**Mathematics**

## Department Information

**Department of Mathematics**

**College of Science**

**Office: North Science 335**

**Phone: (510) 885-3414**

**E-mail:** **mathcs@csueastbay.edu**

**Website:** **http://www20.csueastbay.edu/csci/departments/math-cs/index.html **

**Student Service Center: North Science 337**

**Phone: (510) 885-4011**

*Professor Emeritus*

Stuart Smith, Ph.D. University of California, Berkeley

*Professors*

Kevin E. Callahan, Ph.D. University of California, San Diego

Julie S. Glass, Ph.D. University of California, Santa Cruz

Kathleen Hann, Ph.D. University of California, Davis

Gary E. Lippman, Ph.D. University of California, Riverside

Massoud Malek, Ph.D. University of Houston

Donald L. Wolitzer, Ph.D. Northeastern University

*Associate Professors*

Julia Olkin, Ph.D. Rice University

Chung-Hsing OuYang, Ph.D. University of California, Berkeley

Shirley Yap, Ph.D. University of Pennsylvania

*Assistant Professors*

Ehsan Kamalinejad, Ph.D. University of Toronto (Canada)

*Graduate Coordinator*: Donald L. Wolitzer

## Program Description

The Mathematics and Computer Science Department offers graduate study leading to the degree of Master of Science in Mathematics. The goal of the faculty is to provide excellent instruction in advanced mathematics and to maintain a supportive environment for graduate students. Students who complete the program should be equipped for careers in community college teaching or positions in industry that require knowledge of mathematics beyond the undergraduate level. The M.S. degree in Mathematics can also serve as preparation for advanced study toward a Ph.D. degree in mathematics or a related field.

Our program features small classes that allow for close contact between students and faculty. Many graduate classes are offered in the late afternoon or early evening, making it possible for working students to attend. Courses toward the M.S. degree may also be taken during the summer quarter. Students may begin their studies in any one of the four quarters.

Students interested in the M.S. degree program in Mathematics should speak with the Mathematics Graduate Coordinator.

### Student Learning Outcomes

Students who graduate with an M.S. in Mathematics will be able to:

- apply the fundamental definitions and theorems of pure mathematics;
- apply the fundamental definitions and theorems of applied mathematics;
- apply advanced techniques of mathematical analysis;
- apply techniques of advanced algebra;
- apply advanced techniques of geometry and topology;
- use mathematical algorithms.

### Career Opportunities

A number of former Cal State East Bay students currently hold positions as community college mathematics teachers. Others have found the M.S. degree in mathematics to be an ideal preparation for further studies at doctorate-granting institutions and have continued by working towards a Ph.D. degree in mathematics or a related field such as operations research, physics, or economics. A number of these alumni are now professors at four-year institutions. Still others are in mathematics-related careers in industry.

### Faculty

The faculty of the Mathematics and Computer Science Department hold doctorates in a wide variety of areas of specialization and offer courses encompassing a broad range of pure and applied mathematics, including standard graduate mathematics courses as well as courses in new areas. Areas of emphasis include numerical analysis, pure and applied algebra, differential equations, real and complex analysis, topology, geometry, mathematical optimization, computer simulation, probability, statistics, and selected topics in applied mathematics.

### Special Features

Each quarter, a limited number of teaching positions are available to qualified graduate students. These positions, which generally involve teaching one lower division mathematics course per quarter, provide valuable experience, especially for those who intend to become community college teachers. The department also employs qualified students as paper graders.

Mathematics students have access to modern computer equipment, including various mathematical software packages.

The CSUEB Mathematics Club is open to all interested students. This club features lectures by students and faculty and offers a variety of social activities.

### Scholarships

Each year the department awards a number of scholarships for the subsequent year. Scholarship applications may be obtained from the department office during the winter quarter.

