Department of Mathematics
220 Yost Hall
Phone 368-2880; Fax 368-5163
David Singer
The Department of Mathematics offers a variety of programs leading to both undergraduate (Bachelor of Arts and Bachelor of Science in Mathematics and Bachelor of Science in Applied Mathematics) and graduate (Master of Science and Doctor of Philosophy) degrees.
Prospects for employment in mathematics are good. Because of the central role of mathematics in the physical and social sciences, in engineering, and in business, the demand for mathematicians should continue to grow. Applied mathematicians are in demand in industry and government. A student with an undergraduate major in mathematics, including some computer science and statistics courses, and with some concentrated work in an allied field, has excellent career opportunities. There is a strong demand for high school teachers in mathematics.
The bachelor's degree in mathematics furnishes a strong background for graduate study in many areas (e.g., computer science, medicine, law, economics, etc.). The master's degree is sufficient for many areas of non-academic employment. The Ph.D. is necessary for college teaching.
Alejandro D. de Acosta, Ph.D. (University of California, Berkeley)
Professor
Probability; stochastic processes
Nessan Fitzmaurice, Ph.D. (Cornell University)
Assistant Professor
Dynamical systems; turbulence; numerical simulations
David Gurarie, Ph.D. (Hebrew University, Jerusalem, Israel)
Associate Professor
Mathematical physics; partial differential equations; harmonic analysis, applied mathematics
Otomar Hajek, RNDr., CSc. (Caroline University, Prague, Czechoslovakia)
Professor of Mathematics and Systems Engineering
Ordinary differential equations; mathematical control theory
Michael G. Hurley, Ph.D. (Northwestern University)
Associate Professor
Differentiable dynamical systems
Steven H. Izen, Ph.D. (Massachusetts Institute of Technology)
Associate Professor
Mathematics of imaging; image reconstruction
Katherine Kime, Ph.D. (University of Wisconsin, Madison)
Assistant Professor
Distributed parameter systems; control theory
Peter Kotelenez, Ph.D. (Universitat Bremen)
Associate Professor
Probability theory, stochastic processes, particle systems
Joel Langer, Ph.D. (University of California, Santa Cruz)
Associate Professor
Differential geometry; calculus of variations
Dong Hoon Lee, Ph.D. (Tulane University)
Professor
Topological groups; Lie groups and algebras
Marshall J. Leitman, Ph.D. (Brown University)
Professor
Integral equations; continuum physics
Arthur E. Obrock, Ph.D. (Washington University )
Associate Professor
Complex analysis
David A. Singer, Ph.D. (University of Pennsylvania)
Professor and Chairman
Riemannian geometry; differential topology
Stanislaw J. Szarek, Ph.D. (Mathematical Institute, Polish Academy of Science)
Professor
Functional analysis
Charles Wells, Ph.D. (Duke University)
Professor
Category theory; connections with computer science
Elisabeth Werner, Ph.D. (Universite Pierre et Marie Curie, Paris IV)
Assistant Professor
Functional analysis, convexity
Wojbor A. Woyczynski, Ph.D. (Wroclaw University, Poland)
Professor
Probability; stochastic processes
Ta-Sun Wu, Ph-D. (Tulane University)
Professor
Topological dynamics; topological groups; ergodic theory, Lie groups, and symmetric spaces
Kenneth Loparo, Ph.D. (Case Western Reserve University)
Associate Professor of Systems Engineering and Mechanical Engineering and Mathematics
Nonlinear chaotic and stochastic systems, analysis and control
Mihajilo D. Mesarovic, Ph.D. (Serbian Academy of Science)
Cady Staley Professor of Engineering and Mathematics
Bruce M. Ikenaga, Ph.D. (Cornell University)
Adjunct Assistant Professor
Group cohomology; algebraic topology, computer algebra systems
Raymond K. Neff, Sc.D. (Harvard University)
Adjunct Professor, V.P. for Information Services
Margaret Robinson, M.A. (State University of New York at Stony Brook)
Adjunct Instructor
A Bachelor of Arts degree in mathematics, a Bachelor of Science in Mathematics, and a Bachelor of Science in Applied Mathematics degrees are available to students at Case Western Reserve University.
All undergraduate mathematics degrees are based on a four-course sequence in calculus and differential equations and a five-course Mathematics Core in analysis and algebra.
The B.A. in mathematics requires CMPS 131 and at least 38 hours of mathematics, including
- MATH 121, 122, 223, and 224, or an equivalent sequence;
- The Mathematics Core
- MATH 307, 308, 321
- at least one of MATH 322, 323
- at least one of MATH 324, 425;
- three approved technical electives, no more than one of which can be from outside the department.
High school teaching certification is available in the B.A. program in mathematics through a joint program with John Carroll University. See the undergraduate adviser for details.
The Bachelor of Science in Mathematics degree requires at least 50 hours of mathematics courses, including
- MATH 121, 122, 223, and 224, or an equivalent sequence;
- The Mathematics Core
- MATH 307, 308, 321
- at least one of MATH 322, 323
- at least one of MATH 324, 425;
- 21 hours (normally seven courses) of approved technical electives, no more than 9 hours of which may be from outside the department.
The B.S. in Applied Mathematics requires at least 50 hours of mathematics and related subjects, in addition to a professional core. The detailed requirements are as follows:
- MATH 121, 122, 223, and 224, or an equivalent sequence;
- The Mathematics Core
- MATH 304, 307, 308, 321
- at least one of MATH 322, 323
- at least one of MATH 324, 425;
- Six technical electives as follows:
- Four courses chosen from the lists below, specific to each track. At least two of these four courses must be MATH courses.
- Two other courses of MATH at the 300 level or higher, except 470, 471.
- A professional core, as specified for each track. The professional core does not count toward the satisfaction of any of requirements 1, 2, or 3.
Total number of required credit hours for all tracks is 125, except the computer science track, which requires 128 credit hours.
