Mathematics

Chair: Wallace Goldberg

Graduate Advisor: Nick Metas

Department Office: Kiely Hall 237, 997-5800

Department Website: http://www.qc.cuny.edu/Academics/Degrees/DMNS/Math

Students in the master’s program can choose a program of study to prepare them for PhD programs in mathematics, for teaching at a pre-university level, for a career in probability or statistics, or for actuarial work. For those students who are interested in computer science as well as mathematics, a program can be arranged so that students do approximately one-half of their work in mathematics and one-half in computer science, each area complementing the other.

Faculty

Goldberg, Wallace, Chair, Professor, PhD 1974, Polytechnic Institute of New York: applied mathematics, differential equations

Metas, Nick, Graduate Advisor, Assistant Professor, PhD 1966, Massachusetts Institute of Technology: functional analysis, injective Banach spaces

Braun, Martin, Professor, PhD 1968, New York University: qualitative theory of differential equations, mathematical models

Dodziuk, Jozef, Professor, PhD 1973, Columbia University: geometric analysis

Emerson, William R., Professor, PhD 1967, University of California at Berkeley: number theory, combinatorics, and topological group theory

Hanusa, Christopher, Assistant Professor, Phd 2005, University of Washington: combinatorics & graph theory

Jiang, Yunping, Professor, PhD 1990, City University of New York: dynamical systems

Kahane, Joseph, Professor, PhD 1963, Columbia University: combinatorics, applied mathematics

Kramer, Kenneth B., Professor, PhD 1973, Harvard University: algebraic number theory

Lee, Dan, Assistant Professor, PhD 2005, Stanford University: differential geometry

Maller, Michael J., Professor, PhD 1978, University of Warwick: dynamical systems and analysis

Miller, Russell G., Associate Professor, PhD 2000, University of Chicago: logic, computability theory

Mitra, Sudeb, Associate Professor, PhD 1999, Cornell University: complex analysis, geometric function theory, Riemann surfaces, Teichmüller spaces

Ovchinnikov, Alexey, Assistant Professor, PhD 2007, North Carolina State University: differential algebra

Ralescu, Stefan S., Professor, PhD 1981, Indiana University at Bloomington: statistics, non-parametric inference, probability theory

Rothenberg, Ronald I., Associate Professor, PhD 1964, University of California at Davis: operations research, probability and statistics, applied mathematics

Sabitova, Maria, Assistant Professor, PhD 2005, University of Pennsylvania: algebraic number theory

Saric, Dragomir, Assistant Professor, PhD 2001, The City University of New York: Teichmuller theory

Sisser, Fern S., Associate Professor, PhD 1977, Columbia University: optimization

Sultan, Alan, Professor, PhD 1974, Polytechnic Institute of New York: topological measure theory

Terilla, John, Assistant Professor, PhD 2001, University of North Carolina at Chapel Hill: deformation theory, mathematical physics

Weiss, Norman J., Professor, PhD 1966, Princeton University: harmonic analysis on Euclidean spaces and Lie groups

Wilson, Scott, Assistant Professor, PhD 2005, Stony Brook University, algebraic topology

Zakeri, Saeed, Assistant Professor, PhD 1999, State University of New York at Stony Brook: dynamical systems

Requirements for Matriculation in the Master of Arts Programs

These requirements are in addition to the general requirements for admission.

1. To be admitted to the program, a candidate must have at least 25 credits in advanced courses in mathematics and related fields (such as computer science and physics). At least 12 credits must be in mathematics, including advanced calculus and linear algebra, with an average of at least B in the mathematics courses. Applicants not meeting these requirements must secure special permission of the department, and may be required to take courses to remove the deficiencies without receiving graduate credit.

2. At least two of the written recommendations must be from the applicant’s undergraduate instructors and must deal with the ability of the applicant to pursue graduate work in mathematics.

3. The applicant must have the approval of the department’s committee of the graduate program.

4. The applicant’s plan of study must be approved by the department.

Requirements for the Master of
Arts Degree

These requirements are in addition to the general requirements for the Master of Arts degree.

