Department of Mathematics

• Department of Mathematics
  • FACULTY
  • ASSOCIATE FACULTY
  • ADJUNCT FACULTY
  • UNDERGRADUATE PROGRAMS
    • Degree Requirements
    • Bachelor of Arts Degree in Mathematics
    • Bachelor of Science in Mathematics Degree
    • Bachelor of Science in Applied Mathematics Degree
    • Non-Major Undergraduate Programs in Mathematics
      • Minor in Mathematics - All undergraduates
      • High School Teaching Certification
  • GRADUATE PROGRAMS
    • Doctor of Philosophy and Master of Science in Mathematics
  • RESEARCH AND TEACHING
  • UNDERGRADUATE COURSES
  • GRADUATE COURSES
  • BACHELOR OF SCIENCE IN MATHEMATICS DEGREE
  • BACHELOR OF SCIENCE IN APPLIED MATHEMATICS

Department of Mathematics

220 Yost Hall
Phone 368-2880; Fax 368-5163
Dong Hoon Lee

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, there should be continuing demand for mathematicians. Applied mathematicians are in demand in industry and government. A student with an undergraduate major in mathematics, including some computer science, 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.

The Math Tutoring Center, located in Yost 321A, provides a place within the Mathematics Department where students could work together and receive help as needed. Along with individual assistance, the Math Tutoring Center also conducts supplemental instruction sessions for Math 121, 122, 125 and 126. In these sessions, upperclassmen work with small groups of students on the class material.

FACULTY

Dong Hoon Lee, Ph.D. (Tulane University)

Alejandro D. de Acosta, Ph.D. (University of California, Berkeley)

David Gurarie, Ph.D. (Hebrew University, Jerusalem, Israel)

Michael G. Hurley, Ph.D. (Northwestern University)

Steven H. Izen, Ph.D. (Massachusetts Institute of Technology)

Peter Kotelenez, Ph.D. (Universitat Bremen)

Joel Langer, Ph.D. (University of California, Santa Cruz)

Marshall J. Leitman, Ph.D. (Brown University)

Arthur E. Obrock, Ph.D. (Washington University )

David A. Singer, Ph.D. (University of Pennsylvania)

Stanislaw J. Szarek, Ph.D. (Mathematical Institute, Polish Academy of Science)

Charles Wells, Ph.D. (Duke University)

Elisabeth Werner, Ph.D. (Universite Pierre et Marie Curie, Paris IV)

Ta-Sun Wu, Ph.D. (Tulane University)

ASSOCIATE FACULTY

Kenneth Loparo, Ph.D. (Case Western Reserve University)

Mihajilo D. Mesarovic, Ph.D. (Serbian Academy of Science)

ADJUNCT FACULTY

Margaret Robinson, M.A. (State University of New York at Stony Brook)

UNDERGRADUATE PROGRAMS

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.

Degree Requirements

Bachelor of Arts Degree in Mathematics

(1) Mathematics Requirements

The B.A. degree in Mathematics requires at least 38 hours of mathematics courses, including

A 3-credit hour course in computer science (CMPS 131 or other approved course). High school teaching certification is available in the B.A. program in mathematics through a joint program with John Carroll University. The requirements are:

Students interested in this program should contact the director of teacher certification for further information about eligibility and requirements.

Bachelor of Science in Mathematics Degree

(1) Mathematics Requirements

The B.S. degree in Mathematics requires at least 50 hours of mathematics courses, including

(2) Non-mathematics Requirements

The B.S. degree in mathematics requires the following non-mathematics courses:

Bachelor of Science in Applied Mathematics Degree

The B.S. degree in Applied Mathematics requires at least 50 hours of mathematics and related subjects, in addition to a professional core that is specific to the area of application in which the student is interested. A student in this degree program must design a program of study (called a "track") in consultation with his or her academic advisor. This program of study must explicitly list the technical electives and the professional core in the area of application. Some of the tracks offer the possibility of an integrated five year study leading to a B.S. in Applied Mathematics and an M.S. in the area of application. Currently there are four such tracks: Computing and Information Science, Operations Research, Systems Engineering - Systems, Systems Engineering - Control Theory. The general academic requirements for Integrated B.S./M.S. programs must be followed. (Since the graduate courses required for the M.S. degree are determined by the respective department, each student in the dual-degree program should have a secondary advisor in that department, starting no later than the junior year, and such consult with this advisor concerning requirements for the M.S. degree.)