Starting from the junior year, students in the Applied Mathematics Program will have to use dual academic advisers, one adviser from the department of mathematics, and one adviser from the department corresponding to the track the student has chosen.
There are four programs of study, each one leading to a different master's degree. They are:
- M.S. in Computing and Information Sciences (Department of Computer Engineering)
- M.S. in Operations Research (Department of Operations Research)
- M.S. in Systems Engineering, Systems Track (Department of Systems Engineering)
- M.S. in Systems Engineering, Control Theory Track (Department of Systems Engineering)
The master's degree requires 18 credit hours, in addition to the B.S. requirements. The required graduate courses are determined by the respective department. All students have to comply with the academic requirements for Integrated B.S./ M.S. programs, as well as with the academic requirements for the respective department.
- Four of the following courses. At least two must be MATH courses. At least one numerical analysis course must be chosen:
- MATH 410, (CMPS 440), Automata Theory
- MATH 343, (CMPS 343), Theoretical Computer Science
- MATH 413, (OPRE 514), Graph Theory
- MATH 431, Introduction to Numerical Analysis
- PHIL 306, Mathematical Logic
- CMPS 454, Analysis of Algorithms
- CMPS 541, Mathematical Linguistics
- Professional Core
- EEAP 282, Introduction to Microprocessors
- ECMP 333, Introduction to Data Structures
- ECMP 337, Systems Programming
- ECMP 338, Introduction to Operating Systems
- ECMP 280, Logic Design of Digital Systems
- Choose four of the following. At least two must be MATH courses:
- MATH 331, Numerical Linear Algebra
- MATH 423, Introduction to Real Analysis I
- MATH 491, Probability I
- MATH 492, Probability II
- MATH 495, Combinatorics
- STAT 403, Regression Techniques
- STAT 406, Methods of Experimental Design
- STAT 408, Survey Sampling
- STAT 484, Multivariate Statistical Analysis
- STAT 487, Stochastic Processes in Engineering and Science
- STAT 489, Theory of Queues
- Professional Core:
- STAT 380, Introduction to Probability and Statistics
- OPRE 428, Statistical Methods for Operations Research
- OPRE 411, Linear Programming
- One of MATH 413, STAT 487, STAT 489, or an approved 400- or 500-level course.
- Select four of the following:
- MATH 401, Abstract Algebra I
- MATH 402, Abstract Algebra II
- MATH 410, Automata and Formal Languages
- MATH 413, Graphs Theory
- MATH 415, Group Representation Theory
- MATH 423, Introduction to Real Analysis I
- MATH 428, Fourier Analysis
- MATH 431, Introduction to Numerical Analysis
- MATH 435, Intermediate Ordinary Differential Equations
- MATH 436, Mathematical Control Theory
- MATH 445, Introduction to Partial Differential Equations
- MATH 465, Differential Geometry
- MATH 491, Probability I
- Professional Core:
- ESYS 212, Signals and Systems
- ESYS 304, Control Engineering I
- ESYS 313, Signal Processing
- ESYS 306, Control Engineering II
- Select four of the following:
- MATH 410, Automata and Formal Languages
- MATH 413, Graph Theory
- MATH 401, Abstract Algebra I
- MATH 423, Introduction to Real Analysis I
- MATH 431, Introduction to Numerical Analysis
- MATH 435, Intermediate Ordinary Differential Equations
- MATH 445, Introduction to Partial Differential Equations
- MATH 447, Integral Equations
- MATH 469, Calculus of Variations
- MATH 491, Probability I
- MATH 495, Combinatorics
- STAT 385, Statistical Methods
- Professional Core:
- STAT 380, Introduction to Probability and Statistics
- ESYS 315, Decision Analysis
- ESYS 416, Optimization Theory & Techniques
- ECMP 251, Numerical Methods I
A minimum of 17 hours including:
- MATH 125, 126 (or equivalent sequence)
- MATH 201
- Two approved MATH electives
The quantitative reasoning requirement of the Western Reserve Core may be satisfied by completion of one course in mathematics (except MATH 101).
Bachelor of Arts Degree
Major in Mathematics
FRESHMAN
Fall Semester
MATH 121, Calculus for Science and Engineering I (4)
Science core course (3-4)
Core Sequence II, III or IV (3)
MATH 150, Mathematics from a MathematicianŐs Perspective (3)
ENGL 150, Expository Writing (3)
PHED 101, Physical Education Activities (0)
Spring Semester
MATH 122, Calculus for Science and Engineering II (4)
CMPS 131, Elementary Computer Programming (3)
Science core course (3-4)
Core Sequence II, III or IV (3)
Core Sequence II, III or IV (3)
PHED 102, Physical Education Activities (0)
SOPHOMORE
Fall Semester
MATH 223, Calculus for Science and Engineering III (3)
MATH 307, Abstract and Linear Algebra I (3)
Core Sequence II, III or IV (3)
Course in selected minor held (3)
Electives (3)
Spring Semester
MATH 224, Elementary Differential Equations (3)
MATH 308, Abstract and Linear Algebra II (3)
Core Sequence II, III or IV (3)
Course in selected minor field (3)
Elective (3)
JUNIOR
Fall Semester
MATH 321, Fundamentals of Analysis I(3)
Approved elective in mathematics (3)
Course in selected minor held (3)
Electives (6)
Spring Semester
MATH 322, Fundamentals of Analysis II (3)
or MATH 323, Advanced Calculus (3)
Course in selected minor field (3)
MATH 324, Introduction to Complex Analysis (3)
or MATH 425, Complex Analysis I (3)
Electives (6)
SENIOR
Fall Semester
Course in selected minor field (3)
Approved elective in mathematics (3)
Electives (9)
Spring Semester
Course in selected minor field (3)
Approved elective in mathematics (3)
Electives (9)
FRESHMAN
Fall Semester
Open elective or humanities/social science (3-0-3)b,c
CHEM 105, Principles of Chemistry I (3-0-3) or CHEM 107, Properties and Structure of Matter I (3-0-3)
