220 Yost Hall
Phone 216-368-2880; Fax 216-368-5163
James C. Alexander, Chair

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.

James C. Alexander, Ph.D. (Johns Hopkins University)
Levi Kerr Professor and Chair
Dynamics, applied mathematics

Christopher Butler, M.S. (Case Western Reserve University)
Instructor
Teaching of mathematics

Daniela Calvetti, Ph.D. (University of North Carolina)
Professor
Numerical linear algebra, numerical methods for image processing, orthogonal polynomials and quadruture rules, large-scale eigenvalue computations.

Teresa Contenza, Ph.D. (University of Kentucky)
Lecturer
Mathematics

Peter Garfield, Ph.D. (University of Washington)
Visiting Professor
Mathematics

David Gurarie, Ph.D. (Hebrew University, Jerusalem, Israel)
Professor
Mathematical physics; differential equations; geophysical modeling; harmonic analysis

Michael G. Hurley, Ph.D. (Northwestern University)
Professor
Differentiable dynamical systems, dynamics of cellular automata and dynamics of numerical methods.

Steven H. Izen, Ph.D. (Massachusetts Institute of Technology)
Associate Professor
Mathematics of imaging; image reconstruction

Peter Kotelenez, Ph.D. (Universitat Bremen)
Professor
Probability theory, stochastic processes, particle systems

Joel Langer, Ph.D. (University of California, Santa Cruz)
Professor
Differential geometry; geometric variational problems and geometric evolution equations.

Marshall J. Leitman, Ph.D. (Brown University)
Professor
Applied Mathematics: Continuum physics, Integral equations, Functional Analysis, Mechanics of Materials.

Elizabeth Meckes, Ph.D. (Stanford Univeristy)
Assistant Professor
Probability

Mark Meckes, Ph.D. (Case Western Reserve Univeristy)
Assistant Professor
Functional Analysis and Convexity Theory.

David A. Singer, Ph.D. (University of Pennsylvania)
Professor
Differential geometry; dynamical systems and variational problems.

Stanislaw J. Szarek, Ph.D. (Mathematical Institute, Polish Academy of Science)
Professor
Functional Analysis and its Applications, Asymptotic Geometric Analysis.

Peter J. Thomas, Ph.D. (University of Chicago)
Professor
Applied Mathematics; Computational Biology

Elisabeth Werner, Ph.D. (Universite Pierre et Marie Curie, Paris IV)
Associate Professor
Functional analysis, convexity theory and affine differential geometry.

ASSOCIATE FACULTY

Colin McLarty, Ph.D. (Case Western Reserve University)
Associate Professor of Philosophy
Logic, philosophy of mathematics

Marvin E. Goldstein, Ph.D. (University of Michigan)
Adjunct Professor; Chief Scientist, NASA-Lewis Research Center
Fluid mechanics, heat transfer

Christophe Geuzaine, Ph.D. (University of Liege, Belgium)
Adjunct Professor
Mathematics

Stewart Robinson, Ph.D. (Duke University)
Adjunct Professor; Clin. Assoc. Professor
Mathematics

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.

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

(2) Non-mathematics Requirements
A 3-credit hour course in computer science (ENGR 131 or other approved course).

Teaching Certification
High school teaching certification is available in the B.A. program in mathematics through a joint program with John Carroll University. The requirements are:

(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 Mathematics and Physics
Students with strong interests in both Mathematics and Physics may be interested in the joint Bachelor of Science degree in Mathematics and Physics, which is described under the Department of Physics in this Bulletin.

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.)

A minor in mathematics is available to all University 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, 327, 330, 338, 343, 345, 361, 363, 380, or any 400-level MATH course (only one of 201, 308).

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

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 advisor.

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.

MATH 110. Introduction to Mathematical Communication and Software (1)
Mathematical text editors. Mathematical composition and exposition. Posting mathematical material on the Web. Basics of computer symbolic manipulation (Mathematica). Computer vector/matrix manipulation and applications (MATLAB). Basic computer statistical methods (Minitab). Integration of output from computer calculations into text.

MATH 120. 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. Not open to students with credit for MATH 121 or MATH 125. Prereq: Three years of high school mathematics.

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. Prereq: 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. Prereq: 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. Prereq: Placement by the department.

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. Prereq: MATH 123 and placement by the 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 sciences. Students planning to take more than two semesters of introductory mathematics should take MATH 121. Prereq: Three and one half years of high school mathematics.