Listed below are specific technical electives of the four B.S./M.S. tracks.

COMPUTING AND INFORMATION SCIENCES TRACK

Four of the following courses, of which at least two must be MATH courses. At least one numerical analysis course must be chosen. MATH 410, MATH/CMPS 343, MATH 413/OPRE 514, MATH 431, PHIL 306, CMPS 454, CMPS 541.

OPERATIONS RESEARCH TRACK

Four of the following courses, at least two of which must be MATH courses. MATH 331, MATH 423, MATH 491, MATH 492, MATH 495, MATH 487, MATH 489, STAT 403, STAT 406, STAT 408, STAT 484.

SYSTEMS ENGINEERING - CONTROL THEORY TRACK

Four of the following MATH courses. 401, 402, 410, 413, 415, 423, 428, 431, 435, 436, 445, 465, 491,

SYSTEMS ENGINEERING - SYSTEMS TRACK

Four of the following MATH courses 401, 410, 413, 423, 431, 435, 445, 447, 469, 491, 495.

The professional core requires 12 credit hours of course work specific to the area of application. Listed below are the professional cores for the four B.S./M.S. tracks.

COMPUTING AND INFORMATION SCIENCES TRACK

Four of the following courses: EEAP 282, ECMP 280, ECMP 333, ECMP 337, ECMP 338.

OPERATIONS RESEARCH TRACK

MATH 380, OPRE 428, OPRE 411, and one of MATH 413, 487, 489 or another approved 400-level course.

Systems Engineering - Control Theory Track

The following four ESYS courses: 212, 304, 313, 306.

SYSTEMS ENGINEERING - SYSTEMS TRACK

MATH 380, ESYS 315, ESYS 416, ECMP 251.

Non-mathematics Requirements

The B.S. degree in applied mathematics requires the following non-mathematics courses.

Non-Major Undergraduate Programs in Mathematics

Minor in Mathematics - All undergraduates

A minor in mathematics is available to all CWRU undergraduates. It consists of 17 credit hours of approved course work in mathematics. No more than two courses can be used to satisfy both minor requirements and the requirements of the student's major field (meaning departmental degree requirements, including departmental technical electives and common course requirements of the student's school). The 17 hours must be from among the following MATH courses: 121 or 123 or 125, 122, or 124 or 126, 223 or 227, 224 or 228, 150(*), 201(**), 301, 302, 303, 304, 307, 308(**), 321, 322, 323, 324, 331, 338, 343, 345, 380, or any 400-level(**) MATH course.

High School Teaching Certification

This program is described in the description of the mathematics B.A. degree given above.

GRADUATE PROGRAMS

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.

Doctor of Philosophy and Master of Science in 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 advisor.

RESEARCH AND TEACHING

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)

UNDERGRADUATE COURSES

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. Three years of high school mathematics required.

MATH 105, Mathematics of Finance I, 3

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. Three years of high school mathematics required.

MATH 106, Mathematics for 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 and polynomials, limits, derivatives of algebraic and trigonometric functions. Definite integral, antiderivatives, fundamental theorem of calculus, change of variables. Three and one half years of high school mathematics required.

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. Department consent required.

MATH 124, Calculus II, 4

Review of differentiation. Techniques of integration, and applications of the definite integral. Parametric equations and polar coordinates. Taylor's theorem. Sequences, series, power series. Complex arithmetic. Introduction to multivariable calculus.

Prerequisite: MATH 123

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 sciences. Students planning to take more than two semesters of introductory mathematics should take MATH 121. Three and one half years of high school mathematics required.

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

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. Should be taken in freshman year to count toward a major in mathematics. Three and one half years of high school mathematics required.

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. Less theoretical than MATH 307. May not be taken for credit by mathematics majors. Only one of MATH 201, MATH 307, or MATH 470 may be taken for credit.

Prerequisite: MATH 122 or MATH 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 derivative, 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, Laplace transforms, 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

MATH 227, Calculus III, 3

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: MATH 124

MATH 228, Differential Equations, 3

Elementary ordinary differential equations: first order equations; linear systems; applications; numerical methods of solution.