CMPS 131, Elementary Computer Programming (2-2-3)
MATH 121, Calculus for Science and Engineering I (4-0-4)
ENGL 150, Expository Writing (3-0-3)
PHED 101, Physical Education Activities (0-3-0)
Total (15-5-16)
Spring Semester
Humanities/social science or open elective (3-0-3)b
CHEM 106, Principles of Chemistry II (3-0-3) or CHEM 108, Properties and Structure of Matter II (3-0-3)
CHEM 113, Principles of Chemistry Laboratory (1-3-2)
MATH 122, Calculus for Science and Engineering II (4-0-4)
PHYS 120, General Physics I (4-0-4)
PHED 102, Physical Education Activities (0-3-0)
Total (15-6-16)
SOPHOMORE
Fall Semester
Humanities or Social Science Sequence I (3-0-3)
MATH 223, Calculus for Science and Engineering III (3-0-3)
MATH 307, Abstract and Linear Algebra I (3-0-3)
PHYS 219, General Physics II (4-0-4)
Open elective (3-0-3)
Total (16-0-16)
Spring Semester
Humanities or Social Science Sequence II (3-0-3)
MATH 224, Elementary Differential Equations (3-0-3)
MATH 308, Abstract and Linear Algebra II (3-0-3)
PHYS 220, General Physics III (3-0-3)
Approved elective (3-0-3)
Total (15-0-15)
JUNIOR
Fall Semester
Humanities or Social Science Sequence III (3-0-3)
MATH 321, Fundamentals of Analysis I (3-0-3)
Approved elective (3-0-3)
Approved elective (3-0-3)
Open elective (3-0-3)
Total (15-0-15)
Spring Semester
Humanities or Social Science Sequence IV (3-0-3)
MATH 322, Fundamentals of Analysis II (3-0-3)
or MATH 323, Advanced Calculus (3-0-3)
MATH 324, Introduction to Complex Analysis (3-0-3)
or MATH 425, Complex Analysis I (3-0-3)
Open elective (3-0-3)
Open elective (3-0-3)
Total (15-0-15)
SENIOR
Fall Semester
Humanities or social science elective (3-0-3)
Approved elective (3-0-3)
Approved elective (3-0-3)
Open elective (3-0-3)
Open elective (3-0-3)
Open elective (3-0-3)
Total (18-0-18)
Spring Semester
Humanities or social science elective (3-0-3)
Approved elective (3-0-3)
Approved elective (3-0-3)
Open elective (3-0-3)
Open elective (3-0-3)
Total (15-0-15)
Hours required for graduation: 126.
The Bachelor of Science in Mathematics degree requires a minimum of 50 hours of mathematics courses, which must include MATH 121, 122, 223, 224 or an equivalent sequence, and MATH 307, 308, 321, 322 or 323, 324 or 425 (the Mathematics Core). ŇApproved electivesÓ must be approved by the studentŐs major adviser, and may include no more than three courses from other departments. In addition, the degree allows eleven open electives.
The following courses cannot be counted towards the 50 hours required for the major: MATH 101, 105, 106, 201, 470; STAT 119, 319, 320.
Students wishing to emphasize computing should take MATH 304, 343 and 410, along with suitable courses from the Department of Computer Engineering and Science.
a A suitable open elective is MATH 150, Mathematics from a MathematicianŐs Perspective. This course must be taken during the freshman year to count towards the 50 hours requirement for mathematics courses.
b One of these courses must be a humanities/social science elective.
c Selected students may be invited to take PHYS 125, 126, General Physics I, II Honors (3,3) in place of an open elective (3) and PHYS 120 (4).
FRESHMAN
Fall Semester
Open elective or humanities/social science (3-0-3)b,c
CHEM 105, Principles of Chemistry I (3-0-3) or CHEM 107, Properties and Structure of Matter I (3-0-3)
CHEM 113, Principles of Chemistry Laboratory (1-3-2)
MATH 121, Calculus for Science and Engineering I (4-0-4)
ENGL 150, Expository Writing (3-0-3)
PHED 100, Physical Education Activities (0-3-0)
Total (14-6-15)
Spring Semester
Humanities/social science or open elective (3-0-3)a,b
CHEM 106, Principles of Chemistry II (3-0-3) or CHEM 108, Properties and Structure of Matter II (3-0-3)
CMPS 131, Elementary Computer Programming (2-2-3)
MATH 122, Calculus for Science and Engineering II (4-0-4)
PHYS 120, General Physics I (4-0-4)
PHED 100, Physical Education Activities (0-3-0)
Total (17-3-17)
SOPHOMORE
Fall Semester
Humanities or Social Science Sequence I (3-0-3)
PHYS 219, General Physics II (4-0-4)
MATH 223, Calculus for Science and Engineering III (3-0-3)
MATH 304, Discrete Mathematics (3-0-3)
Technical elective
Total (16-0-17)
Spring Semester
Humanities or Social Science Sequence II (3-0-3)
PHYS 220, General Physics III (3-0-3)
MATH 224, Elementary Differential Equations (3-0-3)
Technical elective
Technical elective
Total (15-0-16)
JUNIOR
Fall Semester
Humanities or Social Science Sequence III (3-0-3)
MATH 307, Abstract and Linear Algebra I (3-0-3)
MATH 321, Fundamentals of Analysis I (3-0-3)
Technical elective
Open elective
Total (15-0-15)
Spring Semester
Humanities or Social Science Sequence IV (3-0-3)
MATH 308, Abstract and Linear Algebra II (3-0-3)
MATH 322, Fundamentals of Analysis II (3-0-3)
or MATH 323, Advanced Calculus (3-0-3)
MATH 324, Introduction to Complex Analysis (3-0-3)
or MATH 425, Complex Analysis I (3-0-3)
Open elective
Total (15-0-15)
SENIOR
Fall Semester
Humanities or social science elective (3-0-3)
Technical elective
Technical elective
Technical elective
Open elective
Total (15-0-15)
Spring Semester
Humanities or social science elective (3-0-3)
Technical elective
Technical elective
Technical elective
Technical elective
Total ( 15-0-15)
Total hours to graduate between 125-128 depending on option.
a A suitable open elective is MATH 150, Mathematics from a MathematicianŐs Perspective. This course must be taken during the freshman year to count towards the 50 hours requirement for mathematics courses.
b One of these courses must be a humanities/social science elective.
c Selected students may be invited to take PHYS 125, 126, General Physics I, II Honors (3,3) instead of an open elective (3) and PHYS 120 (4).