## Options

There are three options available. Option I emphasizes coursework drawn from fundamental branches of mathematics: algebra, topology, and real and complex analysis. Option II, Mathematics Teaching, is intended for those who hold secondary teaching credentials and who intend to pursue a career in secondary education. Option III, Applied Mathematics, is designed to expose students to various aspects of applied mathematics, while allowing some coursework in "pure" mathematics as well. Students who intend to become community college teachers or go on to further graduate study should select Option I or Option III.

### Option I (Pure Mathematics)

#### Admission

To enter the program with "Classified Graduate" status, a student must have completed at least 36 quarter units of acceptable upper division mathematics with a grade point average of "B" or higher. Included among these units must be courses in:

- Analysis
- Abstract algebra
- Linear algebra theory
- Differential equations

A student may be admitted to the program with "Conditionally Classified Graduate" status while making up course or grade point deficiencies. Units taken to meet any course deficiencies may not be applied toward the master's degree, and no more than 20 quarter units taken while in "Conditionally Classified Graduate" status may be applied to the degree.

A "Conditionally Classified Graduate" student who has no course deficiencies, a "B" or higher average in at least 12 quarter units of postbaccalaureate study, and has satisfied the University Writing Skills Requirement, should petition the graduate coordinator for admission to the master's degree program with "Classified Graduate" status.

#### Advancement to Candidacy

A student with "Classified Graduate" status may apply for Advancement to Candidacy after completing at least 16 quarter units toward the master's degree with a "B" or higher average, including at least two 6000-level mathematics courses with a "B" or higher average. Before being Advanced to Candidacy, a student's complete course of graduate study must be approved by the Mathematics Graduate Studies Committee.

#### Degree Requirements

The following departmental requirements must be satisfied:

- The following four courses (or their equivalents) must be completed, either as an undergraduate or as a graduate student:
- MATH 4121 Advanced Algebra (4)
- MATH 4340 Introduction to Complex Variables (4)
- MATH 4350 Theory of Functions of a Real Variable (4)
- MATH 4360 Introduction to Topology (4)

- The 45 quarter units applied to the degree must include:
- At least 24 quarter units of 6000-level courses, of which at least 20 quarter units are mathematics courses. Credit will be given for the seven M.A.T.H. courses (MATH 6015-6065, and 6899), only with the permission of the Mathematics Graduate Committee.
- At least two of the following four courses:
- MATH 6121 Topics in Advanced Algebra I (4)
- MATH 6201 Topology (4)
- MATH 6340 Complex Analysis (4)
- MATH 6350 Real Analysis (4)

- A comprehensive examination must be passed. Details are available in the department office and on the department website

In addition to departmental requirements, every student must also satisfy the university requirements for graduation which are described in the Graduate Degree Information chapter at the beginning of the graduate section of this catalog. These requirements include the 32-unit residence requirement, the five-year rule on currency of subject matter, the minimum number of units of 6000-level courses, the 3.00 GPA, and the University Writing Skills Requirement. For information on meeting the University Writing Skills Requirement, see the Testing Office website at www.csueastbay.edu/testing or call 510.885.3661.

### Option II (Mathematics Teaching)

#### Admission

The M.A.T.H. (Mathematics and Teaching at Hayward) option is available only to holders of teaching credentials, unless special permission is obtained. In order to be admitted to the master's degree program with "Classified Graduate" status, a student must have completed 24 or more quarter units of acceptable upper division mathematics with an average of "B" or higher. A student may be admitted to the program with "Conditionally Classified Graduate" status while making up course or grade point deficiencies. Units taken to meet any course deficiencies may not be applied toward the master's degree, and no more than 20 quarter units taken while in "Conditionally Classified Graduate" status may be applied to the degree. A "Conditionally Classified Graduate" student who has no course deficiencies, a "B" or higher average in at least 12 quarter units of post-baccalaureate study, and has satisfied the University Writing Skills requirement, should petition the graduate coordinator for admission to the master's degree program with "Classified Graduate" status.