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. Prereq: 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. Prereq: Three and one half 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. Less theoretical than MATH 307. May not be taken for credit by mathematics majors. Only one of MATH 201 or MATH 307 may be taken for credit. Prereq: 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. Prereq: 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, numerical methods of solution. Prereq: MATH 223.

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. Prereq: MATH 124 or placement by department.

MATH 228. Differential Equations (3)
Elementary ordinary differential equations: first order equations; linear systems; applications; numerical methods of solution. Prereq: MATH 227.

MATH 234. Differential Equations and Dynamical Systems (3)
An introductory course in discrete and continuous dynamics (difference and differential equations). One dimensional differential equations: dynamics; linear equations, separable equations; numerical methods. Systems of differential equations in two dimensions: dynamics of autonomous systems, numerical methods, solution of constant coefficient linear systems, with and without forcing. Laplace transforms and convolution. Discrete dynamics; introduction to chaos, numerical methods as difference equations. Linear difference equations in one and two dimensions, z-transform, convolution. Prereq: MATH 223.

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). Prereq: 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. Prereq: 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 mathematics. Only one of MATH 201 or MATH 307 may be taken for credit. Prereq: MATH 122.

MATH 308. Introduction to Abstract Algebra II (3)
Continuation of MATH 307. Prereq: MATH 307.

MATH 319. Applied Probability and Stochastic Processes for Biology (3)
Applications of probability and stochastic processes to biological systems. Mathematical topics will include: introduction to discrete and continuous probability spaces (including numerical generation of pseudo random samples from specified probability distributions), Markov processes in discrete and continuous time with discrete and continuous sample spaces, point processes including homogeneous and inhomogeneous Poisson processes and Markov chains on graphs, and diffusion processes including Brownian motion and the Ornstein- Uhlenbeck process. Biological topics will be determined by the interests of the students and the instructor. Likely topics include:  stochastic ion channels, molecular motors and stochastic ratchets,  actin and tubulin polymerization, random walk models for neural spike trains, bacterial chemotaxis, signaling and genetic regulatory networks, and stochastic predator-prey dynamics. The emphasis will be on practical simulation and analysis of stochastic phenomena in biological systems. Numerical methods will be developed using both MATLAB and the R statistical package. Student projects will comprise a major part of the course. Prereq: MATH 224 or MATH 228 or BIOL 300 or BIOL 306.  Cross-listed as BIOL 319, BIOL 419, EECS 319

MATH 321. Fundamentals of Analysis I (3)
Abstract mathematical reasoning in the context of analysis in Euclidean space. Introduction to formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Required for all mathematics majors. Prereq: MATH 223.

MATH 322. Fundamentals of Analysis II (3)
Continuation of MATH 321. Point-set topology in metric spaces with attention to n-dimensional space; completeness, compactness, connectedness, and continuity of functions. Topics in sequences, series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration. Required for all mathematics majors. Prereq: 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. Prereq: 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. Prereq: MATH 224.

MATH 327. Convexity and Optimization (3)
Introduction to the theory of convex sets and functions and to the extremes in problems in areas of mathematics where convexity plays a role. Among the topics discussed are basic properties of convex sets (extreme points, facial structure of polytopes), separation theorems, duality and polars, properties of convex functions, minima and maxima of convex functions over convex set, various optimization problems. Prereq: MATH 223 or consent.

MATH 330. Scientific Computing: Fundamentals and Applications (3)
An introductory survey to Scientific Computing, from principles to applications. Topics include accuracy and efficiency, conditioning and stability, numerical solution of linear and nonlinear systems, optimization, interpolation, quadrature rules, numerical solutions of ODEs and PDEs. Coreq: MATH 224.

MATH 338. Introduction to Dynamical Systems (3)
Nonlinear discrete dynamical systems in one and two dimensions. Chaotic dynamics, elementary bifurcation theory, hyperbolicity, symbolic dynamics, structural stability, stable manifold theory. Prereq: MATH 223.

MATH 343. Theoretical Computer Science (3)
Introduction to mathematical logic, different classes of automata and their correspondence to different classes of formal languages, recursive functions and computability, assertions and program verification, denotational semantics. MATH/EECS 343 and MATH 410 cannot both be taken for credit. Prereq: MATH 304 and EECS 340. Cross-listed as EECS 343.

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. Prereq: MATH 224.