The department offers programs leading to the Master of Science and Doctor of Philosophy degrees. At the Master's level there are two degrees: the degree of Master of Science in Mathematics and the degree of Master of Science in Applied Mathematics.
The Ph.D. program is designed for students who intend to pursue a career in either pure or applied mathematics. The candidate must pass qualifying examinations in approved subjects; demonstrate a reading knowledge of an approved foreign language; and must present a doctoral dissertation representing significant original research. Candidates for the M.S. degree must complete 27 semester hours of approved courses and successfully pass a comprehensive examination. Throughout the student's graduate career in the department, his or her work will be closely supervised by a faculty adviser.
The Department of Mathematics at Case Western Reserve University is an active center for mathematical research. Faculty conduct research in algebra, applied mathematics, analysis, geometry and topology, and probability.
Mathematics (MATH)
MATH 101. Elementary Functions and Analytic Geometry (3).
Polynomial, rational, exponential, logarithmic, and trigonometric functions (emphasis on computation, graphing and location of roots) straight lines and conic sections. Primarily a precalculus course for the student without a good background in trigonometric functions and graphing and/or analytic geometry. Prerequisite: Three years of high school mathematics. (This course is not open to any CWRU student who has credit for or is currently taking MATH 121 or MATH 125.)
MATH 105. Mathematics of Finance I (3).
The first semester of a two-semester sequence in mathematics designed for students majoring in accounting. Simple interest, discounts, compound interest, annuities, amortization, applications to bonds, capital budgeting and depreciation, life insurance, and investments. Some aspects of probability and statistics and other topics. Prerequisite: Three years of high school mathematics.
MATH 106. Mathematics of Finance II (3).
A continuation of MATH 105. Prerequisite: MATH 105.
MATH 121. Calculus for Science and Engineering I (4).
Functions. Analytic geometry of lines & polynomials. Limits. Derivatives of algebraic and trigonometric functions. Definite integral, antiderivatives, fundamental theorem of calculus, change of variables. Required course for students seeking the Bachelor of Science Degree. Also recommended for students seeking a Bachelor of Arts Degree in one of the sciences. Prerequisite: Three and one-half years of high school mathematics.
MATH 122 Calculus for Science and Engineering II (4).
(Continuation of MATH 121.) Exponentials and logarithms, growth and decay. Inverse trigonometric functions. Related rates. Basic techniques of integration. Area and volume. Polar coordinates, parametric equations. Taylor polynomials and Taylor's theorem. Prerequisite: MATH 121.
MATH 123. Calculus I (4).
Limits, continuity. Derivatives of algebraic and transcendental functions, including applications. Basic properties of integration. Techniques of integration, and applications of the definite integral. Prerequisite: consent of department.
MATH 124. Calculus II (4).
Parametric equations and polar coordinates. Taylor's theorem. Sequence, series, power series. Complex arithmetic. An introduction multivariable calculus. Prerequisite: consent of department.
MATH 125. Mathematics I (4).
Discrete and continuous probability; differential and integral calculus of one variable; graphing, related rates, maxima and minima. Integration techniques, numerical methods; volumes, areas. Applications to the physical, life, and social science. First half of a two-semester sequence for Liberal Arts Students students; required for Lambda Core, does NOT satisfy the Case Core Requirements. Students planning to take more mathematics should take MATH 121. Prerequisite 3 1/2 years of high school mathematics, including trigonometry, exponentials, logarithms, and analytic geometry.
MATH 126. Mathematics II (4).
Continuation of MATH 125 covering differential equations, multivariable calculus, discrete methods. Partial derivatives, maxima and minima for functions of two variables, linear regression. Differential equations; first and second-order equations, systems, Taylor series methods; Newton's method; difference equations. Prerequisite: MATH 125; required for Lambda Core.
MATH 150. Mathematics from a Mathematician's Perspective (3).
An interesting and accessible mathematical topic not covered in the standard curriculum is developed. Students are exposed to methods of mathematical reasoning and historical progression of mathematical concepts. Introduction to the way mathematicians work and their attitude toward their profession. Must be taken in freshman year to count toward a major in mathematics. Prerequisite: 3 1/2 years of high school mathematics.
MATH 201. Introduction to Linear Algebra (3).
Matrix operations, systems of linear equations, vector spaces, subspaces, bases and linear independence, eigenvalues and eigenvectors, diagonalization of matrices, linear transformations, determinants. Emphasis on computation, less theoretical than MATH 307. May not be taken for credit by mathematics majors. Only one of MATH 201, MATH 308, or MATH 470 may be taken for credit. Prerequisite: MATH 122 or 126.
MATH 223. Calculus for Science and Engineering III (3).
Introduction to vector algebra; lines and planes; functions of several variables: partial derivatives, gradients, chain rule, directional derivatives, maxima/minima. Multiple integrals, cylindrical and spherical coordinates. Derivatives of vector valued functions, velocity and acceleration. Vector fields, line integrals, Green's theorem. Prerequisite: MATH 122.
MATH 224. Elementary Differential Equations (3).
A first course in ordinary differential equations. First-order equations and applications, linear equations with constant coefficients, linear systems, Fourier series, numerical methods of solution. Prerequisite: MATH 223.
MATH 225. Discrete and Continuous Models (3).
Discrete and continuous mathematical models. Differential equations, difference equations, dynamical systems. Optimization, probabilistic models. Optional topics: graph theory, game theory, combinatorics. Prerequisite: MATH 126 or consent of instructor. Required for Lambda Core.
