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:

Core Courses (15 credits)

Course Title Crs.
MATH 6201Modern Algebra3
MATH 6261Real Variables I3
MATH 6150Linear Algebra3
MATH 6202 Modern Algebra II 3
MATH 6301Functions of a Complex Variable.3

Pure Mathematics Specialization (45 credits)

Course Title Crs.
MATH 6540Introduction to Topology3
MATH 6271Mathematical Analysis I3
MATH 6262 Functions of Real Variables II 3
MATH 6551Algebraic Topology I 3
MATH 6272Mathematical Analysis II3
MATH 8309Complex Analysis II3
MATH 6460Functional Analysis I3
MATH 8469Functional Analysis II3
MATH 8465Spectral Theory and D.E 3
Courses in Specialization Area 12
Elective Courses 6

Discrete Mathematics Specialization (45 credits)

Course Title Crs.
MATH 8001Graph Theory3
MATH 6881 Linear Programming3
MATH 8005 Enumerative Combinatorics3
MATH 8015Discrete Algorithms 3
MATH 8051 Convex Polytopes3
MATH 8021 Algebraic Combinatorics3
MATH 8031 Combinatorial Optimization3
MATH 6656Applied Algebra I3
MATH 8041Matroid Theory 3
Courses in Specialization Area 12
Elective Courses 6

Computational Mathematics Specialization (45 credits)

Course Title Crs.
MATH 6601 Probability and Statistics I3
MATH 6271 Mathematical Analysis I3
MATH 6681 Data Structures I3
MATH 6602Probability and Statistics II 3
MATH 6272 Mathematical Analysis II3
MATH 6682 Data Structures II3
MATH 6680 Computational Analysis I3
MATH 6690 Computational Analysis II3
Courses in Specialization Area 15
Elective Courses 6

Seminarios (6 Créditos)

Course Title Crs.
Seminars6
Doctoral Dissertation3

Ph.D. Total Credits: 69

Qualifying Exams

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

MA Degree - Syllabi and Old Exams

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:
  1. Demonstrated a competent knowledge of Spanish and English. A reading knowledge of French, German, or Russian is strongly recommended.
  2. Maintained a B average or better in formal course work, with grades of at least B in the courses related to the chosen option.
  3. Passed the written qualifying examinations at the doctoral level, as described in Section V of the Departmental Rules.
  4. Passed the oral examination for advancement to the Ph.D. candidacy.
  5. Satisfied the specific requirements of the option committee governing the field of principal interest.
  6. 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.