MATH 350. Domain Theoretic Methods for Artificial Intelligence (3)
Resolution for propositional logic and completeness via Zorn’s Lemmq, Domain theory and topology through three-value logic. Default reasoning and extensions. Clausal logic for Scott domains and Smyth power domains. Power defaults theory and the semantics of nonmonotonic reasoning and disjunctive logic programming. Prereq: EECS 343, EECS 391, MATH 307, or PHIL 306. Cross-listed as EECS 358.

MATH 361. Geometry I (3)
An introduction to the various two-dimensional geometries, including Euclidean, spherical, hyperbolic, projective, and affine. The course will examine the axiomatic basis of geometry, with an emphasis on transformations. Topics include the parallel postulate and its alternatives, isometries and transformation groups, tilings, the hyperbolic plane and its models, spherical geometry, affine and projective transformations, and other topics. We will examine the role of complex and hypercomplex numbers in the algebraic representation of transformations. The course is self-contained, however.Prereq: MATH 224.

MATH 363. Knot Theory (3)
An introduction to the mathematical theory of knots and links, with emphasis on the modern combinatorial methods. Reidemeister moves on link projections, ambient and regular isotopies, linking number tricolorability, rational tangles, braids, torus knots, seifert surfaces and genus, the knot polynomials (bracket, X, Jones, Alexander, HOMFLY), crossing numbers of alternating knots and amphicheirality. Connections to theoretical physics, molecular biology, and other scientific applications will be pursued in term projects, as appropriate to the background and interests of the students. Prereq: MATH 223.

MATH 378: Computational Neuroscience (3)
Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems. Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural circuits, as well as a cable theory, passive and active compartmental modeling, numerical integration methods, models of plasticity and learning, models of brain systems, and their relationship to artificial and neural networks. Term project required. Students enrolled in MATH 478 will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Prereq: MATH 223 & MATH 224 or BIOL 300 & BIOL 306, or consent of department. Cross-listed as EBME 478, EECS 478, MATH 478, NEUR 478

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 chains, entropy. Prereq: MATH 122 or MATH 126.

MATH 381. Introduction to Mathematical Methods in Finance (3)
Mathematical finance in discrete time. Single period market models. Arbitrage. Risk-neutral valuation of contingent claims. Complete markets. Summary of results from probability theory and stochastic processes in discrete time. Conditional expectation. Discrete parameter martingales. Multiperiod market modes. Equivalent martingale measures. Risk-neutral valuation. Hedging strategies. Complete markets. The Cox-Ross-Rubinstein model. European options. American options. The Black-Scholes model. Binomial approximation. The pricing formula for European call options. Prereq: MATH 380.

MATH 399. Special Topics (3)
Special Topics in Mathematics

MATH 401. Abstract Algebra I (3)
Basic properties of groups, rings, modules and fields. Isomorphism theorems for groups; Sylow theorem; nil potency 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. Prereq: MATH 308.

MATH 402. Abstract Algebra II (3)
A continuation of MATH 401. Prereq: MATH 401.

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 Skolem theorem. Cross-listed as PHIL 406.

MATH 408. Introduction to Cryptology (3)
Introduction to the mathematical theory of secure communication. Topics include: classical cryptographic systems; one-way and trapdoor functions; RSA, DSA, and other public key systems; Primality and Factorization algorithms; birthday problem and other attack methods; elliptic curve cryptosystems; introduction to complexity theory; other topics as time permits. Prereq: MATH 303.

MATH 410. Automata and Formal Languages (3)
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. MATH 410 and MATH/EECS 343 cannot both be taken for credit. Prereq: MATH 304. Cross-listed as EECS 440.

MATH 413. Graph Theory (3)
Building blocks of a graph, trees, connectedness, transversability connectedness, transversability, matching, coverings, planarity, and NP-complete problems; various applications and algorithms. Prereq: MATH 201 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. Prereq: 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.) Coreq: 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.) Prereq: MATH 321.

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. Prereq: 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. Prereq: 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. Prereq: MATH 322.

MATH 427. Convexity and Optimization (3)
(See MATH 327.) Cross-listed as OPRE 427.

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. Prereq: MATH 224.