MATH 227. Calculus III (4).
Vector algebra and geometry. Linear maps and matrices. Calculus of vector valued functions. Derivatives of functions of several variables. Multiple integrals. Vector fields and line integrals. Prerequisite: consent of instructor.
MATH 228. Differential Equations (3).
Elementary ordinary differential equations: first-order equations; linear systems: applications; numerical methods of solution. Prerequisite: consent of instructor.
MATH 301. Undergraduate Reading Course (1-3).
Students must obtain the approval of a supervising professor before registration. More than one credit hour must be approved by the undergraduate committee of the department.
MATH 302. Problem Solving Seminar (1).
A seminar devoted to methods of solving problems in various areas of mathematics and statistics. Content varies. Students may take this course for credit up to four times.
MATH 303. Elementary Number Theory (3).
Primes and divisibility, theory of congruences, and number-theoretic functions. Diophantine equations, quadratic residue theory, and other topics determined by student interest. Emphasis on problem solving (formulating conjectures and justifying them). Prerequisite: One year of calculus or consent of instructor.
MATH 304. Discrete Mathematics (3).
A general introduction to basic mathematical terminology and the technique of abstract mathematics in the context of discrete mathematics. Topics introduced are mathematical reasoning, including Boolean connectives, deduction, mathematical induction, sets, functions and relations, algorithms, graphs, combinatorial reasoning. Required for the Lambda Core.
MATH 307. Abstract and Linear Algebra I (3).
The first semester of an integrated, two semester theoretical course in abstract and linear algebra, studies on an axiomatic basis. The major structures studied are groups, rings, fields, modules, vector spaces and inner-product spaces. Topics include homomorphisms and quotient structure, the theory of polynomials, canonical forms for linear transformations and the Principal Axes Theorem. This course is required of all students majoring in mathematics. Students who take MATH 307 and 308 may not take MATH 201 or 470. Prerequisite: MATH 122
MATH 308. Abstract and Linear Algebra II (3).
Continuation of MATH 307.
MATH 321 Fundamentals of Analysis I (3).
Abstract mathematical reasoning in the context of analysis of functions of one real variable. Formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Point-set topology in metric spaces, especially the real line; completeness, compactness, connectedness, and continuity of functions. The basic theorems of one-variable calculus. Required for all mathematics majors. Prerequisite or corequisite: MATH 223 or equivalent.
MATH 322 Fundamentals of Analysis II (3).
Continuation of MATH 321. Further study of analysis in metric spaces with attention to real n-dimensional space. Topics in sequences and series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration together with introduction to measure spaces and Lebesgue integration. Fundamental theorem of Calculus. Prerequisite MATH 321.
MATH 323 Advanced Calculus (3).
A systematic approach to the differential and integral calculus of functions of several variables. Sets and topology in Euclidean spaces. Continuity. Differentiability. Riemann integration in Euclidean spaces. Inverse and implicit function theorems.
Introduction to manifolds. Prerequisite MATH 321.
MATH 324. Introduction to Complex Analysis (3).
Properties, singularities, and representations of analytic functions, complex integration. Cauchy's theorems, series, residues, conformal mapping and analytic continuation. Riemann surfaces. Relevance to the theory of physical problems. Prerequisite: MATH 224.
MATH 331. Numerical Linear Algebra (3).
Techniques of computational linear algebra for applied mathematicians, engineers, and scientists. Prerequisite MATH 201 or MATH 307, MATH 223.
MATH 338. Introduction to Dynamical Systems (3).
An introduction to nonlinear discrete dynamical systems in one and two dimensions. Chaotic dynamics, elementary bifurcation theory, hyperbolicity, symbolic dynamics, structural stability, stable manifold theory. Prerequisite MATH 224.
MATH 343. Theoretical Computer Science (3).
(Also listed as CMPS 343.) Introduction to mathematical logic. Different classes of automata and their correspondence to the different classes of formal languages. Recursive functions and computability. Assertions and program verification. Denotational semantics. Prerequisite: MATH 304.
MATH 345. Introduction to Applied Mathematics (3).
Mathematical formulation of problems, development of various methods of solution, and interpretation of results, Boundary value problem. Sturm-Liouville problems, complex analysis, transform methods. Prerequisite MATH 224 or equivalent.
Math 384. Uncertainty in Science and Engineering (3)
(Also listed as STAT 333.) Phenomenon of uncertainty appears in engineering and science for various reasons and can be modeled in different ways. The course will integrate the mainstream ideas in statistical data analysis with models of uncertain phenomena stemming from the three distinct viewpoints: algorithmic/computational complexity; classical probability theory; and chaotic behavior of nonlinear systems. Descriptive statistics, estimation procedures and hypotheses testing (including design of experiments) will be covered in this introductory statistics course. Mathematica notebooks and simulations will be used throughout the course. Prerequisite: MATH 122 or equivalent or consent of instructor.
MATH 401. Abstract Algebra I (3).
Basic properties of groups, rings, modules, and fields. Isomorphism theorems for groups; simplicity of certain alternating groups; Sylow theorem; nilpotency and solvability of groups; Jordan-Hölder Theorem; principal ideal domains; Gauss lemma and Eisenstein's Criterion; finitely generated modules over principal ideal domains with applications to abelian groups and canonical forms for matrices; categories and functors; tensor product of modules; linear and quadratic forms; field extensions: fundamental theorem of Galois theory; solvability of equations by radicals. Prerequisite MATH 308.
MATH 402. Abstract Algebra II (3).
A continuation of MATH 401. Prerequisite MATH 401.
MATH 405. Linear Analysis Control (3).
A second course in matrix theory and its application to systems of linear differential equations. Similarity; direct sum decomposition theorems; canonical forms for matrices; orthogonal transformations; functions of matrices; Cayley-Hamilton theorem; generic properties of matrices. Applications to linear systems of differential equations with constant and with time varying coefficients; global geometry of linear systems; structure of hyperbolic systems. Prerequisites: MATH 224, and MATH 201 or MATH 470 or equivalent.
