# Doctor of Philosophy in Mathematics

## ABOUT THE PROGRAM

### General Objective

Doctor of Philosophy in Mathematics Program is a doctoral level graduate program that enables students to: acquire advanced knowledge in pure and applied mathematics; enhance their research skills to help them produce quality research output; and become prolific mathematicians, and future leaders in academe, industry and professional organizations. The program will allow UP Baguio to contribute to contribute to the urgent need to train and develop more PhD Mathematics graduates who will go on to teach and engage in Mathematics and Math-related research. At the end of the program, graduates of PhD Mathematics are expected to become competent educators and professionals who can conduct advanced and high-quality research, who are able to review researches in their areas of specialization, and who will help improve the quality of programs in their institutions. This will contribute to the upgrading of tertiary and graduate mathematics instruction and research in the Philippines.

## COURSE STRUCTURE FOR PROGRAM OPTIONS

### Regular Ph. D. Mathematics Program

A three-year program designed for students who have earned their Master of Science in Mathematics degree (or in an allied field).

Required Courses | 9 units |

Electives | 18 units |

Graduate Seminar | 1 unit |

Qualifying Examination | - |

Candidacy Examination | - |

Dissertation | 12 units |

Total |
40 |

#### Required Courses for Regular Ph. D. Mathematics (3 units each)

Math 223,237,240

### Straight Ph. D. Mathematics Program

A four-year program designed for students who have earned their Bachelor of Science in Mathematics degree (or in allied field) who meet certain qualifications (consult the Graduate Program Office).

Required Courses | 21 units |

Electives | 24 units |

Graduate Seminar | 1 unit |

Qualifying Examination | - |

Candidacy Examination | - |

Dissertation | 12 units |

Total |
58 |

#### Required Courses for Straight Ph. D. Mathematics (3 units each)

Math 221,222,223,232,237,240

### PROGRAM REQUIREMENTS

- Core Courses
- 27 units
**-**MM 201, 202, 203, 220, 222, 230, 240, 291.1, and 291.2

- 27 units
- Elective Courses
- 9 units - MM 250, 290, 292, 293, 294, 295, 296, 298*, and 299

* ***May be taken for credit twice; student may take a maximum of two different special topics.*

- Comprehensive Examination

### REQUIRED CORE COURSES

#### MATH 221 – ABSTRACT ALEGBRA I

Groups, subgroups, homomorphins, cosets, normality, quotient groups. Special topics include direct products, generators and relations, finitely generated Abelian groups, Sylow theorems, classification of finite groups, nilpotent and solvable series, normal and subnormal series. Applications include frieze groups, wallpaper groups, permutation groups and symmetry groups. 3 units

#### MATH 222 – LINEAR ALEGBRA

Linear equations, vector spaces, linear transformations, matrices, determinants, invariant subspaces, direct sum decompositions, canonicals forms, rational and Jordan forms inner product spaces and bilinear forms. Emphasizes proofs. Applications in various disciplines. 3 units

#### MATH 232 – REAL ANALYSIS

Rigorous introduction to classical real analysis. Brief review of real numbers and a discussion of the topology of metric spaces. Includes detailed discussion of the following topics: the analysis of sequences and series; continuity, differentiation and Taylor ’s theorem; Riemann and Lebesgue integration; measure theory. 3 units

#### MATH 234 – COMPLEX ANALYSIS

Analytic functions, integration, power series, residue and applications, Mittag-Leffler’s theorem, conformal mapping. 3 units Prereq: MATH 232

#### MATH 223 – ABSTRACT ALGEBRA II

Rings and Fields: integral domains, ideal and factor rings, ring homomorphisms, polynomial rings factorization of polynomials, extension fields, and finite fields. Galois Theory: Fundamental theorem of Galois theory, splitting fields, algebraic closure, and normality, Galois group of polynomial, separability, cyclic, cyclotomic and radical extensions. 3 units Prereq: MATH 221

#### MATH 237 – FUNCTIONAL ANALYSIS

Banach spaces, review of Lebesgue integration and Lp spaces. Foundations of linear operator theory, nonlinear operators. The contraction mapping principle. Nonlinear compact operators and monotonicity. The Schauder fixed point theorem. The Spectral Theorem. 3 units Prereq: MATH 234

#### MATH 240 – TOPOLOGICAL STRUCTURES

Metric spaces, topological spaces, bases, continuous function. Product spaces, compactness and completeness, sequences and natural countability, separability and metrization.