The Department of Mathematics offers to the student the opportunity to obtain the Master of Arts degree either in pure mathematics or with a concentration in applied mathematics.

Master of Arts in Pure Mathematics

1. A candidate for this degree is required to complete MATH 621, 628, 701, 702, and 703. A total of 30 credits required for the degree must be in mathematics, except that, with the approval of the Mathematics Department, a limited number of appropriate courses in physics or computer science may be substituted for mathematics courses. It is required that the program be completed with an average of B or better.

2. Each candidate for the degree must pass an oral examination.

Master of Arts with a Concentration in Applied Mathematics

1. A candidate for this degree is required to complete 30 credits in an approved sequence of graduate-level courses in mathematics and related fields. All students must achieve a solid grounding in the three areas of probability and statistics, analytic methods, and numerical methods. This can be achieved by taking the following courses: MATH 621, 624, 625, 628, and 633; or by demonstrating competence in specific areas to the satisfaction of the department; or by taking an alternative program of courses selected with the advisement and approval of the graduate advisor. A list of current courses and suggested programs of study will be made available. Students may obtain permission to design programs tailored to their individual needs. It is required that the master’s program be completed with an average of B or better.

2. Each candidate will be required to pass a written examination in an area of specialization to be approved by the Mathematics Department.

3. Students will be encouraged to obtain practical experience in applied mathematics by working for private businesses or governmental agencies participating in the Queens College Cooperative Education program.

Program for the Master of Science in Education Degree

Requirements for Matriculation

These requirements are in addition to the general requirements for admission. To be admitted to the program a candidate must have:

1. A cumulative index and mathematics index of at least B, as well as a B index in education are required for matriculated status. Students who do not meet the above requirements may be permitted to enter as probationary matriculants. Probationary status will be removed when the first 12 credits of approved coursework have been completed with a minimum average of B.

2. At least 21 credits in college-level mathematics courses. These courses must include intermediate calculus and linear algebra, with an average of at least B. Note that before taking the mathematics courses that go toward the master’s degree, students must have a total of 36 credits in college-level mathematics.

3. Two letters of recommendation.

Requirements for the Degree

1. Candidates in this program have two advisors, one in the Department of Secondary Education & Youth Services and one in the Department of Mathematics. The education advisor should be consulted first to plan out the required coursework.

2. Students must take 15 credits in mathematics and 15 credits in secondary education. Note that the coursework in mathematics usually includes study in the history of mathematics, probability and statistics, and geometry. Students must consult their advisor to plan an appropriate course of study.

3. Students are required to pass an oral examination in mathematics. This exam is given by two of the student’s professors and is based on the content of the two courses. The student may decide on the professors and submits a request to the mathematics advisor who then schedules the oral examination.

Courses in Mathematics

MATH 503. Mathematics from an Algorithmic Standpoint. 3 hr.; 3 cr. Prereq.: One year of calculus. An algorithmic approach to a variety of problems in high school and college mathematics. Experience in programming is not necessary. Topics may include problems from number theory, geometry, calculus, combinatorics, probability, and games and puzzles. Students will learn to program in the powerful Mathematica language and use this capability to conduct research in the above areas. Prior experience in programming is not necessary. (Students may not receive credit for this course and MATH 213W.) This course may not be credited toward the Master of Arts in Mathematics, except with the special permission of the mathematics department chair. Spring

MATH 505. Mathematical Problem-Solving. 3 hr.; 3 cr. Prereq. or coreq.: One year of college mathematics. This course presents techniques and develops skills for analyzing and solving problems mathematically and for proving mathematical theorems. Students will learn to organize, extend, and apply the mathematics they know and, as necessary, will be exposed to new ideas in areas such as geometry, number theory, algebra, combinatorics, and graph theory. This course may not be credited toward the Master of Arts in Mathematics, without the special permission of the department chair.