MATH 406. Introduction to Logic and Model Theory (3).
Model theory and the closely related proof theory, with emphasis upon their relevance to computer science. The relationship between formal logical systems and their models. An introduction to the elementary concepts and theorems of Model Theory, such as completeness, compactness, and Lowenheim-Skolem theorems. A study of the sometimes subtle distinctions between finite and infinite. First order theories.
MATH 407. Category Theory (3).
(Also listed as PHIL 407.) Introduction to category theory and its connections with logic and the foundations of mathematics. Category theory provides tools to describe a wide array of mathematical structures. In fact, it can be used to define many kinds of structure without using set theory. Thus it can provide independent foundations for various branches of mathematics. Topics covered include categories, functors, cartesian closed categories, toposes, and category theoretic axioms for a category of sets. A further topic will be either the foundations of geometry or applications in computer science. Prerequisite: PHIL 306 or MATH 308 or MATH 304.
MATH 410. Automata and Formal Languages (3).
(Also listed as CMPS 440.) Finite automata, Turing and Post machines, and pushdown automata. The languages generated, accepted, and decided by these machines. Closure properties. Decidability and undecidability. Regular expressions. Right linear, unrestricted, and context-free grammars. Prerequisite: MATH 304 or consent of instructor.
MATH 413. Graph Theory (3).
(Also listed as OPRE 514.) Building blocks of a graph, trees, connectedness, transversality, matching, coverings, planarity, and NP-complete problems; various applications and algorithms. Prerequisite: MATH 201 or 470, or consent of instructor.
MATH 415. Group RepresentationTheory (3).
Representation and character theory of finite groups and certain (infinite) compact groups. Fundamental concepts and methods of the theory together with examples which are useful, particularly in quantum chemistry or physics. Suitable for undergraduates and graduates who have some acquaintance with linear algebra and group theory. Prerequisite: MATH 308 or equivalent, or consent of instructor.
MATH 421. Fundamentals of Analysis I (3).
(See MATH 321.) Additional work required. (May not be taken for credit by graduate students in the Department of Mathematic and Statistics.) Prerequisite or co-requisite: MATH 223.
MATH 422. Fundamentals of Analysis II (3).
(See Math 322.) Additional work required. (May not be taken for credit by graduate students in the Department of Mathematics and Statistics.) Prerequisite: MATH 321 or Math 421.
MATH 423. Introduction to Real Analysis I (3).
General theory of measure and integration. Measures and outer measures. Lebesgue measure on n-space Integration. Convergence theorems. Product measures and Fubini's theorem. Signed measures. Hahn-Jordan decomposition, Radon-Nikodym theorem, and Lebesgue decomposition. L2 spaces. Lebesgue differentiation theorem in n-space. Prerequisite: MATH 322 or consent of instructor.
MATH 424. Introduction to Real Analysis II (3).
Measures on locally compact spaces. Riesz representation theorem. Elements of functional analysis. Normed linear spaces. Hahn-Banach, Banach-Steinhaus, open mapping, closed graph theorems. Weak topologies. Banach-Alaoglu theorem. Function spaces. Stone-Weierstrass and Ascoli theorems. Basic Hilbert space theory. L2 theory of Fourier Series. Additional topics: Haar measure on locally compact groups. Prerequisite: MATH 423.
MATH 425. Complex Analysis I (3).
Analytic functions. Integration over paths in the complex plane. Index of a point with respect to a closed path; Cauchy's theorem and Cauchy's integral formula; power series representation; open-mapping theorem; singularities; Laurent expansion; residue calculus; maximum modulus theorem. Harmonic functions; mean value property; Poisson's formula; conformal mapping; the Riemann mapping theorem. More theoretical and at a higher level than MATH 324. Prerequisite: MATH 321 or consent of instructor.
MATH 428. Fourier Analysis (3).
Modern methods of harmonic analysis, which is the main tool in the study of oscillatory phenomena. Single and multiple Fourier series and integrals, general orthogonal systems, introduction to the harmonic analysis on groups, spherical functions, and singular integrals. Prerequisite: MATH 321 or 224, or consent of instructor.
MATH 431. Introduction to Numerical Analysis (3).
Numerical analysis for scientists and engineers. Floating point arithmetic and the propagation of roundoff errors. The direct solution of linear algebraic systems. Polynomial interpolation and splines. Methods of numerical differentiation and quadrature. Acceleration of convergence. Solution of non-linear algebraic equations. Prerequisite MATH 224 and a good working knowledge of a high level computer language (C, Pascal, Fortran etc.).
MATH 432. Numerical Solution of Partial Differential Equations (3).
Numerical solution of differential equations for scientists and engineers. Solution of ordinary differential equations by multistep and one step methods. Stability, consistency and convergence. Time stepping for parabolic partial differential equations; matrix and Fourier stability analysis. Finite difference discretizations of elliptic problems and iterative solution techniques for the resulting algebraic systems. Multigrid methods. Introduction to the finite element and spectral methods for boundary value problems. Introduction to hyperbolic systems. Prerequisite: MATH 431 or the equivalent and a good working knowledge of a high level computer language (C, Pascal, Fortran, etc.)
MATH 434. Optimization of Dynamical Systems (3).
(Also listed as ESYS 421.)Fundamentals of dynamic optimization with applications to control. Variational treatment of control problems and the Maximum Principle. Structures of optimal systems; regulators, terminal controllers, time-optimal controllers. Singular controls. Computational aspects. Selected applications. Prerequisite: ESYS 408 or consent of instructor.
MATH 435. Intermediate Ordinary Differential Equations (3).
Existence, uniqueness, continuation of solutions of ODE. Linear systems, fundamental matrix. Dependence on initial data and parameters (Bellman's lemma, nonlinear variation of parameters).
Stability for linear and nonlinear equations, linearization. Poincare-Bendixson theory, autonomous oscillations. Prerequisites: MATH 224; and MATH 201, 308, or 470; or consent of instructor.
