# Mathematics Ph.D.

## Introduction

The objectives of the mathematics Ph.D. program are to prepare students for a career in academia, industry, or teaching. At the end of their studies, students will possess the ability to solve problems and communicate solutions in rigorous mathematical language, to communicate mathematical concepts effectively, and to conduct independent research.

Entering graduate students are advised initially by an assigned faculty mentor. Within the first two years, and typically after passing the preliminary examinations, the student selects a Ph.D. advisor in the area of the student's research interest.

Each graduate student is expected to consult with their advisor to formulate a plan of study and research. The student's advisor ultimately will be the student's thesis advisor.

Ph.D. students are expected to obtain their Ph.D. degree within six years. Students admitted to the Ph.D. program may receive a master's degree en route to the Ph.D.

## Advancement to Candidacy

### Course Requirements

A three-course sequence in each of the three fields of algebra, analysis, and geometry-topology (manifolds) will be offered each year. Preliminary examinations will be given for each core sequence at the beginning, middle, and end of each academic year.

First-level passage of a preliminary examination satisfies the core sequence requirement for that field. Ph.D. students are required to complete the full core sequence in the field associated with the preliminary examination in which they do not achieve a first-level pass. The core sequences are as follows:

MATH 200 | Algebra I | 5 |

MATH 201 | Algebra II | 5 |

MATH 202 | Algebra III | 5 |

MATH 204 | Analysis I | 5 |

MATH 205 | Analysis II | 5 |

MATH 206 | Analysis III | 5 |

MATH 208 | Manifolds I | 5 |

MATH 209 | Manifolds II | 5 |

MATH 210 | Manifolds III | 5 |

#### The following course:

MATH 288A | Pedagogy of Mathematics | 2 |

#### Students are also required to complete six additional courses in mathematics.

No more than three courses may be independent study or thesis research courses. Sample courses include:

MATH 203 | Algebra IV | 5 |

MATH 207 | Complex Analysis | 5 |

MATH 211 | Algebraic Topology | 5 |

MATH 212 | Differential Geometry | 5 |

MATH 213A | Partial Differential Equations I | 5 |

MATH 213B | Partial Differential Equations II | 5 |

MATH 214 | Theory of Finite Groups | 5 |

MATH 215 | Operator Theory | 5 |

MATH 216 | Advanced Analysis | 5 |

MATH 217 | Advanced Elliptic Partial Differential Equations | 5 |

MATH 218 | Advanced Parabolic and Hyperbolic Partial Differential Equations | 5 |

MATH 219 | Nonlinear Functional Analysis | 5 |

MATH 220A | Representation Theory I | 5 |

MATH 220B | Representation Theory II | 5 |

MATH 222A | Algebraic Number Theory | 5 |

MATH 222B | Algebraic Number Theory | 5 |

MATH 223A | Algebraic Geometry I | 5 |

MATH 223B | Algebraic Geometry II | 5 |

MATH 225A | Lie Algebras | 5 |

MATH 225B | Infinite Dimensional Lie Algebras | 5 |

MATH 226A | Infinite Dimensional Lie Algebras and Quantum Field Theory I | 5 |

MATH 226B | Infinite Dimensional Lie Algebras and Quantum Field Theory II | 5 |

MATH 227 | Lie Groups | 5 |

MATH 228 | Lie Incidence Geometries | 5 |

MATH 229 | Kac-Moody Algebras | 5 |

MATH 232 | Morse Theory | 5 |

MATH 233 | Random Matrix Theory | 5 |

MATH 234 | Riemann Surfaces | 5 |

MATH 235 | Dynamical Systems Theory | 5 |

MATH 238 | Elliptic Functions and Modular Forms | 5 |

MATH 239 | Homological Algebra | 5 |

MATH 240A | Representations of Finite Groups I | 5 |

MATH 240B | Representations of Finite Groups II | 5 |

MATH 246 | Representations of Algebras | 5 |

MATH 248 | Symplectic Geometry | 5 |

MATH 249A | Mechanics I | 5 |

MATH 249B | Mechanics II | 5 |

MATH 249C | Mechanics III | 5 |

MATH 252 | Fluid Mechanics | 5 |

MATH 254 | Geometric Analysis | 5 |

MATH 256 | Algebraic Curves | 5 |

MATH 260 | Combinatorics | 5 |

MATH 280 | Topics in Analysis | 5 |

MATH 281 | Topics in Algebra | 5 |

MATH 282 | Topics in Geometry | 5 |

MATH 283 | Topics in Combinatorial Theory | 5 |

MATH 284 | Topics in Dynamics | 5 |

MATH 285 | Topics in Partial Differential Equations | 5 |

MATH 286 | Topics in Number Theory | 5 |

MATH 287 | Topics in Topology | 5 |

### Foreign Language Requirements

**Beginning with the cohort entering fall 2022, there is no longer a foreign language requirement for the Ph.D. in Mathematics. **Although there is no longer a foreign language requirement, some students may need to read papers in unfamiliar languages to carry out their research. When this is the case, students are strongly encouraged to reach out to faculty for guidance.

### Teaching Requirement

Ph.D. students must complete a minimum of three quarters as a teaching assistant (TA). All TAs are required to participate in the department's teaching assistant training program.

TA appointments are usually made at 50 percent time (an assigned workload of approximately 220 hours per quarter). TAs are under the supervision of the faculty member responsible for the course. TAs are covered by a collective bargaining agreement between the University of California and the United Auto Workers (UAW).

Instructors and their TA(s) will meet at the beginning of the quarter to complete the Notification of TA Duties form in order to identify the agreed upon tasks. The performance of these tasks will form the basis of the end-of-quarter performance evaluation and will use the following criteria: quality of work; accuracy and attention to detail; interaction with students, peers, and instructor; knowledge of subject; and dependability. The specific allocation of TA duties is subject to change, depending on enrollments and the number of teaching assistantships in the department allocation. The general duties vary, depending on the course assigned and level of the course.