MATH 509. Set Theory and Logic. 3 hr.; 3 cr. Prereq.: One year of calculus or permission of the instructor. May not be credited toward the Master of Arts in Mathematics. Propositional logic and truth tables. Basic intuitive ideas of set theory: cardinals, order types, and ordinals. Fall

MATH 518. College Geometry. 3 hr.; 3 cr. Prereq.: One course in linear algebra. Advanced topics in plane geometry, transformation geometry. Not open to candidates for the Master of Arts in Mathematics. Fall

MATH 524. History of Mathematics. 3 hr.; 3 cr. Prereq. or coreq.: MATH 201 (Intermediate Calculus). Not open to candidates for the Master of Arts degree in Mathematics. Fall

MATH 525. History of Modern Mathematics. 3 hr.; 3 cr. Prereq.: MATH 524 or permission of the instructor. May not be credited toward the Master of Arts in Mathematics. Selected topics from the history of nineteenth- and twentieth-century mathematics, e.g., topology, measure theory, paradoxes and mathematical logic, modern algebra, non-Euclidean geometries, foundations of analysis.††

MATH 550. Studies in Mathematics. Prereq.: Permission of the Mathematics Department. Topics will be announced in advance. May be repeated once for credit if topic is not the same. Not open to candidates for the Master of Arts in Mathematics.††

MATH 550.1. 1 hr.; 1 cr.

MATH 550.2. 2 hr.; 2 cr.

MATH 550.3. 3 hr.; 3 cr.

MATH 555. Mathematics of Games and Puzzles. 3 hr.; 3 cr. Prereq.: Two years of calculus or permission of the instructor. May not be credited toward the Master of Arts in Mathematics. Elements of game theory. Analysis of puzzles such as weighing problems, mazes, Instant Insanity, magic squares, paradoxes, etc.††

MATH 601. Discrete Mathematics for Computer Science. 4 hr.; 3 cr. An introduction to discrete mathematics for those incoming computer science master’s degree students who do not have an undergraduate background in discrete mathematics. Topics include elementary set theory, elements of abstract algebra, propositional calculus, and Boolean algebra, proofs, mathematical induction, combinatorics, graphs, and discrete probability theory. (Students may not receive credit for both MATH 601 and either MATH 220 or CSCI 221, or an equivalent course in discrete mathematics. MATH 601 cannot be counted toward an undergraduate major in mathematics or a master’s degree in mathematics.)††

MATH 609. Introduction to Set Theory. 3 hr.; 3 cr. Prereq.: MATH 201 (Intermediate Calculus) or permission of the instructor. Axiomatic development of set theory; relations, functions, ordinal and cardinal numbers, axiom of choice. Zorn’s lemma, continuum hypothesis. Spring

MATH 611. Introduction to Mathematical Probability. 3 hr.; 3 cr. Prereq.: A one-year course in differential and integral calculus (including improper integrals). A first course in probability at an advanced level. Topics to be covered include axioms of probability, combinatorial analysis, conditional probability, random variables, binomial, Poisson, normal, and other distributions, mathematical expectation, and an introduction to statistical methods. Not open to students who have received credit for MATH 241 or 621. May not be counted toward the Master of Arts in Mathematics. Spring

MATH 612. Projective Geometry. 3 hr.; 3 cr. Prereq.: A course in linear algebra. Study of the projective plane.††

MATH 613. Algebraic Structures. 3 hr.; 3 cr. Prereq.: A course in linear algebra. Not open to students who have received undergraduate credit for MATH 333 at Queens College. Groups, rings, polynomials, fields, Galois theory. Spring

MATH 614. Functions of Real Variables. 3 hr.; 3 cr. Prereq.: A course in elementary real analysis or point set topology (equivalent of MATH 310 or 320), or permission of the instructor. Provides a foundation for further study in mathematical analysis. Topics include basic topology in metric spaces, continuity, uniform convergence and equicontinuity, introduction to Lebesgue theory of integration. Fall

MATH 616. Ordinary Differential Equations. 3 hr.; 3 cr. Prereq.: MATH 614 or permission of the chair. Existence and uniqueness of solutions, linear systems, Liapunov stability theory, eigenvalue and boundary value problems.††

MATH 617. Number Systems. 3 hr.; 3 cr. Prereq.: Three semesters of undergraduate analytic geometry and calculus including infinite series. Not open to students who have received undergraduate credit for MATH 317 at Queens College. Axiomatic development of the integers, rational numbers, real numbers, and complex numbers. Fall

