for groups, commutative and noncommutative rings, modules, algebras, fields (including Galois theory), and group representations.**Ma 151 abc Topology and Geometry:**First, second, third terms. Fundamental groups and covering spaces, homology, cohomology and calculation of homology groups, exact sequences. Fibrations, higher homotopy groups and exact sequences of fibrations, structure of differentiable manifolds, degree theory, de Rham cohomology, elements of Morse theory. Geometry of Riemannian manifolds, covariant derivatives, geodesics, curvature, relations between curvature and topology.

Qualifying examinations in the three core areas—analysis, algebra, and geometry/topology—are offered in October and June. These examinations emphasize mastery of the basic concepts and theorems and the ability to apply them to specific cases.

The exams are 1-hour oral exams (possibly shorter
or longer). A list of problems will be given to the examinee one week prior to
the exam. The student is not
required or expected to turn in
written solutions for the Algebra and Topology exams. The Analysis exam require
written solutions to be turned in. During the exam, one or more examiners will ask the student questions
from the list, though a few other (related) questions may also be asked. The student must not discuss the questions with anyone else,
even after taking the oral exam, or give a copy of the exam to other students. The
student may use lecture notes and homework from the courses to prepare, plus the
textbooks, but no other sources (specific instructions on listed on each exam).

If a student does not pass one of the qualifying exams in June of the first year, the student will be given a chance to take the exam in the coming fall. If the student fails again, depending on the performance during the exam, one more chance may be granted."

Email [email protected] to coordinate your qualifying exams and to request practice materials.

**What to do in REGIS?**

Nothing additional. Courses will be automatically counted towards your degree requirements as you complete them.

The PMA Student Programs Office will update your REGIS record once you pass the exams.

Advanced Mathematics Courses

In addition, students are required to complete nine quarters of other advanced mathematics courses Ma 111 and above, at least two of which are in discrete mathematics: combinatorics, complexity, and computability, or logic and set theory. Unless these nine course quarters are given pass/fail only, they must be taken for grades. Reading and research do not normally qualify to meet these requirements.

**What to do in REGIS?**

Nothing additional. Once you complete the courses advanced courses they will automatically be counted towards your degree requirements.

### Candidacy Exam

The Candidacy Exam is primarily a test of the candidate's suitability for research in his or her chosen field. The presentation will describe both the general area of the student's proposed thesis research and the specific problem or problems to be addressed.

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 qualifying exams 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.

**What to do in REGIS?**

Login to Regis, navigate to the Ph. D. Candidacy Tab of your Graduate Degree Progress page, and scroll down to the Candidacy Examination section. Enter your candidacy committee members and a brief description of the topic of your presentation. Click the "Submit Candidacy Committee for Approval" button and this will automatically generate notifications for the Option Rep and your advisor to approve your committee. Enter the date, time, and location of your exam and click "Submit Examination Details." Your committee members will automatically be sent email reminders with the exam details.

After the exam, your committee members will enter their result and any comments they may have. Non-Caltech committee members are instructed to send their results and comments to the math graduate office who will enter the information on their behalf. Once all "pass" results have been entered, the Option Rep will be prompted to recommend you for admission to candidacy. The recommendation goes to the Dean of Graduate Studies who has the final approval to formally admit you to candidacy.

### Thesis Advisory Committee (TAC) - Requirement to begin in the 2018-2019 academic year

After the oral candidacy exam, students will hold annual meetings with their Thesis Advisory Committee (TAC). The TAC will review the research progress and provide feedback and guidance towards completion of the degree. The TAC is normally constituted from the candidacy examiners, but students may propose variations or changes at any time to the option representative. The TAC chair is normally someone other than the research adviser. The TAC chair will typically also serve as the thesis defense chair, but changes may be made by contacting the math graduate office.

**What to do in REGIS?**

Login to Regis, navigate to the Ph. D. Examination Tab of your Graduate Degree Progress page, and scroll down to the Examination Committee section. Enter the names of your Thesis Advisory Committee members. Click the "Submit Examination Committee for Approval" button and this will automatically generate notifications for the Option Rep and the Dean of Graduate Studies to approve your committee.

Enter the date, time and location of your TAC meeting and click "Submit Details." Your committee members will automatically be sent email reminders with the meeting details.

### PhD Defense

The final thesis examination will cover the thesis topic and its relation to the general body of knowledge of mathematics. On or before the first Monday in May of the year in which the degree is to be conferred, candidates for the degree of Doctor of Philosophy must deliver copies of their theses to their advisers, to the Graduate Office, and to the members of the committee. The defense must take place at least three weeks before the degree is to be conferred. Please refer to the Graduate Office and Library webpages for thesis guidelines, procedures, and deadlines.

**What to do in REGIS?**

Login to Regis and navigate to the Ph. D. Examination tab of your Graduate Degree Progress page. By default, the thesis committee is the same as your TAC committee. If you need to change your committee members, please contact the math graduate office.

Enter the following information:

- | Date, time, and location of your exam and click the "Submit Examination Details" button. Your committee members will automatically be sent email reminders with the exam details. |

- | Commencement Information and click the "Submit Commencement Information" button (at least 2 weeks prior to defense) |

- | Marching Information and click the "Submit your Marching Information" button (at least 2 weeks prior to commencement) |

After the exam, your committee members will enter their result and any comments they may have in REGIS. Non-Caltech committee members are instructed to send their results and comments to the math graduate office who will enter the information on their behalf. Once all "pass" results have been entered and your electronic thesis submission has been has been accepted (see Library website), the Division chair, then Dean of Graduate studies, will be prompted to approve your thesis.

Other steps for final defense, including submission to the graduate office and library, are detailed here.