MATH 436. Mathematical Control Theory (3).
Basic examples. Observed linear control systems: controllability, control systems. Optimal control theory: linear-quadratic, time optimal, maximum principle. Prerequisites: MATH 435 or consent of instructor.
MATH 445. Introduction to Partial Differential Equations (3).
Method of characteristic for linear and quasilinear equations. Second order equations of elliptic, parabolic, hyperbolic type; initial and boundary value problems. Method of separation of variables, eigenfunction expansions, Sturm-Liouville theory. Fourier, Laplace, Hankel transforms, Bessel functions, Legendre polynomials. Green's functions. Examples include: heat-diffusion, Laplace's equation, wave equations, one dimensional gas dynamics and others. Appropriate for seniors and graduate students in science, engineering, and mathematics. Prerequisite: MATH 201, 224; or equivalent
MATH 447. Integral Equations (3).
Sturm-Liouville systems and Fredholm integral equations. Eigenvalue-eigenfunction expansions; Green's function; theory of Fredholm equations. Evolutionary systems. Volterra equations. Examples of equations of evolutionary and non-evolutionary types. Linear evolutionary systems: solution by iterative schemes; qualitative behavior stability. Non-linear evolutionary systems. Semigroups of finite difference methods for integrodifferential equations. Appropriate for seniors and graduate students in science and engineering and in mathematics. Prerequisite MATH 224, and 201 or 470; or equivalent
MATH 448. Applied Partial Differential Equations (3).
(Continuation of MATH 445.) Linear and nonlinear partial differential equations, with emphasis on applications. Variational methods (Rayleigh-Ritz); asymptotic and perturbation methods: regular and singular perturbations; boundary layer, multiple scales, method of geometric optics and stationary phase. Applications are given to fluid-gas dynamics, elasticity; electrodynamics; wave scattering. Another range of topics include theory of nonlinear waves and lattices with emphasis on shocks and exactly solvable models (KdV, Toda Lattice etc.). Topics depend upon instructor and may vary from year to year. Appropriate for seniors and graduate students in science, engineering, and mathematics. Prerequisite: MATH 445 or equivalent, or consent of instructor.
MATH 452. Continuum Mechanics (3).
Kinematics of deformation. Tensors. Mathematical and physical formulation of continuum mechanics. Thermodynamical notions. Constitutive relations. Simple nonlinear materials with memory. Role of the classical theories of solids and fluids. Modern developments. Prerequisites: MATH 201 and 224 or consent of instructor.
MATH 461. Introduction to Topology (3).
Metric spaces, topological spaces, and continuous functions. Compactness, connectedness, path connectedness. Topological manifolds; topological groups. Polyhedra; simplical complexes. Fundamental groups. Prerequisite: MATH 224 or consent of instructor.
MATH 462. Algebraic Topology (3).
The fundamental group and covering spaces; van Kampen's theorem. Higher homotopy groups; long-exact sequence of a pair. Homology theory; chain complexes; short and long exact sequences; Mayer-Vietoris sequence. Homology of surfaces and complexes; applications. Prerequisite: MATH 461 or consent of instructor.
MATH 465. Differential Geometry (3).
Manifolds and differential geometry. Tangent vectors; Riemannian metrics; intrinsic and extrinsic geometry of surfaces and curves; structural equations of Riemannian geometry; the Gauss-Bonnet theorem. Prerequisite: MATH 321 or consent of instructor.
MATH 466. Vector Bundles (3).
Theory of vector bundles and fiber bundles. Vector bundles; structure groups; Lie groups and homogeneous spaces; Grassmann and Stiefel manifolds; classifying spaces for vector bundles; characteristic homotopy and homology classes. Prerequisite: MATH 321 or consent of instructor; MATH 201 or 307 recommended.
MATH 467. Differentiable Manifolds (3).
Differentiable manifolds and structures on manifolds. Tangent and cotangent bundles: vector fields; differential forms: tensor calculus; integration and Stokes' theorem. May include Hamiltonian systems and their formulation on manifolds; symplectic structures; connections and curvature; foliations and integrability. Prerequisite: MATH 322 or Math 323 or consent of instructor.
MATH 469. Calculus of Variations (3).
Examples of variational problems; variation of a functional; linear spaces; Frechet derivative; Euler Lagrange equations; Lagrange multipliers; Hamiltonian formulation; canonical coordinates; Noether's theorem; second variation; conjugate points; direct methods; regularity of solutions; Sobolev spaces; other topics depending on audience. Prerequisite: MATH 224 or consent of instructor.
MATH 470. Matrix Theory (3).
Matrix theory with emphasis on techniques useful for applications. Matrices, determinants, orthogonality, some linear algebra, material on canonical forms, applications to systems of equations, differential equations, variational principles. Students may take only one of MATH 201, 308, and 470 for credit. Prerequisite MATH 224.
MATH 471. Advanced Engineering Mathematics (3).
Vector analysis. Fourier series and integrals. Laplace transforms, separable partial differential equations, and boundary-value problems. Bessel and Legendre functions. Emphasis on techniques and applications. Students may not take both MATH 345 and 471 for credit. Prerequisite: MATH 224 or equivalent.
MATH 475. The Mathematics of Imaging in Industry and Medicine (3).
The mathematics of image reconstruction; properties of Radon transform, relation to Fourier transform; inversion methods, including convolution back projection, rho-filtered layergram, algebraic reconstruction technique (ART), and orthogonal polynomial expansions. Reconstruction from fan beam geometry; limited angle technique used in NMR; survey of applications. Prerequisite: one of either PHYS 431, MATH 345, MATH 470, or consent of instructor.
MATH 491. Probability I (3).
Probabilistic concepts. Discrete probability, elementary distributions. Measure-theoretic framework of probability theory. Probability spaces, sigma-algebras, expectations, distributions. Independence. Classical results on almost sure convergence of sums of independent random variables. Kolmogorov's law of large numbers. Recurrence of sums. Weak convergence of probability measure. Characteristic functions. Inversion, P. Levy's continuity theorem. Central limit theorem (Gaussian case). Introduction to the central limit problem. Prerequisite: MATH 322 or MATH 323, or consent of instructor; STAT 380 recommended.
