# Mathematics, Ph.D.

Graduate courses are designed to meet the needs of students doing graduate work in mathematics and in those disciplines that require graduate-level mathematics for their study. Among the fields covered are analysis, algebra, applied and computational mathematics, mathematical biology, geometry and topology, probability, ordinary and partial differential equations, and mathematical logic.

In addition to formal courses, there are seminars for advanced study toward the Ph.D. in various fields of mathematics. Topics will vary from year to year. Each seminar is conducted by a faculty member specializing in the subject studied. Enrollment will be subject to the approval of the instructor in charge.

When accepted into the doctoral program, the student embarks on a program of formal courses, seminars, and individual study courses to prepare for the Ph.D. written examinations, Advancement to Candidacy examination, and dissertation.

##### Requirements

Upon entering the program, students are expected to take MATH 210A, MATH 210B, MATH 210C, MATH 220A, MATH 220B, MATH 220C, MATH 230A, MATH 230B, and MATH 230C, which must be passed with a grade of B or better. Students must complete these sequences by the end of the second year.

By the start of the second year, students must achieve at least two passes at the M.S. level among three exams in Real Analysis, Complex Analysis, and Algebra. By the start of the third year, students must achieve two Ph.D. level passes among three exams in Real Analysis, Complex Analysis, and Algebra.

To satisfy the exam requirements, students may take the Core Assessment Exams (offered in spring of every year) or the Qualifying Exams (offered before the start of the fall quarter) in these areas. Students may not attempt to take an exam in a particular subject area more than three times. A student who passes a Qualifying Examination at the Ph.D. level prior to taking the corresponding course will be exempted from taking the course.

Some students may require additional background prior to entering MATH 210A, MATH 210B, MATH 210C, MATH 230A, MATH 230B, and MATH 230C. This will be determined by assessment prior to the start of the students’ first year by the Vice Chair for Graduate Studies, upon consultation with the Graduate Studies Committee. Such students will be directed into MATH 205 and/or MATH 206, or equivalent, during their first year. These students may pass one Comprehensive Exam in the areas of Algebra or Analysis in lieu of achieving an M.S. pass on one Core Assessment or Qualifying Exam that must be obtained prior to the start of the students’ second year. Comprehensive Exams in Analysis and Algebra will be offered once per year in the spring quarter.

By the end of their second year, students must declare a major specialization from the following areas: Algebra, Analysis, Applied and Computational Mathematics, Geometry and Topology, Logic, or Probability. Students are required to take two series of courses from their chosen area. (Students who later decide to change their area must also take two series of courses from the new area.) Additionally, all students must take two series outside their declared major area of specialization. Special topics courses within certain areas of specialization and courses counted toward the M.S., other than MATH 205A-MATH 205B-MATH 205C , will count toward the fulfillment of the major specialization requirement.

By the beginning of their third year, students must have an advisor specializing in their major area. With the advisor’s aid, the student forms a committee for the Advancement to Candidacy oral examination. This committee will be approved by the Department on behalf of the Dean of the Graduate Division and the Graduate Council and will consist of five faculty members. At least one, and at most two, of the members must be faculty from outside the Department. Before the end of the third year, students must have a written proposal, approved by their committee, for the Advancement to Candidacy examination. The proposal should explain the role of at least two series of courses from the student’s major area of specialization that will be used to satisfy the Advancement to Candidacy requirements. The proposal should also explain the role of additional research reading material as well as providing a plan for investigating specific topics under the direction of the student’s advisor(s). Only one of the courses MATH 210A-MATH 210B-MATH 210C, MATH 220A-MATH 220B-MATH 220C, and MATH 230A-MATH 230B-MATH 230C may count for the course requirement for Advancement to Candidacy Examinations. After the student meets the requirements, the Graduate Studies Committee recommends to the Dean of the Graduate Division the advancement to candidacy for the Ph.D. Students should advance to candidacy by the beginning of their fourth year.

After advancing to candidacy, students are expected to be fully involved in research toward writing their Ph.D. dissertation. Ideally, students should keep in steady contact/interaction with their Doctoral Committee.

Teaching experience and training is an integral part of the Ph.D. program. All doctoral students are expected to participate in the Department’s teaching program.