MATH 618. Foundations of Geometry. 3 hr.; 3 cr. Prereq.: One year of calculus. Historical perspective. Axiomatics: models, consistency, and independence. Rigorous development of both Euclidean geometry and the non-Euclidean geometry of Bolyai and Lobachevski. Spring

MATH 619. Theory of Numbers. 3 hr.; 3 cr. Prereq.: MATH 231 or 237. Prime numbers, the unique factorization property of integers, linear and non-linear Diophantine equations, congruences, modular arithmetic, quadratic reciprocity, continued factions, contemporary applications in computing and cryptography.

MATH 621. Probability. 3 hr.; 3 cr. Prereq.: A semester of intermediate calculus (the equivalent of MATH 201) and an introductory course in probability, or permission of the chair. Binomial, Poisson, normal, and other distributions. Random variables. Laws of large numbers. Generating functions. Markov chains. Central limit theorem. Fall

MATH 623. Operations Research (Probability Methods). 3 hr.; 3 cr. Prereq.: Course in probability theory (such as MATH 241). An introduction to probabilistic methods of operations research. Topics include the general problem of decision making under uncertainty, project scheduling, probabilistic dynamic programming, inventory models, queuing theory, simulation models, and Monte Carlo methods. The stress is on applications. Spring

MATH 624. Numerical Analysis I. 3 hr.; 3 cr. Prereq.: A course in linear algebra (MATH 231 or 237) and either MATH 171 or knowledge of a programming language; coreq.: MATH 201 (Calculus). Numerical solution of nonlinear equations by iteration. Interpolation and polynomial approximation. Numerical differentiation and integration. Fall

MATH 625. Numerical Analysis II. 3 hr.; 3 cr. Prereq.: MATH 624 or its equivalent, including knowledge of a programming language. Numerical solution of systems of linear equations. Iterative techniques in linear algebra. Numerical solution of systems of nonlinear equations. Orthogonal polynomials. Least square approximation. Gaussian quadrature. Numerical solution of differential equations. Spring

MATH 626. Mathematics and Logic. 3 hr.; 3 cr. Prereq.: Intermediate calculus or permission of the department. Propositional calculus, quantification theory, recursive functions, Gödel’s incompleteness theorem. Spring

MATH 628. Functions of a Complex Variable. 3 hr.; 3 cr. Prereq.: One year of advanced calculus (MATH 202) or permission of the instructor. Topics covered include analytic functions, Cauchy’s integral theorem, Taylor’s theorem and Laurent series, the calculus of residues, Riemann surfaces, singularities, meromorphic functions. Spring

MATH 630. Differential Topology. 3 hr.; 3 cr. Prereq.: Advanced calculus. Differentiable manifolds and properties invariant under differentiable homeomorphisms; differential structures; maps; immersions, imbeddings, diffeomorphisms; implicit function theorem; partitions of unity; manifolds with boundary; smoothing of manifolds.††

MATH 631. Differential Geometry. 3 hr.; 3 cr. Prereq.: Advanced calculus. Theory of curves and surfaces and an introduction to Riemannian geometry.††

MATH 632. Differential Forms. 3 hr.; 3 cr. Prereq.: Advanced calculus. A study in a coordinate-free fashion of exterior differential forms: the types of integrands which appear in the advanced calculus.††

633. Statistical Inference. 3 hr.; 3 cr. Prereq.: A semester of intermediate calculus (the equivalent of MATH 201) and either an undergraduate probability course which includes mathematical derivations or MATH 611 or 621. Basic concepts and procedures of statistical inference. Spring

MATH 634. Theory of Graphs. 3 hr.; 3 cr. Prereq.: One semester of advanced calculus. An introduction to the theory of directed and undirected graphs. The four-color theorem. Applications to other fields. Fall

MATH 635. Stochastic Processes. 3 hr.; 3 cr. Prereq.: MATH 611 or 621. A study of families of random variables.††

MATH 636. Combinatorial Theory. 3 hr.; 3 cr. Prereq.: A course in linear algebra. This course will be concerned with techniques of enumeration. Spring

MATH 650. Studies in Mathematics. Prereq.: Permission of the department. The topic will be announced in advance. This course may be repeated for credit provided the topic is not the same.††

MATH 650.1. 1 hr.; 1 cr.

