Mathematics

Department Chairperson : Rohan Attele

Graduate Program Advisor : Dawit Getachew

Graduate Faculty : Victor K. Akatsa, Kapila Rohan Attele, John F. Erickson, Dawit Getachew, Lun-Pin Ho, Daniel J. Hrozencik, Lixing A. Jia, Paul M. Musial, Sharon O’Donnell, Howard A. Silver (Emeritus), Richard J. Solakiewicz, Marjorie M. Stinespring (Emeritus), Asmamaw Yimer, George I. Zazi.

The Department of Mathematics and Computer Science offers a program of study that leads to a Master of Science degree in Mathematics. The program is especially designed to meet the needs of the busy professional. With evening courses, research seminars, and interdisciplinary studies in physical and life sciences, the program offers opportunities for enrichment and professional growth.

The department is especially proud of its record of providing access to students with strong academic potential. Students can learn in a supportive environment and engage in scholarly activities outside the classroom setting. They can participate in research seminars and colloquia or Graduate Students’ Seminar. Each year, the department offers a major lecture series in the Spring semester. The program keeps abreast of the needs of potential employers with an Advisory Board consisting of representatives from industry, City Colleges of Chicago, and Chicago Board of Education.

A limited number of Graduate Assistantships are offered subject to the availability of funds. Graduate Assistants may teach under mentorship of a regular faculty member and attend Case Studies workshops on various teaching scenarios to gain teaching experience.

The Graduate Students’ Handbook describes departmental policies in detail, and has other useful information such as links to useful web sites. Policies for graduate assistants are in the booklet Procedures for Graduate Assistants. Both publications are available from the department.

General Requirements

- Fulfillment of the general requirements for admission to the School of Graduate and Professional Studies.
- A bachelor’s degree from an accredited institution with an undergraduate concentration in mathematics and a B average or better (3.0 or higher from a 4.0 scale) in advanced undergraduate mathematics courses. An undergraduate concentration in mathematics is defined as successful completion of at least four advanced mathematics courses.
- Applicants without the above mentioned requisite concentration or average may gain Conditional Admission.
- Any International Student who wishes to apply to the Masters of Science in Mathematics program must, in addition to fulfilling all other requirements, submit an official copy of scores received on the Mathematics subject test of the Graduate Records Examination (GRE).
- To gain full admission, students with Conditional Admission status are required to complete four approved graduate courses (which could be credited to the degree) with a B average or better and, if needed, their prerequisite courses.
- Completion of at least thirty-three graduate credit hours.
- A maximum of twelve credit hours of approved 4000-level courses completed in graduate student status may be applied towards the degree provided grade of B or better is earned in each. Two of these 4000-level courses (maximum of six hours) may be chosen, with the consent of the graduate adviser, from interdisciplinary courses in mathematical biology, mathematical physics, or computer science.

Specific Requirements

Required courses (9 hours)

MATH 5420/415, 5220/420, and select one course in an area in geometry from the following: MATH 5120/408, 5130/410, 5140/411.

Completion of a Master’s Project (Math 5950/495)-3 hours, orMaster’sThesis (Math 5980*)-6 hours

Completion of three additional elective courses at the 5000-level selected with the approval of the graduate advisor.

Completion of the remaining 9–12 hours to be selected from approved 4000- or 5000-level mathematics courses that are applicable to the Master’s degree. The following 4000-level courses cannot be applied to the Master’s degree: 4040/304, 4050/305, 4060/306, 4070/307, 4010/347, 4020/348, 4000/363, 3005/375, 4920/381, 4940/392. This list is not exhaustive; consult the graduate adviser.

* Needs prior approval from the graduate committee.

Course Offerings

Mathematics (MATH)

NOTE: Students enrolling in mathematics classes must receive at least a grade of C in all its pre-requisite mathematics and computer science courses.

5000/401 CONTINUING EDUCATION TOPICS FOR PRIMARY SCHOOL MATHEMATICS TEACHERS (1-3)

Prerequisite: Standard Teaching Certificate.