Prerequisite: MATH 227

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. Content varies. Students may take this course for credit up to four times.

MATH 303, Elementary Number Theory, 3

Primes and divisibility, theory of congruencies, 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: MATH 122

MATH 304, Discrete Mathematics, 3

A general introduction to basic mathematical terminology and the techniques of abstract mathematics in the context of discrete mathematics. Topics introduced are mathematical reasoning, Boolean connectives, deduction, mathematical induction, sets, functions and relations, algorithms, graphs, combinatorial reasoning.

Prerequisite: CMPS 131 and MATH 122 or MATH 126

MATH 307, Introduction to Abstract Algebra I, 3

First semester of an integrated, two-semester theoretical course in abstract and linear algebra, studied on an axiomatic basis. The major algebraic structures studied are groups, rings, fields, modules, vector spaces, and inner product spaces. Topics include homomorphisms and quotient structures, the theory of polynomials, canonical forms for linear transformations and the principal axis theorem. This course is required of all students majoring in math. Students who take MATH 307 and 308 may not take MATH 201 or 470.

Prerequisite: MATH 122

MATH 308, Introduction to Abstract Algebra II, 3

Continuation of MATH 307.

Prerequisite: MATH 307

MATH 321, Fundamentals of Analysis I, 3

Abstract mathematical reasoning in the context of analysis in real n-dimensional space. Formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Point-set topology in metric spaces with attention to n-dimensional space; completeness, compactness, connectedness, and continuity of functions. Required for all mathematics majors.

Prerequisite: MATH 223

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, series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration

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, Computational Linear Algebra, 3

Techniques of computational linear algebra for applied mathematicians, engineers, and scientists.

Prerequisite: MATH 201 and 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 223

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 and CMPS 340

MATH 345, Introduction to Applied Mathematics, 3

Mathematical formulation of problems, development of various methods of solution, and interpretation of results, boundary value problems. Sturm-Liouville problems, complex analysis, transform methods.

Prerequisite: MATH 224

MATH 380, Introduction to Probability, 3

Combinatorial analysis. Permutations and combinations. Axioms of probability. Sample space and events. Equally likely outcomes. Conditional probability. Bayes' formula. Independent events and trials. Discrete random variables, probability mass functions. Expected value, variance. Bernoulli, binomial, Poisson, geometric, negative binomial random variables. Continuous random variables, density functions. Expected value and variance. Uniform, normal, exponential, Gamma random variables. The De Moivre-Laplace limit theorem. Joint probability mass functions and densities. Independent random variables and the distribution of their sums. Covariance. Conditional expectations and distributions (discrete case). Moment generating functions. Law of large numbers. Central limit theorem. Additional topics (time permitting): the Poisson process, finite state space Markov chairs, entropy.

Prerequisite: MATH 122 or MATH 126

GRADUATE COURSES

MATH 401, Abstract Algebra I, 3

Basic properties of groups, rings, modules and fields. Isomorphism theorems for groups; Sylow theorem; nilpotency and solvability of groups; Jordan-Holder theorem; 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, bilinear and quadratic forms; field extensions; fundamental theorem of Galois theory, solving equations by radicals.

Prerequisite: MATH 308

MATH 402, Abstract Algebra II, 3

A continuation of MATH 401.

Prerequisite: MATH 401

MATH 405, Linear Analysis, 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;

Prerequisite: MATH 224 and MATH 201 or MATH 470

MATH 406, Mathematical Logic and Model Theory, 3

A study of formal logical systems and their models. Propositional logic and quantification. First order theories; consistency, compactness, and the Lowenheim theorem. (cross-listed as PHIL 406)

MATH 407, Category Theory, 3

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 foundation for various branches of mathematics. Topics covered include categories, functors, Cartesian closed categories, toposes and category theoretic axioms for a category of sets.

Prerequisite: PHIL 306 or MATH 308 or MATH 304

MATH 410, Automata and Formal Languages, 3

(Also listed as CMPS 443). 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

MATH 413, Graph Theory, 3

(Also listed as OPRE 514). Building blocks of a graph, trees, connectedness, transversability, matching, coverings, planarity, and NP-complete problems; various applications and algorithms.