MATH 492. Probability II (3).
Conditional expectations. Discrete parameter martingales. Stopping times, optional stopping. Discrete parameter stationary processes and ergodic theory. Discrete time Markov processes. Introduction to continuous parameter stochastic processes. Kolmogorov's consistency theorem. Gaussian processes. Elementary Brownian motion theory (sample path properties, strong Markov property, functional central limit theorem). Prerequisite MATH 491; MATH 423 recommended.
MATH 495. Combinatorics (3).
Permutations, combinations and variations. Principle of inclusion and exclusion. Generating functions. Difference equations. Partitions. Stirling numbers. Eulerian numbers. Ballot problems. Ramsey's theorem. Finite groups. Polya's theorem. DeBruijn's theorem. Graphs. Trees. Finite fields, finite geometries. Orthogonal Latin squares. Hadamard matrices. Block designs. Coding theory. Prerequisite: MATH 321 or consent of instructor.
MATH 503. Algebra Seminar (1-3).
Prerequisite: Consent of instructor.
MATH 504. Algebra Seminar (1-3).
Prerequisite Consent of instructor.
MATH 527. Functional Analysis (3).
Special topics course. Prerequisite: MATH 424 and 425.
MATH 529. Analysis Seminar (1-3).
Prerequisite: consent of instructor.
MATH 530. Analysis Seminar (1-3).
Prerequisite: consent of instructor.
MATH 535. Numerical Analysis Seminar (1-3).
Topic chosen near the beginning of semester. Topics may include multigrid methods and numerical solution of nonlinear partial differential equations by successive approximations, frozen coefficients (Kacanov method), methods of Newton type, and pseudolinear equations. Prerequisite: consent of instructor.
MATH 536. Numerical Analysis Seminar (1-3).
Prerequisite: consent of instructor.
MATH 541. Partial Differential Equations (3).
Special topics course. Prerequisite: MATH 424 or MATH 445.
MATH 553. Applied Mathematics Seminar (1-3).
Prerequisite: consent of instructor.
MATH 554. Applied Mathematics Seminar (1-3).
Prerequisite: consent of instructor.
MATH 563. Topology Seminar (1-3).
Continuing seminar on areas of current interest in topology and geometry. Topics may include minimal submanifolds; hyperbolic geometry and diffeomorphisms of surfaces; global analysis; discrete dynamical systems; gauge theory symplectic geometry; closed geodesics. May be taken more than once for credit. Prerequisite: MATH 461 and consent of instructor.
MATH 564. Topology Seminar (1-3).
Prerequisite: consent of instructor.
MATH 591. Stochastic Processes I (3).
Continuous parameter martingales. Optional stopping. Processes with independent increments. Levy-Ito decomposition. Sample path properties. Markov processes. Strong Markov property. Semigroups and infinitesimal generators. Brownian motion theory and potential theory. Prerequisite: MATH 424 and 492; MATH 445 recommended.
MATH 592. Stochastic Processes II (3).
Stochastic integration. The Ito calculus. Stochastic differential equations. Diffusions. Connection with partial differential equations. Prerequisite MATH 591.
MATH 595. Special Topics in Probability I (3).
Topics of current research interest in probability and stochastic processes. Topics may change from year to year. Prerequisite: consent of instructor.
MATH 597. Probability Seminar (1-3).
Prerequisite: consent of instructor.
MATH 601. Reading and Research Problems (1-36).
Presentation of individual research, discussion, and investigation of research papers in a specialized field of mathematics.
MATH 651. Thesis (M.S.) (credit as arranged).
MATH 701. Dissertation (Ph.D.) (credit as arranged).
Statistics (STAT)
See also Department of Statistics.
STAT 380. Introduction to Probability (3).
Random variables. Algebra of sets. Combinatorial methods. Sampling. Conditional probability. Bayes' Theorem. Independent events. Mathematical expectation. Mean, variance, moments. Chebyshev's inequality. Bernoulli, binomial, geometric, negative binomial, and Poisson distributions. Histogram, ogives, sample mean, sample variance. Uniform, exponential, gamma, chi-square, and normal distributions. Central limit theorem. Law of large numbers. Order statistics. Confidence intervals for percentiles, means, and percentages. Sample size. Prerequisite: MATH 122 or 126 or consent of instructor.
STAT 385. Statistical Methods (3).
For advanced undergraduate or graduate students in engineering, physical sciences, life sciences, social sciences, etc. Provides comprehensive introduction to probability models and statistical methods for analyzing data. It also discusses available statistical packages for computer applications and applies them to real-life data. General objective is to train students in formulating statistical models and in choosing appropriate methods of data analysis to test the validity of these models. Starts with descriptive statistics and moves to probability theory as a framework for statistical inference. Emphasis on the following statistical techniques: analysis of variance, regression analysis, analysis of categorical data, nonparametric statistical analysis, fitting statistical distributions to data. Prerequisite: MATH 122 or consent of instructor.
STAT 487. Stochastic Processes in Engineering and Science (3).
(See OPRE 426.)
STAT 489. Theory of Queues (3).
(Also listed as OPRE 521.) The mathematical model of queues. The distributions of the queue size and the waiting time, and the stochastic law of the busy periods. Single server queues with (i) Poisson input and exponential service times, (ii) Poisson input and general service times, (iii) recurrent input and exponential service times, and (iv) recurrent input and general service times. Many-server queues with (i) Poisson input and exponential service times and (ii) recurrent input and exponential service times. Queues with infinitely many servers in case of (i) Poisson input and general service times and (ii) recurrent input and exponential service times. Loss systems. Erlang's and Palm's formulas. Queues with a finite number of sources. Processes of serving machines. Prerequisites: STAT 487, or consent of instructor.
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