An exploration of both content and pedagogical topics for teachers of primary school mathematics. Topics may include new curriculum programs, materials and teaching strategies, calculators and computers, number systems, operations of arithmetic, and problem solving, as well as topics of current interest. A hands-on approach using manipulative and concrete models will be a focus for the course. May be repeated for up to a maximum of six credit hours. Cannot be used to satisfy requirements for certification or for a bachelor’s or master’s degree.

5005/402 CONTINUING EDUCATIONTOPICS FOR MIDDLE SCHOOL MATHEMATICS TEACHERS (1-3)

Prerequisite: Standard Teaching Certificate

An exploration of both content and pedagogical topics for teachers of middle school or junior high school mathematics. Topics may include new curriculum programs, materials and teaching strategies, calculators and computers, geometry, probability and statistics, relations, equations and problem solving, as well as other topics of current interest. May be repeated for up to a maximum of six credit hours. Cannot be used to satisfy requirements for certification or for a bachelor’s or master’s degree.

5010/301 A REVIEW OF MATHEMATICAL TOPICS (3)

Prerequisite: Bachelor’s degree and consent of the department.

A review of the basic elements of calculus, linear algebra, and the nature of proof. Cannot be used for Master’s degree.

5020/302 RECENT TRENDS IN MATHEMATICS (3)

Prerequisite: Bachelor’s degree and/or consent of the department.

Course for in-service teachers. Changes in the teaching, philosophy, and course content of pre-collegiate mathematics. Cannot be used for Master’s degree.

5040/304 NUMBER THEORY FOR MIDDLE SCHOOL TEACHERS (3)

Prerequisite: MATH 1020/141 or MATH 1200/162 or HON 1300/130 and HON 1310/131; and a 2000-level mathematics course.

Prime numbers. Euclidean algorithm. Greatest common divisors and least common multiples. Modular arithmetic. Diophantine equations. Integration of the content in elementary and middle school. May not be used for graduation credit in mathematics or mathematics secondary education options. Cannot be used for Master’s degree.

5050/305 GEOMETRY FOR MIDDLE SCHOOL TEACHERS (3)

Prerequisite: MATH 1210/141 or MATH 1200/162 or HON 1300/130 and 1310/131; and a 2000-level mathematics course.

Incidence relation, angles, congruence, measurement, compass-and-straightedge constructions, circles, arcs and arc length, polygons, similarity, right-triangle trigonometry, area, volume, coordinate geometry in two and three dimensions. Integration of content in the elementary and middle school. Cannot be used for Master’s degree.

5060/306 MODERN ALGEBRA FOR MIDDLE SCHOOL TEACHERS (3)

Prerequisite: MATH 1210/141 or MATH 1200/162 or HON 1300/130 & 1310/131 or equivalent; and a 200-level mathematics course.

Groups, rings, fields and their applications. Integration of content in the elementary and middle school. Cannot be used for Master’s degree.

5070/307 HISTORY OF MATHEMATICS FOR MIDDLE SCHOOL TEACHERS (3)

Prerequisites: MATH 1400/209 and MATH 4050/305 or equivalent

The historical basis of numeration, operations, geometry, algebra, trigonometry, and calculus. The cultural aspect of mathematics and its contributions to knowledge and learning. Integration of content in the elementary and middle school. Cannot be used for Master’s degree.

5090/309 CALCULUS FOR TEACHERS I (3)

Prerequisite: Bachelor’s degree and consent of the department.

A review of the basic elements of calculus with special emphasis on teaching strategies. Cannot be used for Master’s degree.

5095/310 CALCULUS FOR TEACHERS II (3)

Prerequisite: MATH 5090/309 or consent of the department.

A continuation of MATH 309. Cannot be used for Master’s degree.

5080/311 PROBABILITY AND STATISTICS FOR TEACHERS (3)

Prerequisite: Bachelor’s degree and consent of the department.

Basic elements of probability and statistics with special emphasis on teaching strategies. Cannot be used for Master’s degree.

5120/408 CONCEPTS OF GEOMETRY I (3)

Prerequisite: MATH 4110/342.

