###### Graduate Courses

The Master’s program in mathematics provides both pure and applied tracks of study, with a variety of elective courses from which student can choose topics that interest them most. Students completing this program earn a strong background in both applications and theory, making our students highly qualified for employment in industry, the business world, and academia. Many MA Mathematics courses are offered on weekday evenings, giving those students with employment or other time constraints maximum flexibility.

Contact **Graduate Advisor Prof. Scott Wilson** if you have any questions about the Master’s Program in Mathematics.

##### Requirements for Matriculation in the Master of Arts Programs

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

- 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.
- 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.
- The applicant must have the approval of the department’s Committee of the Graduate Program.
- The applicant’s plan of study must be approved by the department.

##### 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.

##### 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

###### Graduate Course Descriptions

#### Mathematics 500 Level Courses

##### Math 503

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 and numerical analysis, combinatorics and probability, and games and puzzles. This course aims at a better understanding of mathematics by means of concrete, constructive examples of mathematical concepts and theorems. This course may not be credited toward the degree of Master of Arts in Mathematics, except with the special permission of the Chair of the Mathematics Department.

##### Math 505

Prereq. or coreq.: One year of college mathematics.

Not open to students who are taking or who have received credit for MATH205. 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 degree in Mathematics.

##### Math 509

Prereq.: One year of calculus or permission of instructor. Propositional logic and truth tables. Basic intuitive ideas of set theory: cardinals, order types, and ordinals. May not be credited toward the Master of Arts degree in Mathematics.

##### Math 518

MATH 518. College Geometry. 3 hr.; 3 cr. (Syllabus)

Prereq.: One course in linear algebra.

Advanced topics in plane geometry, transformation geometry. Not open to candidates for the Master of Arts degree in Mathematics.

##### Math 524

MATH 524. History of Mathematics. 3 hr.; 3 cr. (Syllabus)

Prereq. or coreq.: Mathematics 201.

Not open to candidates for the Master of Arts degree in Mathematics.

##### Math 525

MATH 525. History of Modern Mathematics. 3 hr.; 3 cr.

Prereq.: Mathematics 524 or permission of instructor.

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. May not be credited toward the Master of Arts degree in Mathematics.

##### Math 555

Prereq.: Two years of calculus or permission of instructor.

Elements of game theory. Analysis of puzzles such as weighing problems, mazes, Instant Insanity, magic squares, paradoxes, etc. May not be credited toward the Master of Arts degree in Mathematics.

##### Math 590

MATH 590. 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 degree in Mathematics.

#### Mathematics 600 Level Courses

##### Math 601

Prereq.: MATH 231

Not open to students who are taking or who have received credit for MATH 301 or 702. Theory of groups, including cyclic and permutation groups, homomorphisms, normal subgroups and quotient groups. Theory of rings, including integral domains and polynomial rings. Additional topics may be discussed

##### Math 602

Prereq.: MATH 301 (or 601)

This is a continuation of MATH 601. Not open to students who are taking or who have received credit for MATH 302 or 702. Advanced topics in group and ring theory. Fields and field extensions

##### Math 605 [Formerly 619]

Prereq.: MATH 231 or 237

Prime numbers, the unique factorization property of integers, linear and non-linear Diophantine equations, congruences, modular arithmetic, quadratic reciprocity, contemporary applications in computing and cryptography. Not open to students who are taking or have received credit for MATH305.

##### Math 609

Prereq.: Mathematics 201 or permission of instructor.

Axiomatic development of set theory; relations, functions, ordinal and cardinal numbers, axiom of choice. Zorns lemma, continuum hypothesis.

##### Math 611

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 Mathematics 241 or 621. May not be counted toward the Master of Arts degree in Mathematics.

##### Math 616 [Formerly 628]

Prereq.: One year of multivariable calculus (MATH 202) or the equivalent.

Not open to students who are taking or have received credit for MATH 316. Topics covered include analytic functions, Cauchy’s Integral Theorem, Taylor?s theorem and Laurent series, the calculus of residues, singularities, meromorphic functions

##### Math 612

MATH 612. Projective Geometry. 3 hr.; 3 cr. (Syllabus)

Prereq.: A course in linear algebra.

Study of the projective plane.

##### Math 614

Prereq.: Course in Elementary Real Analysis or Point Set Topology (equivalent of Mathematics 310 or 320), or permission of 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.

##### Math 615

Prereq.: Mathematics 333 or 613 or permission of instructor.

Modern theory of algebraic integers (generalization of integers), the problem of prime factorization, p-adic numbers, the Riemann zeta function, L-functions, theorem on primes in arithmetic progression

##### Math 616 [Formerly 628]

Prereq.: One year of multivariable calculus (MATH 202) or the equivalent. Not open to students who are taking or have received credit for MATH 316.

Topics covered include analytic functions, Cauchy’s Integral Theorem, Taylor’s theorem and Laurent series, the calculus of residues, singularities, meromorphic functions

##### Math 617

Prereq.: Three semesters of undergraduate analytic geometry and calculus including infinite series.

Axiomatic development of the integers, rational numbers, real numbers, and complex numbers. Not open to students who have received undergraduate credit for Mathematics 317 at Queens College.

##### Math 618

Prereq.: MATH 201.