#### Advancement to Candidacy

A student with "Classified Graduate" status may apply for Advancement to Candidacy after completing at least 16 quarter units of work toward the master's degree with a "B" or higher average.

Before being Advanced to Candidacy, a student's complete course of study must be approved by the Mathematics Graduate Studies Committee.

#### Degree Requirements

The following departmental requirements for the M.S. degree are in addition to the general University requirements:

- Six M.A.T.H core courses (24 units)
- MATH 6015 Algebra for Teachers (4)
- MATH 6025 Geometry for Teachers (4)
- MATH 6035 Analysis for Teachers (4)
- MATH 6045 Mathematics in the Sciences (4)
- MATH 6055 Discrete Mathematics (4)
- MATH 6065 Connections in Mathematics (4)

- Two Teacher Education courses selected from the following (8 units):
- TED 6010 Seminar in Teaching and Learning Mathematics (4)
- TED 6021 Seminar in Diagnosis and Treatment of Learning Difficulties in Mathematics (4)
- TED 6030 Seminar on Problem Solving and Critical Thinking in Mathematics (4)
- TED 6040 Advanced Curriculum and Instruction in Mathematics (4)

- An upper division or graduate-level course offered by the Statistics Department and approved by the Math Graduate Coordinator (4 units)
- One or two upper division or graduate electives approved by the Math Graduate Coordinator (4-8 units)
- MATH 6899 Project (1-5 units)

In addition to departmental requirements, every student must also satisfy the university requirements for graduation which are described in the Graduate Degree Information chapter in this catalog. These requirements include the 32-unit residence requirement, the five-year rule on currency of subject matter, the minimum number of units of 6000-level courses, the 3.00 GPA, and the University Writing Skills Requirement.

### Option III (Applied Mathematics)

#### Admission

To enter the program with "Classified Graduate" status, a student must have completed a course in computer science and at least 36 quarter units of acceptable upper division mathematics, statistics, or computer science courses with a grade point average of "B" or higher. Included among these units must be courses in:

- Analysis
- Abstract algebra
- Linear algebra theory
- Differential equations

A student may enter the program with "Conditionally Classified Graduate" status while making up course or grade point deficiencies. Units taken to meet course deficiencies may not be applied toward the master's degree, and no more than 20 quarter units taken while in "Conditionally Classified Graduate" status may be applied to the degree.

A "Conditionally Classified Graduate" student who has no course deficiencies, a "B" or higher average in at least 12 quarter units of post baccalaureate study, and has satisfied the University Writing Skills Requirement, should petition the department graduate coordinator for a change to "Classified Graduate" status.

#### Advancement to Candidacy

A student with "Classified Graduate" status may apply for Advancement to Candidacy after completing at least 16 quarter units towards the master's degree with a "B" or higher average, including at least two 6000-level mathematics courses with a "B" or higher average.

Before being Advanced to Candidacy, a student's complete course of study must be approved by the Mathematics Graduate Studies Committee. In particular, approval must be obtained for any course(s) taken outside the Mathematics and Computer Science Department.

#### Degree Requirements

The following departmental requirements must be satisfied:

- The following four courses (or their equivalents) must be completed, either as an undergraduate or as a graduate student:
- MATH 3301 Analysis II (4)
- MATH 3401 Introduction to Probability Theory I (4)
- MATH 3750 Numerical Analysis I (4)
- MATH 3841 Linear Programming (4)

- The 45 quarter units applied to the degree must include:
- At least 22 1/2 quarter units of 6000 level courses of which at least 18 are mathematics courses. Credit will be given for the seven MATH courses (MATH 6015-6065, and 6899), only with the permission of the Mathematics Graduate Committee.
- At least two of the following five courses:
- MATH 6100 Applied Algebra (4)
- MATH 6331 Topics in Differential Equations (4)
- MATH 6401 Advanced Probability I (4)
- MATH 6750 Topics in Advanced Numerical Analysis (4)
- MATH 6870 Computer Simulation (4)

- A comprehensive examination must be passed. Details are available in the department office.

