# Master of Science in Mathematics

## ABOUT THE PROGRAM

### General Objective

The Master of Science in Mathematics (MS MATH) Program centers around achieving a balance between pure mathematics and an area of application. It aims to develop the student’s understanding of the content and the methodology of basic mathematical disciplines to prepare him/her for doctoral studies, for research, for careers in industry and government and for teaching junior and senior level undergraduate mathematics courses. Its graduate curriculum is designed to allow the student an in-depth study of standard graduate courses in Analysis and Algebra, after which a choice of electives lead the student to a specialization area of mathematics as a preparation for research.

### HISTORY

In 1991, the then U.P. College Baguio started offering the MA (Mathematics Education) degree program primarily to address the need to upgrade the quality of Mathematics teaching in tertiary and secondary levels. Between 1991 and 2001, the then Discipline of Mathematics has strengthen its faculty profile with three Ph.D. Mathematics degree holders. This consequently improved the discipline’s research capability, degree offerings, and services through training programs for Mathematics teachers in Northern Luzon. In 1997, it was identified by the Commission on Higher Education as a CHED Center of Development in Mathematics, and is the only institution north of Manila that was given this distinction.

In May 2001, the MA (Mathematics Education) program was abolished and the Master of Science in Mathematics (MS in Mathematics) was instituted. The following month, the discipline started implementing the MS in Mathematics degree program. The first batch of graduates of the MS in Mathematics program eventually pursued and finished Ph.D. in Mathematics degrees from universities in Canada, Europe, and Japan.

## COURSE STRUCTURE FOR PROGRAM OPTIONS

### Thesis option

Required Core Courses | 12 units |

Electives Courses | 12 units |

Masters Thesis | 6 units |

Total Number of Units | 30 units |

### Non-Thesis

Required Core Courses | 12 units |

Elective Courses | 21 units |

Comprehensive Examination | Passed |

Total Number of Units | 33 units |

### 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 (12 UNITS)

The master’s program in Mathematics will involve Analysis and Algebra courses. The four required courses (12 units in total) are the usual courses that would provide the theoretical foundation for the graduate students to do research and is a basis for subsequent mathematical studies. Furthermore, the required courses develop the student’s ability to construct rigorous and accurate arguments via proofs.

#### 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

### ELECTIVE COURSES

(Thesis Option: 12u. / Non-Thesis option: 21u.) The electives are chosen by the student to develop their mathematical skills, critical thinking, and to set an advanced training in Mathematics and its applications.

#### 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 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 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 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

#### MATH 250 – MODERN GEOMETRY

Finite geometries, geometric transformations, Non-Euclidean geometry and projective geometry. Various applications will be considered throughout the

course. 3 units

#### 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 298 – SPECIAL TOPICS

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

#### MATH 300 – MASTER’S THESIS

6 units

## PROGRAM CONTENT AND STRUCTURE

## Non-Thesis Option

## First Year

## Second Year

## Third Year

### First Semester

- MATH 221
- MATH 222
- MATH 232

### Second Semester

- MATH 234
- MATH 2__ (Elective)

### First Semester

- MATH 2__ (Elective)
- MATH 2__ (Elective)
- MATH 2__ (Elective)

### Second Semester

- MATH 2__ (Elective)
- MATH 2__ (Elective)
- MATH 2__ (Elective)

**Comprehensive Examination**

## Thesis Option

## First Year

## Second Year

### First Semester

- MATH 221
- MATH 222
- MATH 232

### Second Semester

- MATH 234
- MATH 2__ (Elective)
- MATH 2__ (Elective)

### First Semester

- MATH 2__ (Elective)
- MATH 2__ (Elective)
- MATH 300 Thesis*

### Second Semester

## 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