# Mathematics, MS

## Requirements

The Master of Science in Mathematics program consists of 36 hours of graduate coursework, from the three areas specified below. In accordance with University policy, a GPA of at least 2.75 must be maintained with no more than two passing grades of "C+" or lower being utilized in satisfying the degree requirement.

### Core Coursework (15 hours)

Coursework for the MS degree must include the following five core courses:

MATH 50253 | Abstract Algebra I | 3 |

MATH 50403 | Complex Analysis | 3 |

MATH 50503 | Real Analysis I | 3 |

MATH 60513 | Multivariable Analysis | 3 |

MATH 60223 | Applied Linear Algebra | 3 |

An advanced student may be waived from any or all of these required courses by passing the corresponding Ph.D. preliminary exam(s) or otherwise showing proficiency in the material, as determined by the department.

### Specialized Coursework (12 hours)

In addition to the five core courses, at least four courses are to be chosen from either the Pure Mathematics Option or the Applied Mathematics Option (refer to course listings below).

#### Pure Mathematics Option:

This track of the MS Program is intended to prepare students for careers in academia. Each student selecting this track will take at least four of the pure mathematics courses listed below:

MATH 50323 | Differential Geometry | 3 |

MATH 50703 | Number Theory | 3 |

MATH 60263 | Abstract Algebra II | 3 |

MATH 60313 | Topology | 3 |

MATH 60323 | Algebraic Topology I | 3 |

MATH 60413 | Advanced Complex Analysis | 3 |

MATH 60503 | Real Analysis II | 3 |

MATH 60523 | Measure Theory | 3 |

MATH | Any MATH 70000 level 3 hour course | 3 |

#### Applied Mathematics Option:

This track of the MS Program is intended for students planning to use mathematics in careers outside academia. Each student selecting this track will take at least four applied mathematics courses from the list below:

MATH 50613 | Partial Differential Equations | 3 |

MATH 50623 | Applied Mathematics I | 3 |

MATH 60103 | Graph Theory | 3 |

MATH 60543 | Numerical Analysis | 3 |

MATH 60553 | Modern Fourier Analysis | 3 |

MATH 60603 | Game Theory | 3 |

MATH 60613 | Differential Equations of Mathematical Physics | 3 |

MATH 60633 | Applied Mathematics II | 3 |

MATH 60643 | Dynamical Systems and Applications | 3 |

Up to 6 hours of the applied mathematics courses may be substituted with graduate coursework taken in the departments of Biology, Chemistry, Computer Science or Physics and Astronomy, or from the School of Geology, Energy & the Environment, with approval from the student's graduate advisor in the Department of Mathematics.

### Master's Thesis or Electives (9 hours)

The student may choose either to write a master's thesis (three hours of MATH 70980 and three hours of MATH 70990) and complete three hours of approved elective coursework, or to complete nine hours of approved elective coursework. Based on the recommendation of the department, the dean appoints a Master's Advisory Committee of at least three members, including the thesis advisor as chair. For the student to be eligible for the degree, the Master's Advisory Committee must approve the thesis upon its completion. The thesis need not contain original research but must demonstrate a deep and thorough understanding of some area of mathematics.

## Mathematics, Ph.D.

### Requirements

#### Core Coursework (27 hours):

Coursework for the Ph.D. must include the following nine core courses:

MATH 50253 | Abstract Algebra I | 3 |

MATH 60263 | Abstract Algebra II | 3 |

MATH 50503 | Real Analysis I | 3 |

MATH 60513 | Multivariable Analysis | 3 |

MATH 60223 | Applied Linear Algebra | 3 |

MATH 60313 | Topology | 3 |

MATH 60323 | Algebraic Topology I | 3 |

MATH 60413 | Advanced Complex Analysis | 3 |

MATH 60503 | Real Analysis II | 3 |

Any or all of these courses may be waived for more advanced students by departmental permission.

**Preliminary Examinations**

The program requires substantial training in algebra, real analysis, topology, and complex analysis. The student must pass three of the following four preliminary written exams:

- Real Analysis Exam (based on MATH 50503 and MATH 60513)
- Algebra Exam (based on MATH 30224, MATH 50253 and MATH 60263)
- Topology Exam (based on MATH 60313 and MATH 60323)
- Complex Analysis Exam (based on MATH 50403 and MATH 60413)

The student must pass the Real Analysis Exam, the Algebra Exam, and either the Topology Exam or the Complex Analysis Exam. These exams are administered twice each year and must be passed by the end of the sixth semester.

**Research-Specific Coursework**

After passing the preliminary examinations, the student decides on his/her direction of research and dissertation advisor. Based on the recommendation of the department, the dean appoints a Ph.D. Advisory Committee of at least four members, including the dissertation advisor as chair.

Possible areas of research specialization include real analysis, complex analysis, functional analysis, algebraic geometry, differential geometry, number theory, topology, global analysis and K-theory.

Students must also take at least one semester of three hours of MATH 80880 Graduate Student Seminar, providing training in the oral presentation of research-level mathematics.

**Qualifying Examination**

The student and advisory committee agree on a detailed plan of study to prepare the student for mathematical research. They decide on a syllabus of qualifying topics; after due preparation, the student takes the oral qualifying exam on these topics, administered by the advisory committee. In accordance with University requirements, the exam can only be taken after passing the preliminary examinations and not before the second semester of the second year. If the exam is not passed, at most one re-examination is allowed.

**Admission to Candidacy**

The student advances to candidacy after passing the qualifying examination.

**Research**

The student performs research in his/her area of specialization, which leads to a dissertation, if successful.

**Dissertation (at least 12 hours)**

Admission to candidacy is the prerequisite to enrollment in dissertation research, consisting of an original research project directed by a graduate faculty member at TCU. 6 hours of MATH 90980 Dissertation and 6 hours of MATH 90990 Dissertation are required. According to University policy, the time allowed to complete the dissertation is at most six years after advancement to candidacy. Also in accordance with University rules, the student must submit an Intent to Graduate form at the beginning of the last semester, for which there is a non-refundable fee. For the student to be eligible for the degree, the Ph.D. Advisory Committee must approve the dissertation upon its completion. A final oral examination is required and is open to the public.