MATH 431. Introduction to Numerical Analysis I (3)
Numerical linear algebra for scientists and engineers. Matrix and vector norms, computer arithmetic, conditioning and stability, orthogonality. Least squares problems: QR factorization, normal equations and Singular Value Decomposition. Direct solution of linear system: Gaussian elimination and Cholesky factorization. Eigenvalues and eigenvectors: the QR algorithm, Rayleigh quotient, inverse iteration. Introduction to iterative methods. Students will be introduced to MATLAB. Prereq: MATH 201 or MATH 308.

MATH 432. Numerical Differential Equations (3)
Numerical solution of differential equations for scientists and engineers. Solution of ordinary differential equations by multistep and single step methods. Stability, consistency, and convergence. Stiff equations. Finite difference schemes. Introduction to the finite element method. Introduction to multigrid techniques. The diffusion equation: numerical schemes and stability analysis. Introduction to hyperbolic equations. MATLAB will be used in this course.

MATH 433. Numerical Solutions of Nonlinear Systems and Optimization (3)
The course provides an introduction to numerical solution methods for systems of nonlinear equations and optimization problems. The course is suitable for upper-undergraduate and graduate students with some background in calculus and linear algebra. Knowledge of numerical linear algebra is helpful. Among the topics which will be covered in the course are Nonlinear systems in one variables; Newton’s method for nonlinear equations and unconstrained minimization; Quasi-Newton methods; Global convergence of Newton’s methods and line searches; Trust region approach; Secant methods; Nonlinear least squares. Prereq: MATH 223, MATH 201, MATH 431 or permission.

MATH 434. Optimization of Dynamic Systems (3)
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. Sufficient conditions for optimality. Singular controls. Computational aspects. Selected applications. Cross-listed as EECS 421.

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. Prereq: MATH 201 and 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. Prereq: MATH 445.

MATH 450. Domain Theoretic Methods for Artificial Intelligence (3)
Resolution for propositional logic and completeness via Zorn’s Lemmq, Domain theory and topology through three-value logic. Default reasoning and extensions. Clausal logic for Scott domains and Smyth power domains. Power defaults theory and the semantics of nonmonotonic reasoning and disjunctive logic programming. Prereq: EECS 343, EECS 391, MATH 307, or PHIL 306. Cross-listed as EECS 459X.

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 recommended. Prereq: MATH 224.

MATH 460. Mathematics and the Imagination (3)
This course explores mathematical ideas in geometry, algebra, and combinatorics relating to content areas in the secondary school curriculum. The course is structured around a series of problems and projects not generally covered in the undergraduate curriculum. This course is designed for present and future mathematics teachers in secondary schools. It is offered as an intensive, three-week seminar. Requirements for the class include daily reading assignments and problems taken from the readings. Considerable time will be devoted to group work. Each student will be required to prepare a report and make a 30-minute presentation to the class on a topic relevant to the materials developed in the course.

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. Prereq: 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. Prereq: MATH 461.

MATH 465. Differential Geometry (3)
Manifolds and differential geometry. Vector fields; Riemannian metrics; curvature; intrinsic and extrinsic geometry of surfaces and curves; structural equations of Riemannian geometry; the Gauss-Bonnet theorem. Prereq: 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. Prereq: 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 existence and regularity of solutions; Sobolev spaces; depending on audience. Prereq: 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 MATH 471 for credit. Prereq: 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. Prereq: PHYS 431 and MATH 345 or MATH 471.

MATH 478: Computational Neuroscience (3)
Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems. Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural circuits, as well as a cable theory, passive and active compartmental modeling, numerical integration methods, models of plasticity and learning, models of brain systems, and their relationship to artificial and neural networks. Term project required. Students enrolled in MATH 478 will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Prereq: MATH 223 & MATH 224 or BIOL 300 & BIOL 306, or consent of department. Cross-listed as EBME 478, EECS 478, MATH 478, NEUR 478

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 queuing theory, reliability theory, population growth processes and other biological models. Prereq: MATH 380.

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. Prereq: MATH 423.

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). Prereq: 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. Prereq: MATH 321 or consent.

MATH 499. Special Topics (3)
Special topics in mathematics.

MATH 501. Topics in Algebra (3)
Selected topics from fields, rings, and modules. Prereq: MATH 402.

MATH 527. Functional Analysis (3)
Selected topics in Functional Analysis. Prereq: MATH 424 and MATH 425.

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 601. Reading and Research Problems (1-18)
Presentation of individual research, discussion, and investigation of research papers in a specialized field of mathematics.

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Hours required for graduation: 126.

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.

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 120, 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.