## Degree of Ph.D. in Mathematics

The Purpose of the doctoral program is to prepare specialists capable of doing research and creative, independent, original work in the fields of mathematics represented by the three options. To obtain the Ph.D. degree in mathematics, a student must display a high levei of scholarship shown by the ability to do original research and should possess a broad knowledge of the major fieids of modern mathematics.

### Requirements

Information concerning the degree of Ph.D. can be found in the Graduate Program Rules (2011). The following is a brief summary of the degree requirements (you can also look at this diagram of the PhD curriculum):

#### Core Courses (15 credits)

Course |
Title |
Crs. |

MATH 6201 | Modern Algebra | 3 |

MATH 6261 | Real Variables I | 3 |

MATH 6150 | Linear Algebra | 3 |

MATH 6202 | Modern Algebra II | 3 |

MATH 6301 | Functions of a Complex Variable. | 3 |

#### Pure Mathematics Specialization (45 credits)

Course |
Title |
Crs. |

MATH 6540 | Introduction to Topology | 3 |

MATH 6271 | Mathematical Analysis I | 3 |

MATH 6262 | Functions of Real Variables II | 3 |

MATH 6551 | Algebraic Topology I | 3 |

MATH 6272 | Mathematical Analysis II | 3 |

MATH 8309 | Complex Analysis II | 3 |

MATH 6460 | Functional Analysis I | 3 |

MATH 8469 | Functional Analysis II | 3 |

MATH 8465 | Spectral Theory and D.E | 3 |

Courses in Specialization Area | 12 | |

Elective Courses | 6 |

#### Discrete Mathematics Specialization (45 credits)

Course |
Title |
Crs. |

MATH 8001 | Graph Theory | 3 |

MATH 6881 | Linear Programming | 3 |

MATH 8005 | Enumerative Combinatorics | 3 |

MATH 8015 | Discrete Algorithms | 3 |

MATH 8051 | Convex Polytopes | 3 |

MATH 8021 | Algebraic Combinatorics | 3 |

MATH 8031 | Combinatorial Optimization | 3 |

MATH 6656 | Applied Algebra I | 3 |

MATH 8041 | Matroid Theory | 3 |

Courses in Specialization Area | 12 | |

Elective Courses | 6 |

#### Computational Mathematics Specialization (45 credits)

Course |
Title |
Crs. |

MATH 6601 | Probability and Statistics I | 3 |

MATH 6271 | Mathematical Analysis I | 3 |

MATH 6681 | Data Structures I | 3 |

MATH 6602 | Probability and Statistics II | 3 |

MATH 6272 | Mathematical Analysis II | 3 |

MATH 6682 | Data Structures II | 3 |

MATH 6680 | Computational Analysis I | 3 |

MATH 6690 | Computational Analysis II | 3 |

Courses in Specialization Area | 15 | |

Elective Courses | 6 |

#### Seminarios (6 Créditos)

Course |
Title |
Crs. |

Seminars | 6 | |

Doctoral Dissertation | 3 |

#### Ph.D. Total Credits: 69

### Qualifying Exams

Students must take written quafifying examinations.

- An exam in Real and Complex Analysis, based on the corresponding core courses of the first year.
- An exam in Modern and Linear Algebra, based on the corresponding core courses of the first year.
- An exam specific to the chosen option in one of the following areas:
- Pure Mathematics Option: Real Analysis and Complex Analysis, Topology, Functional Analysis.
- Discrete Mathematics Option: Graph Theory, Linear Programming, Applied Algebra, Enumerative and Algebraic Combinatorics, Discrete Algorithms, Combinatorial Optimization, Convex Polytopes, Matroid Theory.
- Computational Mathematics Option: Mathematical Analysis, Probability and Statistics, Data Structures and Algorithms, Computational Analysis

#### MA Degree – Syllabi

Detailed syllabi may be requested in the Office of Graduate Studies of the Department of Mathematics.

### Admission to Candidacy

Before petitioning to admission to candidacy, a student must have:

- Demonstrated a competent knowledge of Spanish and English. A reading knowledge of French, German, or Russian is strongly recommended.
- Maintained a
*B*average or better in formal course work, with grades of at least*B*in the courses related to the chosen option. - Passed the written qualifying examinations at the doctoral level, as described in Section V of the Departmental Rules.
- Passed the oral examination for advancement to the Ph.D. candidacy.
- Satisfied the specific requirements of the option committee governing the field of principal interest.
- Obtained the consent of a faculty member who will accept the responsibility 0€ directing a dissertation.

### The Ph.D. Dissertation

The dissertation prepared must represent a substantial, original, and independent contribution of the student to the existing mathematical knowledge. It must be defended in an oral examination, by a committee consisting of five members, at least one of whom must belong to the graduate faculty of the department of mathematics of another university. All members of the committee must be specialists in the field of the dissertation.