Prerequisite: MATH 201 or MATH 470 or MATH 308

MATH 415, Group Representation Theory, 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

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 Mathematics.)

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.)

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. Lp spaces. Lebesgue differentiation theorem in N-space.

Prerequisite: MATH 322

MATH 424, Introduction to Real Analysis II, 3

Measures on locally compact spaces. Riesz representation theorem. Elements of functional analysis. Normed linear space. 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. Application to 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; harmonic functions; Poisson's formula; Riemann mapping theorem. More theoretical and at a higher level than MATH 324.

Prerequisite: MATH 321

MATH 428, Fourier Analysis, 3

Introduction to the mathematical aspects of Fourier analysis and synthesis. Accessible to upper level undergraduates and graduate students in the sciences and engineering. Periodic functions. Fourier series. Convergence theorems. The classical sine and cosine series. General orthogonal systems. Multiple Fourier series. Applications. Fourier integrals and Fourier Transforms. L^1 and L^2 theory. Inversion theorems. Classical sine and cosine transforms. Multiple Fourier Transform. Spherical symmetry. Other important transforms. Applications. In addition to the prerequisite listed below, MATH 321 and MATH 201 are recommended.

Prerequisite: MATH 224

MATH 431, Introduction to Numerical Analysis I, 3

Numerical analysis for scientists and engineers. Floating point arithmetic and the propagation of round off errors. The direct solution of linear algebraic systems. Polynomial interpolation and splines. Methods of numerical differentiation and quadrature. Acceleration of convergence. Solution of nonlinear algebraic equations.

Prerequisite: MATH 224

MATH 432, Numerical 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. Introduction to hyperbolic system. Recommend a good working knowledge of a high-level computer language.

Prerequisite: MATH 431

MATH 434, Optimization of Dynamic Systems, 3

(Also listed as ESCI 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: ESCI 408

MATH 435, Intermediate Ordinary Differential Equations, 3

Existence, uniqueness, continuation of solutions of O.D.E. 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.

Prerequisite: MATH 224 and MATH 201 or MATH 307 or MATH 470

MATH 436, Mathematical Control Theory, 3

Basic examples. Observed linear control systems: controllability, control systems. Optimal control theory: linear quadratic, time optimal, maximum principle.

Prerequisite: MATH 435

MATH 445, Introduction to Partial Differential Equations, 3

Method of characteristics for linear and quasilinear equations. Second order equations of elliptic, parabolic, 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 and MATH 224

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 nonevolutionary types. Linear evolutionary systems; solution by iterative schemes; qualitative behavior; stability. Nonlinear evolutionary systems; Semigroups of finite difference methods for integro-differential equations. For seniors and graduate students in science, engineering, and mathematics. MATH 201 or MATH 470 recommended.

Prerequisite: MATH 224

MATH 448, Applied Partial Differential Equations, 3

Continuation of MATH 445. Linear and nonlinear partial differential equations, with emphasis on applications. Variational methods; asymptotic and perturbation methods: regular and singular perturbations; boundary layer, multiple scales, method of geometric optics and stationary phase. Applications to fluid dynamics, elasticity; optics; wave propagation. 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

MATH 452, Continuum Mechanics, 3

Kinematics of deformation. Tensors. Mathematical and physical formulation of continuum mechanics. Thermo-dynamical notions. Constitutive relations. Simple nonlinear materials with memory. Role of the classical theories of solids and fluids. Modern developments. MATH 201 or MATH 470 recommended.

Prerequisite: MATH 224

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

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

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 GaussBonnet theorem.

Prerequisite: MATH 321

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. MATH 201 or 307 recommended.

Prerequisite: MATH 321

MATH 467, Differentiable Manifolds, 3

Differentiable manifolds and structures on manifolds. Tangent and cotangent bundle; 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

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. Other topics such as regularity of solutions; Sobolev spaces; depending on audience.

Prerequisite: MATH 224

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, 307, 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

MATH 475, 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, backprojection, rho-filtered layergram, algebraic reconstruction technique (ART), and orthogonal polynomial expansions. Reconstruction from fan beam geometry limited angle techniques used in nmr; survey of applications.

