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2020-21 UCSC General Catalog
2019-20 UCSC General Catalog

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. adviser in the area of the student's research interest.

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

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.

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:

MATH200 | Algebra I | 5 |

MATH201 | Algebra II | 5 |

MATH202 | Algebra III | 5 |

MATH204 | Analysis I | 5 |

MATH205 | Analysis II | 5 |

MATH206 | Analysis III | 5 |

MATH208 | Manifolds I | 5 |

MATH209 | Manifolds II | 5 |

MATH210 | Manifolds III | 5 |

MATH288A | Pedagogy of Mathematics | 2 |

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

MATH203 | Algebra IV | 5 |

MATH207 | Complex Analysis | 5 |

MATH211 | Algebraic Topology | 5 |

MATH212 | Differential Geometry | 5 |

MATH213A | Partial Differential Equations I | 5 |

MATH213B | Partial Differential Equations II | 5 |

MATH214 | Theory of Finite Groups | 5 |

MATH215 | Operator Theory | 5 |

MATH216 | Advanced Analysis | 5 |

MATH217 | Advanced Elliptic Partial Differential Equations | 5 |

MATH218 | Advanced Parabolic and Hyperbolic Partial Differential Equations | 5 |

MATH219 | Nonlinear Functional Analysis | 5 |

MATH220A | Representation Theory I | 5 |

MATH220B | Representation Theory II | 5 |

MATH222A | Algebraic Number Theory | 5 |

MATH222B | Algebraic Number Theory | 5 |

MATH223A | Algebraic Geometry I | 5 |

MATH223B | Algebraic Geometry II | 5 |

MATH225A | Lie Algebras | 5 |

MATH225B | Infinite Dimensional Lie Algebras | 5 |

MATH226A | Infinite Dimensional Lie Algebras and Quantum Field Theory I | 5 |

MATH226B | Infinite Dimensional Lie Algebras and Quantum Field Theory II | 5 |

MATH227 | Lie Groups | 5 |

MATH228 | Lie Incidence Geometries | 5 |

MATH229 | Kac-Moody Algebras | 5 |

MATH232 | Morse Theory | 5 |

MATH233 | Random Matrix Theory | 5 |

MATH234 | Riemann Surfaces | 5 |

MATH235 | Dynamical Systems Theory | 5 |

MATH238 | Elliptic Functions and Modular Forms | 5 |

MATH239 | Homological Algebra | 5 |

MATH240A | Representations of Finite Groups I | 5 |

MATH240B | Representations of Finite Groups II | 5 |

MATH246 | Representations of Algebras | 5 |

MATH248 | Symplectic Geometry | 5 |

MATH249A | Mechanics I | 5 |

MATH249B | Mechanics II | 5 |

MATH249C | Mechanics III | 5 |

MATH252 | Fluid Mechanics | 5 |

MATH254 | Geometric Analysis | 5 |

MATH256 | Algebraic Curves | 5 |

MATH260 | Combinatorics | 5 |

MATH280 | Topics in Analysis | 5 |

MATH281 | Topics in Algebra | 5 |

MATH282 | Topics in Geometry | 5 |

MATH283 | Topics in Combinatorial Theory | 5 |

MATH284 | Topics in Dynamics | 5 |

MATH285 | Topics in Partial Differential Equations | 5 |

MATH286 | Topics in Number Theory | 5 |

MATH287 | Topics in Topology | 5 |

The foreign language requirement must be satisfied before taking the oral qualifying examination. Graduate students in the Ph.D. program are required to demonstrate knowledge of French, German, or Russian, sufficient to read the mathematical literature in the language. Any member of the mathematics faculty may administer a foreign language examination.

The examination can be either oral or written. It typically requires translation of a text in one of the three foreign languages into English.

The Report on Language Requirement Form must be filled out by the student and the faculty member administering 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 adviser and program coordinator for review and submission to the Graduate Division.

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.

**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 adviser 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

**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 12th 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 adviser 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 adviser, 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 adviser, selects the committee. The chair of the committee must be someone other than the student’s faculty adviser.

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 adviser 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.

**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 adviser 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 adviser 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 adviser 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.

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;
- Satisfy the language requirement;
- 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.

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.

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.

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

- Language examination
Completed by the end of the student’s 3rd year

- Oral qualifying examination (and advancement to candidacy)
Completed no later than student’s 12th quarter

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

Annual meetings with the graduate vice chair and the graduate adviser 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.

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 adviser and program coordinator for review and submission to the Graduate Division.