In addition to departmental requirements, every student must also satisfy the university requirements for graduation which are described in the Graduate Degree Information chapter at the beginning of the graduate section of this catalog. These requirements include the 32-unit residence requirement, the five-year rule on currency of subject matter, the minimum number of units of 6000-level courses, the 3.00 GPA, and the University Writing Skills requirement.

### Upper Division Mathematics, Computer Science, and Statistics Courses Acceptable for M.S. in Mathematics

- Upper division and graduate computer science courses may be used with the approval of the Mathematics Graduate Studies Committee.
- Other upper division mathematics courses may be used with the approval of the Mathematics Graduate Studies Committee. MATH 4012, 4013, 4014, or 4030 will not be approved.
- STAT 3401, 3402 Introduction to Probability Theory I, II (4 each), 3502, 3503 Statistical Inference I, II (4 each), 4401 Introduction to Stochastic Processes (4)

## Graduate Courses

Course Number | Course Information |
---|---|

6000 |
Advanced Topics via a Research Paper (4)Introduction to the skills involved and the topics required in reading a mathematical research paper, chosen by the instructor. Prerequisite: At least 16 units of 3000-level, or higher, mathematics courses, or instructor permission. May be repeated twice for credit with consent of department and when content varies, for a maximum of 8 units. |

6005 |
Teaching Mathematics at the University Level (1)Theory, methodology, and practical experience in the teaching of mathematics at the university level. Includes discussion of lecturing techniques, analysis of tests and supporting material, preparation and grading of examinations, and related topics. Required of departmental teaching associates. Prerequisites: graduate standing and permission of department. May be repeated for credit, but only two units can be used toward the M.S. degree. |

6100 |
Applied Algebra (4)A survey course covering significant areas of applied algebra. Topics might include applied matrix theory, game theory, convexity and inequalities, and/or algebraic coding theory. Prerequisite: MATH 3100 or equivalent. May be repeated once for credit with consent of Mathematics Graduate Studies Committee, for a maximum of 8 units. |

6119 |
Advanced Algebra (4)Theory of groups, including factor groups, Jordan-Holder Theorem, Sylow theorems. Mappings and homomorphisms. Introduction to rings and fields. Students will do independent theoretical work. Topics continued in MATH 6121. Not open to students with credit for Math 4121. Prerequisite: MATH 3122. |

6121 |
Topics in Advanced Algebra I (4)Continuation of MATH 4121. Topics include ideals, commutative rings, modules; field extensions and Galois theory. Prerequisite: MATH 4121. |

6125 |
Introduction to Lie Algebras (4)An introduction to the theory of semisimple Lie algebras. Theorems of Lie, Engel, and Weyl; Cartan's Criterion; the classification of root systems; and abstract theory of weights. Prerequisite: MATH 3100 or consent of instructor. |

6151 |
Advanced Topics in Graph Theory (4)Advanced course in graph theory. Connectivity, planarity, and graph coloring. Advanced topics which may include substructures in graphs and Ramsey Theory, Random Graphs, Spectral Graph Theory. Prerequisites: MATH 3100 and graduate standing. May be repeated once for credit with consent of instructor and when content varies, for a maximum of 8 units. A-F grading only. |

6200 |
Introduction to Topology (4)Topological spaces, metric spaces, continuity, connectedness and compactness. Independent theoretical work. Not open to students with credit for MATH 4360. Prerequisite: MATH 3300. |

6201 |
Topology (4)Continuation of MATH 4360/6200. Topics may include countability and separation axioms, Tychonoff theorem, metrization theorems, homotopy theory. Prerequisite: MATH 4360 or MATH 6200. |