MATH 650.2. 2 hr.; 2 cr.

MATH 650.3. 3 hr.; 3 cr.

MATH 701. Theory of the Integral. 3 hr.; 4.5 cr. Prereq.: MATH 614. The Lebesgue integral in one dimension and in n dimensions, the abstract case. Spring

MATH 702. Modern Abstract Algebra I. 3 hr.; 4.5 cr. Prereq.: MATH 613. A course in the fundamental concepts, techniques, and results of modern abstract algebra. Concepts and topics studied are semi-groups, groups, rings, fields, modules, vector spaces, algebras, linear algebras, matrices, field extensions, and ideals. Spring

MATH 703. Point Set Topology. 3 hr.; 4.5 cr. Prereq.: MATH 614 or 628 or an undergraduate course in topology equivalent to MATH 320. Topological spaces, mappings, connectedness, compactness, separation axioms, product spaces, function spaces. Fall

MATH 704. Functional Analysis. 3 hr.; 4.5 cr. Prereq.: A course in linear algebra and MATH 614. Abstract linear spaces, normed linear spaces, continuous linear transformations, dual spaces. Hahn-Banach theorem, closed graph theorem, uniform boundedness principle, Hilbert spaces, the weak-star-topology, Alaoglu’s theorem, topological linear spaces.††

MATH 705. Theory of Functions of a Complex Variable. 3 hr.; 4.5 cr. Prereq.: MATH 701.††

MATH 706. Advanced Ordinary Differential Equations. 3 hr.; 4.5 cr. Prereq.: MATH 616.††

MATH 707. Partial Differential Equations. 3 hr.; 4.5 cr. Prereq.: MATH 706.††

MATH 708. Combinatorial Topology. 3 hr.; 4.5 cr. Prereq.: MATH 703.††

MATH 709. Set Theory. 3 hr.; 4.5 cr.††

MATH 710. Mathematics and Logic: Advanced Course. 3 hr.; 4.5 cr. Prereq.: MATH 626.††

MATH 711. The Mathematical Structure of Modern Statistics. 3 hr.; 4.5 cr. Prereq.: A course in either probability or statistics.††

MATH 712. Higher Geometry. 3 hr.; 4.5 cr.††

MATH 713. Modern Abstract Algebra II. 3 hr.; 4.5 cr. Prereq.: MATH 702.††

MATH 717. Theory of Approximation I. 3 hr.; 4.5 cr. Prereq.: MATH 614 or permission of the department.††

MATH 718. Theory of Approximation II. 3 hr.; 4.5 cr. Prereq.: MATH 717.††

MATH 790. Independent Research. May be repeated for credit if the topic is changed.

MATH 790.1. 1 hr.; 1 cr.

MATH 790.2. 2 hr.; 2 cr.

MATH 790.3 3 hr.; 3 cr.

MATH 790.4. 4 hr.; 4 cr.

MATH 790.45. 3 hr.; 4.5 cr.

MATH 790.5. 5 hr.; 5 cr.

MATH 791. Tutorial. May be repeated for credit if the topic is changed.

MATH 791.1. 1 hr.; 1 cr.

MATH 791.2. 2 hr.; 2 cr.

MATH 791.3. 3 hr.; 3 cr.

MATH 791.4. 4 hr.; 4 cr.

MATH 791.45. 3 hr.; 4.5 cr.

MATH 791.5. 5 hr.; 5 cr.

MATH 792. Seminar. May be repeated for credit if the topic is changed.

MATH 792.1. 1 hr.; 1 cr.

MATH 792.2. 2 hr.; 2 cr.

MATH 792.3. 3 hr.; 3 cr.

MATH 792.4. 4 hr.; 4 cr.

MATH 792.45. 3 hr.; 4.5 cr.

MATH 792.5. 5 hr.; 5 cr.