{ "college": "uc-irvine", "edition": "2026-2027", "departmentName": "Department of Mathematics", "id": "department-of-mathematics--mathematics-phd", "name": "Mathematics", "award": "Ph.D.", "department": "department-of-mathematics", "sourceUrl": "https://catalogue.uci.edu/schoolofphysicalsciences/departmentofmathematics/mathematics_phd/", "blocks": [ { "kind": "text", "markdown": "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.\n\nIn 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." }, { "kind": "text", "markdown": "### Requirements\n\nWhen 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.\n\n##### Requirements\n\nUpon 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.\n\nBy the start of the second year, students must achieve at least two passes at the M.S. level among four exams in Real Analysis, Complex Analysis, Algebra, and Applied Math. By the start of the third year, students must achieve two Ph.D. level passes among four exams in Real Analysis, Complex Analysis, Algebra, and Applied Math.\n\nTo 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.\n\nSome students may require additional background prior to entering MATH 210A, MATH 210B, MATH 210C. 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, or equivalent, during their first year. These students may pass one Comprehensive Exam in the area of Analysis in lieu of achieving an M.S. pass on one Qualifying Exam that must be obtained prior to the start of the students’ second year. The Comprehensive Exam in Analysis will be offered once per year in the spring quarter.\n\nBy 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.\n\nBy 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.\n\nAfter 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.\n\nTeaching 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.\n\nThe 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.\n\n##### Examinations\n\nPh.D. examinations are given in Algebra, Complex Analysis, Applied Math, 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.\n\n##### Area Requirements\n\nPh.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." }, { "kind": "courselist", "rows": [ { "type": "comment", "text": "Algebra" }, { "type": "course", "code": "MATH 230A & MATH 230B & MATH 230C", "title": "Algebraand Algebraand Algebra (core)", "indent": false }, { "type": "course", "code": "MATH 232A & MATH 232B & MATH 232C", "title": "Algebraic Number Theoryand Algebraic Number Theoryand Algebraic Number Theory", "indent": false }, { "type": "course", "code": "MATH 233A & MATH 233B & MATH 233C", "title": "Algebraic Geometryand Algebraic Geometryand Algebraic Geometry", "indent": false }, { "type": "comment", "text": "Analysis" }, { "type": "course", "code": "MATH 210A & MATH 210B & MATH 210C", "title": "Real Analysisand Real Analysisand Real Analysis (core)", "indent": false }, { "type": "course", "code": "MATH 220A & MATH 220B & MATH 220C", "title": "Analytic Function Theoryand Analytic Function Theoryand Analytic Function Theory (core)", "indent": false }, { "type": "course", "code": "MATH 260A & MATH 260B & MATH 260C", "title": "Functional Analysisand Functional Analysisand Functional Analysis", "indent": false }, { "type": "course", "code": "MATH 295A & MATH 295B & MATH 295C", "title": "Partial Differential Equationsand Partial Differential Equationsand Partial Differential Equations", "indent": false }, { "type": "course", "code": "MATH 296", "title": "Topics in Partial Differential Equations", "indent": false }, { "type": "comment", "text": "Applied and Computational Mathematics" }, { "type": "course", "code": "MATH 290A & MATH 290B & MATH 290C", "title": "Methods in Applied Mathematicsand Methods in Applied Mathematicsand Methods in Applied Mathematics (core)", "indent": false }, { "type": "course", "code": "MATH 225A & MATH 225B & MATH 225C", "title": "Introduction to Numerical Analysis and Scientific Computingand Introduction to Numerical Analysis and Scientific Computingand Introduction to Numerical Analysis and Scientific Computing", "indent": false }, { "type": "course", "code": "MATH 226A & MATH 226B & MATH 226C", "title": "Computational Differential Equationsand Computational Differential Equationsand Computational Differential Equations", "indent": false }, { "type": "course", "code": "MATH 227A & MATH 227B", "title": "Mathematical and Computational Biologyand Mathematical and Computational Biology", "indent": false }, { "type": "course", "code": "MATH 295A & MATH 295B & MATH 295C", "title": "Partial Differential Equationsand Partial Differential Equationsand Partial Differential Equations", "indent": false }, { "type": "comment", "text": "Geometry and Topology" }, { "type": "course", "code": "MATH 218A & MATH 218B & MATH 218C", "title": "Introduction to Manifolds and Geometryand Introduction to Manifolds and Geometryand Introduction to Manifolds and Geometry (core)", "indent": false }, { "type": "course", "code": "MATH 222A", "title": "Topics in Several Complex Variables and Complex Geometry", "indent": false }, { "type": "course", "code": "MATH 240A & MATH 240B & MATH 240C", "title": "Differential Geometryand Differential Geometryand Differential Geometry", "indent": false }, { "type": "course", "code": "MATH 245A & MATH 245C & MATH 245C", "title": "Topics in Geometric Analysisand Topics in Geometric Analysisand Topics in Geometric Analysis", "indent": false }, { "type": "course", "code": "MATH 250A & MATH 250B & MATH 250C", "title": "Algebraic Topologyand Algebraic Topologyand Algebraic Topology", "indent": false }, { "type": "comment", "text": "Logic" }, { "type": "course", "code": "MATH 280A & MATH 280B & MATH 280C", "title": "Mathematical Logicand Mathematical Logicand Mathematical Logic (core)", "indent": false }, { "type": "course", "code": "MATH 281A & MATH 281B & MATH 281C", "title": "Set Theoryand Set Theoryand Set Theory", "indent": false }, { "type": "course", "code": "MATH 282A & MATH 282B & MATH 282C", "title": "Model Theoryand Model Theoryand Model Theory", "indent": false }, { "type": "comment", "text": "Probability" }, { "type": "course", "code": "MATH 210A & MATH 210B & MATH 210C", "title": "Real Analysisand Real Analysisand Real Analysis", "indent": false }, { "type": "course", "code": "MATH 270A & MATH 270B & MATH 270C", "title": "Probabilityand Probabilityand Probability (core)", "indent": false }, { "type": "course", "code": "MATH 271A & MATH 271B & MATH 271C", "title": "Stochastic Processesand Stochastic Processesand Stochastic Processes (core)", "indent": false }, { "type": "course", "code": "MATH 274", "title": "Topics in Probability", "indent": false } ] } ] }