6210 |
Convex Polytopes/Combinatorial Geometry (4)Convex sets including convex hulls, supporting hyperplanes and duality. Convex polytopes, including simple, simplicial, and cyclic polytopes. Combinatorial theory, including Euler's Relations, Dehn-Somerville Relations and Upper Bound Theorem. Prerequisites: MATH 3100 and 3300 or consent of instructor. |

6235 |
Introduction to Knot Theory (4)Introduction to the theory of knots and links. Reidemeister moves, knot invariants, including 3-colorings, linking number, Alexander polynomial, Kauffman bracket and Jones polynomial. Applications in biology and/or chemistry will be discussed, time permitting. Additional work required for graduate level credit. Prerequisite: MATH 3121. |

6250 |
Topics in Differential Geometry and Topology (4)Topics in differential geometry and topology such as manifolds, bundles, differential forms, curvature, theorems of Sard-Smale, Poincaré-Hopf, Gauss-Bonnet, de Rham, and Hodge. Prerequisites: MATH 3100, 3301, or consent of instructor. |

6251 |
Symplectic Geometry (4)Introduction to Symplectic Geometry. Symplectic linear algebra, groups, Lie algebras, and manifolds. Darboux-Weinstein theorem, relation to optics and Hamiltonian dynamics, moment maps, and geometric quantization. Prerequisites: MATH 3100 and 3300, or consent of instructor. A-F grading only. |

6260 |
Computation and Complexity (4)(See CS 6260 for course description.) |

6331 |
Topics in Differential Equations (4)Topics selected from the theory of ordinary and partial differential equations. Prerequisites: MATH 3100, 3331, 3301, or instructor's permission. May be repeated two times for credit with consent of Mathematics Graduate Studies Committee and when content varies, for a maximum of 12 units. |

6339 |
Introduction to Complex Variables (4) Introduction to theory of functions of complex variables. Advanced synthesis of theory and practice. Prerequisite: Math 3300. Not open to students with credit for Math 4340. |

6340 |
Complex Analysis (4)Cauchy integral formula, Mittag-Leffler's theorem, Weierstrass' factorization theorem, normal families, Riemann mapping theorem, and selected topics. Prerequisite: MATH 4340. |

6349 |
Theory of Functions of a Real Variable (4)Pointwise and uniform convergence. Taylor series, Riemann integration, sets of measure zero, Lebesgue's theorem on the Riemann integral, the metric space of continuous functions, and selected topics. Students will do independent theoretical work. Not open to students with credit for MATH 4350. Prerequisite: MATH 3300. |

6350 |
Real Analysis (4)Theory of Lebesgue measure and integration on the real line. Selected topics and applications. Prerequisite: MATH 4350 or MATH 6349. |

6401 |
Advanced Probability I (4)(See STAT 6401 for course description.) |

6501,6502 |
Mathematical Statistics I, II (4, 4)(See STAT 6501, 6502 for course description.) |

6600 |
Advanced Number Theory (4)Topics in number theory such as algebraic number fields, continued fractions, geometry of numbers, theory of partitions, distribution of primes, factoring algorithms and quadratic forms. Prerequisites: MATH 3121 and 3600 or consent of instructor. May be repeated once for credit with consent of the Mathematics/Computer Science department chair, for a maximum of 8 units. |

6750 |
Topics in Advanced Numerical Analysis (4)Topics selected from approximation theory; spline theory; numerical linear algebra; the algebraic eigenvalue problem; numerical solutions to non-linear systems of equations, partial differential equations, and boundary value problems. Prerequisites: MATH 4750 and 3301 or instructor's permission. May be repeated two times for credit with consent of Mathematics Graduate Studies Committee and when content varies, for a maximum of 12 units. Cross-listed with CS 6750. |

6840 |
Advanced Topics in Linear Optimization (4)Topics selected from network algorithms, integer programming, game theory, and other areas related to linear programming. Prerequisite: MATH 3841. May be repeated once for credit with consent of Mathematics Graduate Studies Committee, for a maximum of 8 units. A-F grading only. |