The candidate must demonstrate independent, creative research in Mathematics by writing and defending a dissertation that makes a new and valuable contribution to mathematics in the candidate’s area of concentration. Upon Advancement to Candidacy a student must form a Thesis Committee, a subcommittee of the Advancement Examination Committee, consisting of at least three faculty members and chaired by the student’s advisor. The committee guides and supervises the candidate’s research, study, and writing of the dissertation; conducts an oral defense of the dissertation; and recommends that the Ph.D. be conferred upon approval of the Doctoral Dissertation. The normative time for completion of the Ph.D. is six years, and the maximum time permitted is seven years. Completion of the Ph.D. degree must occur within nine quarters of Advancement to Ph.D. candidacy.

##### Examinations

Ph.D. examinations are given in Algebra, Complex Analysis, and Real Analysis. All students seeking the Ph.D. must successfully complete two examinations before the end of the third year of entering the graduate program. Only two attempts are allowed for a Ph.D. student on each exam.

##### Area Requirements

Ph.D. students will choose from one of six areas of specialization in the Mathematics Department, which determines course work requirements. Each area of specialization will have a core course, which the Department will do its best to offer each year. The Department will offer other courses every other year, or more frequently depending on student demands and other Department priorities.

Algebra | |

MATH 230A- 230B- 230C | Algebra and Algebra and Algebra (core) |

MATH 232A- 232B- 232C | Algebraic Number Theory and Algebraic Number Theory and Algebraic Number Theory |

MATH 233A- 233B- 233C | Algebraic Geometry and Algebraic Geometry and Algebraic Geometry |

MATH 235A | Mathematics of Cryptography |

Analysis | |

MATH 210A- 210B- 210C | Real Analysis and Real Analysis and Real Analysis (core) |

MATH 220A- 220B- 220C | Analytic Function Theory and Analytic Function Theory and Analytic Function Theory (core) |

MATH 260A- 260B- 260C | Functional Analysis and Functional Analysis and Functional Analysis |

MATH 295A- 295B- 295C | Partial Differential Equations and Partial Differential Equations and Partial Differential Equations |

MATH 296 | Topics in Partial Differential Equations |

Applied and Computational Mathematics | |

MATH 290A- 290B- 290C | Methods in Applied Mathematics and Methods in Applied Mathematics and Methods in Applied Mathematics (core) |

MATH 225A- 225B- 225C | Introduction to Numerical Analysis and Scientific Computing and Introduction to Numerical Analysis and Scientific Computing and Introduction to Numerical Analysis and Scientific Computing |

MATH 226A- 226B- 226C | Computational Differential Equations and Computational Differential Equations and Computational Differential Equations |

MATH 227A- 227B | Mathematical and Computational Biology and Mathematical and Computational Biology |

MATH 295A- 295B- 295C | Partial Differential Equations and Partial Differential Equations and Partial Differential Equations |

Geometry and Topology | |

MATH 218A- 218B- 218C | Introduction to Manifolds and Geometry and Introduction to Manifolds and Geometry and Introduction to Manifolds and Geometry (core) |

MATH 222A | Several Complex Variables and Complex Geometry |

MATH 240A- 240B- 240C | Differential Geometry and Differential Geometry and Differential Geometry |

MATH 245A- 245C- 245C | Topics in Geometric Analysis and Topics in Geometric Analysis and Topics in Geometric Analysis |

MATH 250A- 250B- 250C | Algebraic Topology and Algebraic Topology and Algebraic Topology |

Logic | |

MATH 280A- 280B- 280C | Mathematical Logic and Mathematical Logic and Mathematical Logic (core) |

MATH 281A- 281B- 281C | Set Theory and Set Theory and Set Theory |

MATH 282A- 282B- 282C | Model Theory and Model Theory and Model Theory |

MATH 285 | Topics in Mathematical Logic |

Probability | |

MATH 210A- 210B- 210C | Real Analysis and Real Analysis and Real Analysis |

MATH 270A- 270B- 270C | Probability and Probability and Probability (core) |

MATH 271A- 271B- 271C | Stochastic Processes and Stochastic Processes and Stochastic Processes (core) |

MATH 274 | Topics in Probability |