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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 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 200Algebra I

5

MATH 201Algebra II

5

MATH 202Algebra III

5

MATH 204Analysis I

5

MATH 205Analysis II

5

MATH 206Analysis III

5

MATH 208Manifolds I

5

MATH 209Manifolds II

5

MATH 210Manifolds III

5

The following course:

MATH 288APedagogy 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 203Algebra IV

5

MATH 207Complex Analysis

5

MATH 211Algebraic Topology

5

MATH 212Differential Geometry

5

MATH 213APartial Differential Equations I

5

MATH 213BPartial Differential Equations II

5

MATH 214Theory of Finite Groups

5

MATH 215Operator Theory

5

MATH 216Advanced Analysis

5

MATH 217Advanced Elliptic Partial Differential Equations

5

MATH 218Advanced Parabolic and Hyperbolic Partial Differential Equations

5

MATH 219Nonlinear Functional Analysis

5

MATH 220ARepresentation Theory I

5

MATH 220BRepresentation Theory II

5

MATH 222AAlgebraic Number Theory

5

MATH 222BAlgebraic Number Theory

5

MATH 223AAlgebraic Geometry I

5

MATH 223BAlgebraic Geometry II

5

MATH 225ALie Algebras

5

MATH 225BInfinite Dimensional Lie Algebras

5

MATH 226AInfinite Dimensional Lie Algebras and Quantum Field Theory I

5

MATH 226BInfinite Dimensional Lie Algebras and Quantum Field Theory II

5

MATH 227Lie Groups

5

MATH 228Lie Incidence Geometries

5

MATH 229Kac-Moody Algebras

5

MATH 232Morse Theory

5

MATH 233Random Matrix Theory

5

MATH 234Riemann Surfaces

5

MATH 235Dynamical Systems Theory

5

MATH 238Elliptic Functions and Modular Forms

5

MATH 239Homological Algebra

5

MATH 240ARepresentations of Finite Groups I

5

MATH 240BRepresentations of Finite Groups II

5

MATH 246Representations of Algebras

5

MATH 248Symplectic Geometry

5

MATH 249AMechanics I

5

MATH 249BMechanics II

5

MATH 249CMechanics III

5

MATH 252Fluid Mechanics

5

MATH 254Geometric Analysis

5

MATH 256Algebraic Curves

5

MATH 260Combinatorics

5

MATH 280Topics in Analysis

5

MATH 281Topics in Algebra

5

MATH 282Topics in Geometry

5

MATH 283Topics in Combinatorial Theory

5

MATH 284Topics in Dynamics

5

MATH 285Topics in Partial Differential Equations

5

MATH 286Topics in Number Theory

5

MATH 287Topics 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 and end of each academic year. The exams will be designed and graded by recent instructors of each core sequence.

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 their faculty advisor, 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:

  1. Complete the preliminary examinations and core sequences in accordance with the requirements outlined above;
  2. Pass the qualifying examination;
  3. Have a dissertation reading committee approved by the Mathematics Department and the Graduate Division;
  4. 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:

  1. Preliminary examinations and course sequence requirements

    Completed by the end of the student’s 2nd year

  2. Oral qualifying examination (and advancement to candidacy)

    Completed no later than student’s 4th year

  3. 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.