Doctorado en Filosofía en Matemáticas

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 level of scholarship shown by the ability to do original research and should possess a broad knowledge of the major fields of modern mathematics.


Information concerning the degree of Ph.D. can be found in the Graduate Program Rules. The following is a brief summary of the degree requirements:

Cursos Medulares (15 créditos)

Course Title CrCréditos.
MATH 6201Álgebra Moderna I3
MATH 6261Funciones de Variables Reales I3
MATH 6150Álgebra Lineal3
MATH 6202Álgebra Moderna II 3
MATH 6301Funciones de una Variable Compleja I3

Opción en Matemáticas Puras (45 créditos)

Course Title Créditos.
MATH 6540Introducción a la Topología3
MATH 6271Análisis Matemático I3
MATH 6262 Funciones de Variables Reales II 3
MATH 6551Topología Algebraica 3
MATH 6272Análisis Matemático II3
MATH 8309Análisis Complejo II3
MATH 6460 Análisis Funcional I3
MATH 8469 Análisis Funcional II3
MATH 8465Teoría Espectral y Ecuaciones Diferenciales 3
Cursos de Especialización 12
Cursos Electivos 6

Opción en Matemáticas Discretas (45 créditos)

Course Title Crs.
MATH 8001Teoría de Grafos3
MATH 6881 Programación Lineal3
MATH 8005 Combinatoría Enumerativa3
MATH 8015Algoritmos Discretos 3
MATH 8051 Politopos Convexos3
MATH 8021 Algebraic Combinatorics3
MATH 8031 Optimización Combinatoria3
MATH 6656Álgebra Aplicada I 3
MATH 8041Teoría de Matroides 3
Cursos de Especialización 12
Cursos Electivos 6

Opción en Matemáticas Computacionales (45 créditos)

Course Title Crs.
MATH 6601 Probabilidad y Estadística I3
MATH 6271 Análisis Matemático I3
MATH 6681 Estructura de Datos I3
MATH 6602Probability and Statistics II 3
MATH 6272 Análisis Matemático II3
MATH 6682 Estructura de Datos II3
MATH 6680 Análisis Computacional I3
MATH 6690 Análisis Computacional II3
Cursos de Especialización 15
Cursos Electivos 6

Seminarios (6 Créditos)

Course Title Crs.
Disertación Doctoral3

Ph.D Total de Créditos: 69

Qualifying Exams

Ph.D. students must take three qualifying 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

Syllabi y Exámenes viejos

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 competence in 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.
  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.