Prerequisite: PHYS 431 or MATH 345 or MATH 471

MATH 487, Stochastic Processes in Engineering and Science, 3

Review of basic probability concepts. Discrete-time Markov chains. Transition probability matrices. Classification of states. Stationary distributions. Limiting behavior. Random walk; application to the gambler's ruin problem. Branching processes; application to population growth models. Examples of continuous time Markov chains. Poisson and compound Poisson processes. Birth and death processes. Limiting behavior. Renewal processes. Examples are drawn from queueing theory, reliability theory, population growth processes and other biological models.

Prerequisite: MATH 380

MATH 489, Theory of Queues, 3

The mathematical model of queues. The distribution 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.

Prerequisite: MATH 487

MATH 490, Stochastic Processes, 3

Stochastic processes at an intermediate level; more inclusive and at a higher mathematical level than MATH 487. Theory of discrete time Markov chains with countable state space. Construction of invariant measures. Coupling and limiting behavior. Continuous time Markov chains; the minimal construction, stability, the backward and forward equations, semigroup of transition matrices and their generators, stationary and limiting distributions. Renewal theory and regenerative processes. Elementary Brownian Motion theory: reflection principle, escape from a strip, Brownian motion with drift.

Prerequisite: MATH 380 and MATH 321

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 measures. Inversion, Levy's continuity theorem. Central limit theorem. Introduction to the central limit problem.

Prerequisite: MATH 322

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. Brownian motion theory (sample path properties, strong Markov property, Martingales associated to Brownian motion, functional central limit theorem).

Prerequisite: MATH 491

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

MATH 501, Topics in Algebra, 1-3

Selected topics from fields, rings, and modules.

Prerequisite: MATH 402

MATH 527, Functional Analysis, 3

Selected topics in Functional Analysis.

Prerequisite: MATH 424 and MATH 425

MATH 529, Analysis Seminar, 1-3

MATH 531, Advanced Numerical Analysis, 1-3

Special topics course.

Prerequisite: MATH 431

MATH 541, Partial Differential Equations, 3

Special topics course.

Prerequisite: MATH 424 and MATH 445

MATH 553, Applied Mathematics Seminar, 1-3

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.

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 MATH 492

MATH 592, Stochastic Processes II, 3

Einstein-Smoluchowski and Ornstein-Uhlenbeck models of Brownian motion. Stochastic integration. The Ito and Stratonovich integrals. The stochastic Ito calculus and stochastic differential equations (SDE's). Stochastic flows in a random environment. Approximation of SDE's by random ordinary differential equations. SDE's as diffusion processes and partial differential equations. Comparison theorems and positivity results for SDE's. Stationary solutions and stability properties of SDE's.

Prerequisite: MATH 591

MATH 595, Special Topics in Probability, 1-3

Topics of current research interest in probability and stochastic processes. Topics may change from year to year.

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.), 1-36

MATH 701, Dissertation (Ph.D.), 1-36

(*)To count toward a minor in mathematics, MATH 150 must be taken in the freshman or sophomore year.

(**)Only one of 201, 308, 470 can be taken for credit.

BACHELOR OF ARTS DEGREE
MAJOR IN MATHEMATICS

BACHELOR OF SCIENCE IN MATHEMATICS DEGREE

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.

"Approved electives" must be approved by the student's major advisor 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.

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 Mathematician's 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 the honors sequence, PHYS 123, 124, 223, in place of PHYS 121, 122, 221.

d) These two courses must be one of the following sequences: ASTR 201-202, CHEM 105-106, CHEM 107-108, GEOL 110 and one of GEOL 115, 210

e) BIOC 314, BIOL 111, CHEM 113, GEOL 119, PHYS 203 are appropriate.

BACHELOR OF SCIENCE IN APPLIED MATHEMATICS

Total hours to graduate between 125-128 depending on option.

a A suitable open elective is MATH 150, Mathematics from a Mathematician's 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 the honors sequence, PHYS 123, 124, 223, in place of PHYS 121, 122, 221.
d These two courses must be one of the following sequences: ASTR 201-202, CHEM 105-106, CHEM 107-108, GEOL 110 and one of GEOL 115, 210
e BIOC 314, BIOL 111, CHEM 113, GEOL 119, PHYS 203 are appropriate.

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