## Graduate Course Descriptions

### 500-level

*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 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.* Mathematical Problem-Solving. 3 hr.; 3 cr.

*Prere**q. 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.

*MATH 509.* Set Theory and Logic. 3 hr.; 3 cr.

*Prer**eq.:* 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.* College Geometry. 3 hr.; 3 cr.
(Syllabus)

*Prer**eq.:* 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.* History of Mathematics. 3 hr.; 3 cr.
(Syllabus)

*Prer**eq. or coreq.:* Mathematics 201.

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

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

*Prer**eq.:* 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 550.* Studies in Mathematics.

*Prer**eq.:* 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.*

*MATH 555.* Mathematics of Games and Puzzles. 3 hr.; 3 cr.

*Prer**eq.:* 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.*

### 600-level

*MATH 601.* Discrete Mathematics for Computer Science. 4 hr.; 3 cr.

An introduction to discrete mathematics for those incoming Computer Science masters 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 Mathematics 601 and either Mathematics 220 or Computer Science 221, or
an equivalent course in discrete
mathematics. Mathematics 601 cannot be counted toward an undergraduate major in mathematics or a masters degree in
mathematics.*

*MATH 609.* Introduction to Set Theory. 3 hr.; 3 cr.

*Pre**req.:* 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.* Introduction to Mathematical Probability. 3 hr.; 3 cr.

*Pr**ereq.:* 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 612.* Projective Geometry. 3 hr.; 3 cr.
(Syllabus)

*Pr**ereq.:* A course in linear algebra.

Study of the projective plane.

*MATH 613.* Algebraic Structures. 3 hr.; 3 cr.

*Pr**ereq.:* A course in linear algebra.

Groups, rings, polynomials, fields, Galois theory.
*Not open to students who have received undergraduate credit for Mathematics 333 at Queens College.*

*MATH 614.* Functions of Real Variables. 3 hr.; 3 cr.

*Pr**ereq.:* 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.* Algebraic Number Theory. 3 hr.; 3 cr.

*P**rereq.:* 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.* Ordinary Differential Equations. 3 hr.; 3 cr.

*Prereq.:* Mathematics 614 or permission of 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.

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.* 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.

*MATH 619.* Theory of Numbers. 3 hr.; 3 cr.

*Prereq.:* Mathematics 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 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.* Operations Research (Probability Methods). 3 hr.; 3 cr.

*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.* Numerical Analysis I. 3 hr.; 3 cr.

*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.* Numerical Analysis II. 3 hr.; 3 cr.

*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.* 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 628.* Functions of a Complex Variable. 3 hr.; 3 cr.

*Prereq.:* One year of advanced calculus (Mathematics 202) or permission of instructor.

Topics covered include analytic functions, Cauchys Integral Theorem, Taylors theorem and Laurent series, the
calculus of residues, Riemann surfaces,
singularities, meromorphic functions.

*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.

*MATH 633.* Statistical Inference. 3 hr.; 3 cr.

*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.* 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.

*MATH 635.* Stochastic Processes. 3 hr.; 3 cr.

*Prereq.:* Mathematics 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.

*MATH 650.* 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.*

### 700-level

*MATH 701.* Theory of the Integral. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 614.

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

*MATH 702.* Modern Abstract Algebra I. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 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.

*MATH 703.* Point Set Topology. 3 hr.; 4 1/2 cr.

*Prereq.:* Advanced calculus.

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

*MATH 704.* Functional Analysis. 3 hr.; 4 1/2 cr.

*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.* Theory of Functions of a Complex Variable. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 701.

*MATH 706.* Advanced Ordinary Differential Equations. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 616.

*MATH 707.* Partial Differential Equations. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 706.

*MATH 708.* Combinatorial Topology. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 703.

*MATH 709.* Set Theory. 3 hr.; 4 1/2 cr.

*MATH 710.* Mathematics and Logic: Advanced Course. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 626.

*MATH 711.* The Mathematical Structure of Modern Statistics. 3 hr.; 4 1/2 cr.

*Prereq.:* A course in either probability or statistics.

*MATH 712.* Higher Geometry. 3 hr.; 4 1/2 cr.

*MATH 713.* Modern Abstract Algebra II. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 702.

*MATH 717.* Theory of Approximation I. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 614 or permission of department.

*MATH 718.* Theory of Approximation II. 3 hr.; 4 1/2 cr.

*Prereq.:* Mathematics 717.

*MATH 790.* Independent Research.

*May be repeated for credit if the topic is changed.*

*MATH 791.* Tutorial.

*May be repeated for credit if the topic is changed.*

*MATH 792.* Seminar.

*May be repeated for credit if the topic is changed.*