### Pre-Qualifying Requirements

**Preliminary Examinations**

Preliminary examinations are given for each core sequence in the fields of algebra, analysis, and geometry-topology at the beginning, middle, and end of each academic year. The exams will be designed and graded by a committee of three members.

A first-level pass signifies that the student has the basic knowledge to start research with a thesis advisor in this particular area. A second-level pass signifies that the student has a very good understanding of the basic concepts, but not necessarily enough to conduct independent research.

Ph.D. students must obtain a first-level pass on at least one of the three written preliminary examinations and a second-level pass on at least one other. Students must complete the full three-course sequence in the field associated with the preliminary examination in which they did not achieve a first-level pass. Students may take the preliminary examinations as often as they wish.

Ph.D. students should complete the preliminary examinations and core sequence requirements by the end of their second year in order to make satisfactory progress. If a graduate student does not fulfill these requirements by the end of their second year, they may be placed on academic probation, depending on their progress in the program. If a graduate student has not fulfilled these requirements by the end of their third year, they are subject to dismissal from the program.

Topics for the preliminary examinations include:

- Algebra
- Linear algebra
- Group theory
- Ring and module theory
- Field theory
- Galois theory

- Analysis
- Basic analysis
- General topology
- Metric spaces
- Measure and integration
- Complex analysis
- Functional analysis

- Geometry-topology (manifolds)
- Manifold and tangent bundle
- Differential forms and integration on manifolds
- Fundamental group and covering space
- (Co)homology
- Differential geometry

### Qualifying Examination

**Oral Qualifying Examination**

All graduate students in the Ph.D. program are required to take an oral examination, called the oral qualifying examination, for advancement to candidacy for the Ph.D. degree. Students typically complete this examination between their 7th and 9th quarter in residence.

Students will demonstrate that they have a sufficient understanding of their Ph.D. thesis problem. Any student who has not passed their oral exam by the end of the fourth year may be subject to academic probation or dismissal from the program.

The Report on Qualifying Examination Form must be filled out by the qualifying examination committee immediately following the examination. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division. The student may request to see a copy of the report.

If the student fails the examination, a re-examination can be given within the next three months. The membership of the examining committee usually remains fixed.

**Qualifying Examination Committee Composition**

The examining committee consists of the student’s faculty advisor, at least two other faculty members from the Mathematics Department, and at least one outside tenured faculty member from either another discipline at UCSC or another academic institution (involved in research and graduate education of the same or different discipline). The student, in consultation with the student’s faculty advisor, selects the committee. The chair of the committee must be someone other than the student’s faculty advisor.

The Graduate Division must approve the committee. The Committee Nomination of Ph.D. Qualifying Examination Form must be completed and submitted at least one month prior to the requested exam date. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division.

The committee decides on the topics for the examination, which should be broad enough to encompass a substantial body of knowledge in the area of the student’s interest. The written list of topics to be included in the examination, along with a short bibliography, is prepared by the student. A copy is given to each committee member and a copy is put into the student’s permanent records.

### Post-Qualifying Requirements

**Dissertation Reading Committee Composition**

A Ph.D. student, in consultation with the graduate vice chair, is responsible for selecting a dissertation reading committee. The committee consists of the student’s advisor and at least two other members of the mathematics faculty. In special circumstances, a committee member may be chosen from another department and/or from another institution. The student’s advisor is the chair of the committee.

The Graduate Division must approve the committee. The Nominations for Dissertation Reading Committee Form must be completed and submitted prior to advancement to candidacy. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division.

A new form must be submitted for approval if changes to the dissertation reading committee must be made.

### Advancement to Candidacy

To make satisfactory progress, a Ph.D. student should advance to candidacy by the end of their fourth year. A Ph.D. student who has not advanced to candidacy by the end of the fourth year will be placed on academic probation or be subject to dismissal from the program.

Students must complete the following in order to advance to candidacy:

- Complete the preliminary examinations and core sequences in accordance with the requirements outlined above;
- Pass the qualifying examination;
- Have a dissertation reading committee approved by the Mathematics Department and the Graduate Division;
- Have no incomplete grades (I) on their record.

An advancement to candidacy fee will be billed to the student’s account. The student will be officially advanced the following term after all of these requirements are met.

## Dissertation

### Dissertation

Each graduate student in the Ph.D. program is required to write a Ph.D. dissertation or thesis on a research topic in mathematics. The Ph.D. dissertation should contain original research results that are publishable in a peer-reviewed journal. All members of the student’s dissertation committee must read and approve the dissertation.

More information about dissertation submission can be found at the Graduate Division website.

### Dissertation Defense

After the dissertation has been approved, the student has an option of making a public oral presentation of the mathematical results contained in the dissertation—the “thesis defense.” A recommendation by the dissertation committee will be made to the Mathematics Department and to the Graduate Council on the granting of the Ph.D. degree.

### Academic Progress

Ph.D. students are expected to adhere to the below degree timetable:

- Preliminary examinations and course sequence requirements
Completed by the end of the student’s 2nd year

- Oral qualifying examination (and advancement to candidacy)
Completed no later than student’s 9th quarter

- Dissertation defense
Completed no later than the end of the 6th year

Annual meetings with the graduate vice chair and the graduate advisor and program coordinator are conducted with each student on a one-on-one basis. These meetings serve to notify the student of their current progress within the program and outline expectations for the continuation of normative progress toward the Ph.D. degree.

### Applying for Graduation

Ph.D. students must complete the Application for the Doctor of Philosophy degree form by the appropriate quarter’s deadline listed in the current Academic and Administrative Calendar.

The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division.