Not open to students who are taking or have received credit for MATH 318. The course is an exploration of Euclid?s fifth postulate, often referred to as the parallel postulate. Development of the basics of Euclidean geometry with a focus on understanding the role of the fifth postulate. Development and exploration of hyperbolic geometry, a non-Euclidean geometry.

##### Math 620

Coreq.: MATH 201.

Not open to students who are taking or who have received credit for MATH 320. The basic concepts and fundamental results of point-set topology. The course includes a review of sets and functions, as well as the study of topological spaces including metric spaces, continuous functions, connectedness, compactness, and elementary constructions of topological spaces.

##### Math 621

Prereq.: A semester of intermediate calculus (the equivalent of Mathematics 201) and an introductory course in probability, or permission of Chair.

Binomial, Poisson, normal, and other distributions. Random variables. Laws of large numbers. Generating functions. Markov chains. Central limit theorem.

##### Math 623

Prereq.: Course in probability theory (such as Mathematics 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.

##### Math 624

Prereq.: A course in linear algebra (231 or 237) and either Mathematics 171 or knowledge of a programming language;

Coreq.: Mathematics 201 (Calculus). Numerical solution of nonlinear equations by iteration. Interpolation and polynomial approximation. Numerical differentiation and integration.

##### Math 625

Prereq.: Mathematics 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.

##### Math 626

MATH 626. Mathematics and Logic. 3 hr.; 3 cr. (Syllabus)

Prereq.: Intermediate calculus or permission of department.

Propositional calculus, quantification theory, recursive functions, Godels incompleteness theorem.

##### Math 630

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

Prereq.: Advanced calculus.

Theory of curves and surfaces and an introduction to Riemannian geometry.

##### Math 632

Prereq.: Advanced calculus.

A study in a coordinate-free fashion of exterior differential forms: the types of integrands which appear in the advanced calculus.

##### Math 633

Prereq.: A semester of intermediate calculus (the equivalent of Mathematics 201) and either an undergraduate probability course which includes mathematical derivations or Mathematics 611 or 621.

Basic concepts and procedures of statistical inference.

##### Math 634

Prereq.: One semester of advanced calculus.

An introduction to the theory of directed and undirected graphs. The Four-Color Theorem. Applications to other fields.

##### Math 635

Prereq.: Mathematics 611 or 621.

A study of families of random variables.

##### Math 636

Prereq.: A course in linear algebra.

This course will be concerned with techniques of enumeration.

##### Math 640

Prereq.: A course in probability. Coreq.: A course in multivariable calculus and linear algebra.

Not open to students who are taking or who have received credit for MATH 340. Topics include introducing common random variable models, the central limit theorem, law of large numbers, random variable convergence. Topics may also include order statistics, probability inequalities, Slutsky’s Theorem, Markov chains and stochastic gradient descent. Probability computation using modern software.

##### Math 641

Coreq.: MATH 640 or the equivalent.

Not open to students who are taking or who have received credit for MATH 341. Point estimation, confidence sets and hypothesis testing from both the Frequentist and Bayesian perspectives. Topics may also include power calculations, multiple comparisons, model selection and randomized experimentation.

##### Math 642

Prereq: A course in linear algebra, and course in probability, and a course in programming (CSCI 111 or the equivalent)

Not open to students who are taking or who have received credit for MATH 342W. Recommended corequisites include ECON 382, 387, MATH 341, MATH 343 or their equivalents. Philosophy of modeling with data. Prediction via linear models and machine learning including support vector machines and random forests. Probability estimation and asymmetric costs. Underfitting vs. overfitting and model validation. Formal instruction of data manipulation, visualization and statistical computing in a modern language.

##### Math 643

Prereq.: MATH 641 or the equivalent. Coreq.: MATH 642 or the equivalent

Not open to students who are taking or who have received credit for MATH 343. Topics may include the Score and generalized likelihood ratio tests, chi-squared tests, Kolmogorov-Smirnov test, basic linear model theory, ridge and lasso, Metropolis-within-Gibbs sampling, permutation tests, the bootstrap and survival modeling. Special topics.

##### Math 690

MATH 690. Studies in Mathematics.

Prereq.: Permission of department.

The topic will be announced in advance. This course may be repeated for credit provided the topic is not the same.

#### Mathematics 700 Level Courses

##### Math 701

Prereq.: Mathematics 614.

The Lebesgue integral in one dimension and in n dimensions, the abstract case.

##### Math 702

Prereq.: MATH 301 (or 601).

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.

##### Math 703

Prereq.: Advanced calculus.

Topological spaces, mappings, connectedness, compactness, separation axioms, product spaces, function spaces.

##### Math 704

Prereq.: A course in linear algebra and Mathematics 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, Alaoglus theorem, topological linear spaces.

##### Math 705

Prereq.: Mathematics 701.

##### Math 706

Prereq.: Mathematics 616.

##### Math 707

Prereq.: Mathematics 706.

##### Math 708

Prereq.: Mathematics 703.

##### Math 709

##### Math 710

Prereq.: Mathematics 626.

##### Math 711

Prereq.: A course in either probability or statistics.

##### Math 712

##### Math 713

Prereq.: Mathematics 702.

##### Math 717

Prereq.: Mathematics 614 or permission of department.

##### Math 718

Prereq.: Mathematics 717.

##### Math 790

May be repeated for credit if the topic is changed.

##### Math 791

May be repeated for credit if the topic is changed.

##### Math 792

May be repeated for credit if the topic is changed.