Axiom systems, Euclidean geometry, non-Euclidean geometry, theory of incidence, theory of order, affine geometries, similarity and congruence, models of geometries, distance and measurement, ruler and compass constructions.

5130/410 PROJECTIVE GEOMETRY (3)

Prerequisite: MATH 4110/342.

Analytic and axiomatic projections, theory of conic sections, Pascal’s and Branchon’s theorems, linear transformations.

5140/411 DIFFERENTIAL GEOMETRY (3)

Prerequisite: MATH 4410/358.

Theory of curves and surfaces in Euclidean space, Frenet-Serret formulas, Gaussian curvature, geodesics.

5180/461 POINT SET TOPOLOGY (3)

Prerequisite: Consent of the department.

Concept of topology, topological and metric spaces, continuity, connectedness, generalized limits, separation concepts.

5210/427 NUMBER THEORY (3)

Prerequisite: MATH 4210/327 or equivalent.

Selected topics in classical, analytic, and algebraic number theory.

5220/420 MODERN ALGEBRA I (3)

Prerequisite: MATH 4250/361.

Group theory: group action, Sylow theorems, simple and solvable groups. Field Theory: Galois correspondence, radical extensions, algebraic and transcendental extensions, finite fields.

5230/421 MODERN ALGEBRA II (3)

Prerequisite: MATH 5220/420.

Continuation of MATH 5220/420, Commutative algebra: Integrality, Hilbert Basis Theorem, modules over PDI; non-commutative rings: Jacobson radical, Artin-Wedderburn theorem.

5250/520 COMMUTATIVE ALGEBRA (3)

Prerequisites: Math 5220/420 and 5230/421

An introduction to commutative rings and modules over commutative rings. Chain conditions, Noetherian and Artinian rings. Localization. Finitely generated algebras over a field, varieties. Further topics may include discrete valuation rings and Dedekind domains, completions, Nullstellensatz.

5310/454 MODERN APPLIED ALGEBRA (3)

Prerequisite: MATH 4250/361.

Algebraic theory with applications to theoretical computing. Topics include finite state automata, Turing machines, computability, the theoretical limits of computers, and coding theory.

5320 INTRODUCTION TO COMPUTATIONAL AGEBRAIC GEOMETRY (3)

Prerequisite: Math 5220/420.

Geometry of curves/surfaces defined by polynomial equations. Emphasis on concrete computations with polynomials using computer packages, interplay between algebra and geometry as well as algebra and biology. Algebra and topology presented as needed.

5410/426 THEORY AND APPLICATION OF INFINITE SERIES (3)

Prerequisite: MATH 4410/358.

Infinite series of real and complex terms including the summability methods of Abel and Cesaro; tests for convergence of series of positive constants including those of Cauchy, Maclaurin, Gauss; alternating series; conditional convergence and Riemann’s theorem; absolute and uniform convergence; theorems of Weierstrass, Cauchy, Abel, Levi and Sternitz; introduction to the theory of divergent series.

5420/415 ANALYSIS I (3)

Prerequisite: MATH 4410/358.

Functions of bounded variation, Riemann-Stieltjes integral, topology of the real line, measure theory, measurable functions, Lebesgue integral, other selected topics.

5430/416 ANALYSIS II (3)

Prerequisite: MATH 5420/415.

Continuation of MATH 5420/415: Radon-Nikodym theorem, L^{ P} spaces, Riesz Representation theorem, functions of several variables, inverse and
implicit function theorems, smooth manifolds, tangent and cotangent bundles, vector
bundles, differential forms, pullback, wedge product, integration, Poincare lemma.

5440/422 COMPLEX VARIABLES I (3)

Prerequisite: MATH 4450/356 or 4410/358.

Complex numbers and polygenic and monogenic functions, theory of residues, Taylor and Laurent series, Cauchy-Riemann and Laplace equations.

5445 COMPLEX ANALYSIS II (3)

Prerequisites: Math 5440/422.