6841 |
Nonlinear Optimization (4)Optimality conditions and solution procedures for unconstrained and constrained optimization problems. Prerequisite: MATH 3841. |

6842 |
Advanced Topics in Optimization (4)Topics selected from quasi-Newton methods for multi-variable unconstrained optimization; nonlinear least squares; quadratic programming; constrained optimization with nonlinear constraints; convex optimization. Prerequisite: Math 3750 and Math 3841, or permission of instructor. May be repeated once for credit with consent of Mathematics Graduate Studies Committee, for a maximum of 8 units. |

6865 |
Mathematical Modeling (4)Discrete and continuous mathematical models. General introduction to the use of difference and differential equations, probability and statistics, and matrices for solving realistic problems. Computer simulation. Emphasis on effective written reports. Additional graduate applications module. Prerequisites: MATH 2101 and MATH 2304. Not open to students with credit for MATH 3865. Cross-listed with STAT 6865. |

6870 |
Computer Simulation (4)(See CS 6870 for course description.) |

6875 |
Topics in Mathematical Physics (4)Advanced mathematics theory and methods with applications to physics. Prerequisite: MATH 1305. Co-requisite: MATH 2304. May be repeated for credit when content varies, for a maximum of 8 units. |

6900 |
Independent Study (1-4) |

6910 |
University Thesis (1-6)Development and writing of a formal research paper for submission to the university in the specified bound format. Supervision by a departmental committee, at least one of whom must be a Cal State East Bay faculty member. Oral defense normally required. (See also, "University Thesis Writing Guide," www.csueastbay.edu/thesiswritingguide.) Prerequisite: graduate standing. Maximum of 6 units per student. |

6935 |
Mathematical Logic (4)Content of MATH 4100 with a Graduate Module. Propositional calculus and its completeness. Boolean algebras. Functional calculi of various orders. Theorems of Godel and Henkin. Prerequisite: Graduate standing. Not open to students with credit for MATH 4100. A-F grading only. |

Course Number | Course Information |
---|---|

6015 |
Algebra for Teachers (4)Polynomials, groups, fields, and rings from an advanced standpoint as they relate to the high school algebra curriculum. Discussion of strategies to help secondary students develop their algebraic thinking skills. Prerequisite: permission of instructor. |

6025 |
Geometry for Teachers (4)Rigorous development of a non-Euclidean geometry, such as spherical, projective, or hyperbolic geometry. Models and technology used where appropriate. Discussion of implementation strategies for teaching geometry and proof techniques for high school students. Prerequisite: permission of instructor. |

6035 |
Analysis for Teachers (4)A rigorous development of calculus. The real line, functions, limits, continuity, differential and integral calculus. Technology used to develop an intuitive understanding of calculus which can be implemented in the high school classroom. Prerequisite: permission of instructor. |

6045 |
Mathematics in the Sciences (4)Mathematics as found throughout the sciences. The mathematics used to model phenomena in biology, chemistry and/or physics. Students discover some of this mathematics through scientific experiments. Prerequisite: permission of instructor. |

6055 |
Discrete Mathematics (4)Topics in discrete mathematics relating to the high school curriculum such as combinatorics, number theory, and graph theory. Prerequisite: permission of instructor. |

6065 |
Connections in Mathematics (4)Topics which illustrate connections between different fields and applications of mathematics such as neural networks, tomography, coding theory, symmetry groups, optimization theory, and applications found in differential equations or complex analysis. Prerequisite: permission of instructor. |

6899 |
Project (1-5)Development of an original product which is summarized in a written abstract. Both the project and the abstract are submitted to the department which specifies their formats. Supervision by a departmental committee, at least one of whom must be a Cal State East Bay faculty member. Oral defense may be required. Prerequisite: graduate status. Maximum of 5 units per student. |

6900 |
Independent Study (1-4) |