CSUH Catalog 2004-2005: Mathematics (Graduate)

Mathematics

	Department Information
	Program Description
	Options
	Graduate Courses

Department Information

Department of Mathematics and Computer Science
College of Science
Office: North Science 335
Phone: (510) 885-3414


	E-mail: mathcs@csuhayward.edu http://www.mcs.csuhayward.edu

Student Service Center: North Science 337
Phone: (510) 885-4011

Professor Emeritus: Charles M. Marut

Professors: James S. Daley, Kathleen Hann, Edward L. Keller (Associate Chair), Gary E. Lippman, Massoud Malek, Russell L. Merris, Christopher L. Morgan, William R. Nico (Computer Science Coordinator), Edna E. Reiter (Chair), Istvan Simon, Stuart Smith, William Thibault, Bruce E. Trumbo, Donald L. Wolitzer, Ytha Y. Yu

Associate Professors: Jagdish Bansiya, Edward A. Billard, Kevin A. Brown, Kevin E. Callahan, Julie S. Glass, Dan Jurca, Thomas W. Roby, Farzan Roohparvar, Assim Sagahyroon

Assistant Professors: Leann Christianson, Roger W. Doering, Levent Ertaul, Madhavi Gandhi, Lynne L. Grewe, Hilary J. Holz, C. Matthew Johnson, Chung-Hsing OuYang, David Yang

Lecturers: Paula Albert, Jack A. Carter, Francis Conlan, Michael A. Contino, Dorothy E. Fujimura, Phil Gonsalves, Ching-Cheng Lee, David L. Pugno, Denise Sargent-Natour, Jean Simutis

Graduate Coordinator: Donald L. Wolitzer

Please consult the 2005-2006 online catalog for any changes that may occur.

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. Most 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.

Career Opportunities
A number of former Cal State Hayward 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 is 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 CSUH 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 covering a large portion of the fees 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

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
	•	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:


A.	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)
B.	The 45 quarter units applied to the degree must include:
	1.	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.
	2.	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)
C.	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 and Post-baccalaureate Studies 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.

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:


	A.	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)
	B.	Two Teacher Education courses selected from the following (8 units):
		T ED 6010 Seminar in Teaching and Learning Mathematics (4) T ED 6021 Seminar in Diagnosis and Treatment of Learning Difficulties in Mathematics (4) T ED 6030 Seminar on Problem Solving and Critical Thinking in Mathematics (4) T ED 6040 Advanced Curriculum and Instruction in Mathematics (4)
	C.	An upper division or graduate-level course offered by the Statistics Department and approved by the Math Graduate Coordinator (4 units)
	D.	One or two upper division or graduate electives approved by the Math Graduate Coordinator (4-8 units)
	E.	MATH 6899 Project (1-5 units)



	•	Analysis
	•	Abstract algebra
	•	Linear algebra
	•	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:



A.	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)
B.	The 45 quarter units applied to the degree must include:
	1.	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 M.A.T.H. courses (MATH 6015-6065, and 6899), only with the permission of the Mathematics Graduate Committee.
	2.	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)
C.	A comprehensive examination must be passed. Details are available in the department office.


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 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


The course prefix for the following courses is MATH.
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. May be repeated for credit, but only two units can be used toward the M.S. degree. Prerequisites: graduate standing and permission of department.
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.
6105	Multilinear Algebra (4) Introduction to partitions of positive integers; inner product spaces, including such topics as unitary, hermitian, normal matrices; certain "combinatorial" properties of permutation groups. Applications to matrix representations of finite groups and topics in tensor spaces. Graduate applications module. Prerequisites: MATH 3100 and 3121. Not open to students with credit for MATH 4105.
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.
6130	Finite Group Representations and Applications (4) Reducible and irreducible representations, Maschke's theorem, characters, Schur's lemmas, orthogonality theorems, the group algebra, induced representations and Frobenius reciprocity, Young tableaux and representations of the symmetric group, applications in chemistry and physics. Prerequisites: MATH 3100 and 3121 or consent of instructor.
6201	Topology (4) Continuation of MATH 4360. Topics may include countability and separation axioms, Tychonoff theorem, metrization theorems, homotopy theory. Prerequisite: MATH 4360.
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.
6260	Computational 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. May be repeated for credit with consent of Mathematics Graduate Studies Committee. Prerequisites: MATH 3100, 3331, 3301 or instructor's permission.
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.
6341	Elliptic Curves (4) Introduction to the geometry and arithmetic of elliptic curves. Elliptic integrals and functions, theta functions, automorphic functions, and modular forms. Algebraic curves over finite fields. Elliptic curve factorization algorithms and cryptosystems. Prerequisites: MATH 4340 or consent of the instructor.
6350	Real Analysis (4) Theory of Lebesgue measure and integration on the real line. Selected topics and applications. Prerequisite: MATH 4350.
6401, 6402	A dvanced Probability I, II (4, 4) (See STAT 6401, 6402 for course description.)
6501, 6502	M athematical Statistics I, II (4, 4) (See STAT 6501, 6502 for course description.)
6510	Analysis of Variance (4) (See STAT 6510 for course description.)
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. May be repeated for credit with consent of Mathematics Graduate Studies Committee. Cross-listed with CS 6750. Prerequisites: MATH 4750 and 3301 or instructor's permission.
6841	Nonlinear Optimization (4) Optimality conditions and solution procedures for unconstrained and constrained optimization problems. Prerequisite: MATH 3841.
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.)
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 Hayward faculty member. Oral defense normally required. Prerequisite: graduate standing. Maximum of 6 units per student. (See also, "University Thesis Writing Guide" available in WA 859.)
Mathematics Education (M.A.T.H. Option Courses)
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 Hayward faculty member. Oral defense may be required. Prerequisite: graduate status. Maximum of 5 units per student.
6900	Independent Study (1-4)