Quotients, local compactness, complete metrices, Baire category. Banach Fixed point theorem. Manifolds. 3 units Prereq: COI

### ELECTIVE COURSES

#### MATH 213 – THEORY OF DIFFERENTIAL EQUATIONS

Existence and uniqueness, linear systems of differential equations, non-linear systems; stability theorems, Sturm-Liouville theory, applications. 3 units

#### MATH 214 – DYNAMICAL SYSTEMS

Introduction to dynamical systems and applications, symmetric matrices, matrix norm, eigenvalues, dynamical interpretation, multi-input-output system,impulse and step matrices, chaos. 3 units Prereq: MATH 213

#### MATH 215 – INTRODUCTION TO MATHEMATICAL MODELING

Fundamental concepts of mathematical modeling, differential equation models, optimization models, probabilistic models. 3 units Prereq: MATH 213

#### MATH 216 – APPLIED PARTIAL DIFFERENTIAL EQUATIONS

Parabolic (heat), hyperbolic (wave), and elliptic (steady-state) partial differential equations. Solution techniques are demonstrated, including separation of variables and integral forms. 3 units Prereq: MATH 213

#### MATH 217 – INTEGRAL EQUATIONS

Fredholm integral equations, Volterra integral equations, integro-differential equations, singular integral equations, nonlinear integral equations, integral of irrational functions, series representations, error and gamma functions. 3 units Prereq: COI

#### MATH 218 – INTRODUCTION TO APPLIED MATHEMATICS

Calculus of variations. The Rayleigh-Ritz-Galerkin method and finite elements, Fourier series and orthogonal expansions. Fourier integrals. 3 units Prereq: COI

#### MATH 219 – DELAY DIFFERENTIAL EQUATIONS

#### MATH 224 - MATRIX ANALYSIS

#### MATH 225 – NUMBER THEORY with APPLICATIONS

Theory of congruence, Fermat’s theorem, number theoretic functions, primitive roots and indices, quadratic reciprocity law, perfect numbers, Fermat conjecture. Applications. 3 units Prereq: COI

#### MATH 235 – APPLIED COMPLEX VARIABLE THEORY

Complex function theory, contour integration and residue, integral transform, analytic continuation, applications of complex variables to potential theory, Fourier and Laplace transforms, ordinary and partial differential equations, number theory, chaotic dynamical systems, etc. 3 units Prereq: MATH 234

#### MATH 236 – NUMERICAL ANALYSIS

Construction, analysis and implementation of numerical algorithms. Solution of nonlinear equations, linear system and differential equations. Interpolations and functional approximation. Numerical differentiation, numerical integration, numerical optimization. Eigenvalues and eigenvectors. 3units Prereq: COI

#### MATH 238 – SEMIGROUP THEORY AND APPLICATIONS

#### MATH 239 – NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS

#### MATH 241 – ALGEBRAIC TOPOLOGY

#### MATH 245 – COMPUTATIONAL TOPOLOGY WITH APPLICATIONS

#### MATH 250 – MODERN GEOMETRY

#### MATH 251 – DIFFERENTIAL GEOMETRY

#### MATH 255 – APPLIED COMBINATORICS

Inclusion and exclusion principle, generating functions, recurrence relations, cycles and trees, optimization and matching, switching functions, and coding theory. 3 units

#### MATH 256 – GRAPH THEORY

Elements of Graph Theory, covering circuits and graph coloring, trees and searching, network algorithnms, Polya’s enumeration formula, games with graph. 3 units Prereq:MATH 255

#### MATH 260 – PROBABILITY AND APPLICATIONS

Probability spaces and random variables, probability distributions and distribution functions, mathematical expectation, elementary distribution theory, random sampling, estimation, experimental design. 3 units

#### MATH 280 – LINEAR AND NONLINEAR OPTIMIZATION

#### MATH 296 – SELECTED TOPICS IN APPLIED ANALYSIS

#### MATH 298 – SPECIAL TOPICS

Selected topics of current interest (may be taken twice). 3 units Prereq: COI

## Faculty Information

**Addawe, Joel M.**

Ph.D. Mathematics

University of the Philippines, 2012

**Alangui, Wilfredo V.**

Ph.D. Mathematics Education

University of Auckland, 2010

**Bacani, Jerico B.**

Dr.rer.nat. Mathematik

Karl-Franzens-Universitaet Graz, 2013

**Collera, Juancho A.**

Ph.D. Mathematics

Queen's University, 2012

**Gueco, Edna N.**

Ph.D. Mathematics

University of the Philippines, 2012

**Macansantos, Priscilla S.**

Ph.D. Mathematics

University of Delaware. 1996

**Peralta, Gilbert R.**

Dr.rer.nat. Mathematik

Karl-Franzens-Universitaet Graz, 2014