Harmonic functions and the Dirichlet problem. Introduction to Riemann surfaces. Negative curvature and Picard’s Big Theorem. Further topics may include Hardy spaces, Corona theorem, a deeper study of Riemann surfaces, the uniformization theorem.

5470 FUNCTIONAL ANALYSIS (3)

Prerequisite: MATH 5420/415

Hahn-Banach theorem, weak topologies; operators on Hilbert and Banach spaces, normal, self-adjoint, and compact operators; geometric and spectral analysis of linear operators; generalized functions. At instructor’s discretion applications to Fourier series, numerical analysis, probability, or differential equations will be discussed.

5510/433 ADVANCED ORDINARY DIFFERENTIAL EQUATIONS (3)

Prerequisite: MATH 2550/271 and 4410/358.

Boundary value problems, linear differential equations with periodic coefficients, nonlinear differential equations, perturbation theory, Poincare’s method.

5520/431 PARTIAL DIFFERENTIAL EQUATIONS (3)

Prerequisite: MATH 2550/271 and 4410/358.

Classical solutions of first and second order partial differential equations, Bessel and Legendre functions, orthogonal functions, solutions of boundary value problems by the separation of variables and integral transformations, Laplace’s equation, wave equation.

5550/441 ADVANCED NUMERICAL METHODS (3)

Prerequisite: MATH 5550/441, MATH 4410/358.

Difference equations, iteration, Aitken’s delta square method, Steffensen’s method, Bairstow’s method, Bernoulli’s method, and the quotient-difference algorithm. Additional topics may include Mesh-free methods, finite element methods, spectral methods, Galerkin type methods, and fast Fourier transform methods.

5560/443 INTERPOLATION AND APPROXIMATION (3)

Prerequisite: MATH 5550/441.

Interpolation via polynomials, orthogonal families of polynomials, spline interpolation, least squares methods, and Fourier methods. Additional topics may include radial basis function methods, and convolution kernel based methods.

5610/471 MATHEMATICAL STATISTICS I (3)

Prerequisite: MATH 4600/315 or equivalent.

Random variables, conditional probability, multidimensional distributions, functions of random variables, central limit theorem, limiting distributions.

5620/472 MATHEMATICAL STATISTICS II (3)

Prerequisite: MATH 5610/471.

Test of hypothesis, point estimation, confidence intervals, sufficient statistics, Rao Blackwell theorem, multivariate analysis.

5710 MATHEMATICS CURRICULUM IN GRADES 7-12 (3)

Prerequisite: Consent of Graduate Advisor.

The history of curriculum, content, and trends in mathematics of grades 7-12; study and analysis of major reports, recommendations, and theories in mathematics curricula and teaching and learning; issues and trends in assessment practices; study and analysis of reform movements and their effect in curricular designs, practices, and beliefs of mathematics and mathematics teaching and learning.

5780 RESEARCH & PRACTICES IN MATHEMATICS EDUCATION (3)

Prerequisite: Consent of Graduate Advisor

Survey and analysis of research in the field of mathematics education; issues, research, and practices in the teaching and learning of mathematics; critiques and conduct of research in the teaching and learning of mathematics.

5810/451 MATHEMATICAL MODELS AND APPLICATIONS (3)

Prerequisite: High school mathematics teaching experience and consent of the department.

Modern application of mathematics in such fields as economics, industrial management, psychology, political science, biology, ecology, and geography for high school classroom utilization.

5920/492 GRADUATE SEMINAR (3)

Prerequisite: Twenty-one graduate level credit hours in mathematics and consent of the department.

Conducted by graduate faculty of department. The course may be repeated under a different topic with the permission of the department.

5950/495 MASTER’S PROJECT (3)

Prerequisite: Approval of Graduate Advisor.

Research under the supervision of a regular faculty member leading to the completion of a Master’s Project.

5980 MASTER’S THESIS (3)

Prerequisite: Approval of Graduate Advisor.

Research under the supervision of a regular faculty member leading to the completion of a Master’s thesis. (A thesis is a document whose precise format is prescribed by the Graduate School. Refer to the Graduate Students’ Handbook for the thesis formalism.)

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