sists of examinations on three different subjects. All examinations are general proficiency examinations which may or may not be connected to specific courses. The topics for each subject are spelled out in a syllabus, periodically updated, which is available to the student. Students are expected to take the qualifying examination by the end of the second year of study and to successfully pass all parts of it by the beginning of their sixth semester in the Ph.D. program. Each track requires examination in a different set of subjects. More specifically:
  • Write an acceptable thesis that constitutes an original contribution to mathematical knowledge. It is the responsibility of the student to find a thesis advisor who is willing to help plan a program and guide his or her research. This should be done immediately after passing the qualifying examination. A copy of a student’s thesis is to be available no later than 10 days prior to the final oral examination (see below), and the student is required to deliver an expository lecture on the subject of his or her thesis sometime prior to the final oral examination. This lecture is open to all students and faculty.
  • Pass a final oral examination consisting of a defense of the thesis. The examination committee, which consists of not fewer than four members of the faculty, including one whose appointment is outside the mathematics department, is responsible for certifying that the material presented in the thesis meets acceptable scholarly standards. The examination may also include an inquiry into the student’s competence in the major and related fields. All faculty members are welcome to attend.
  • Course work requirements
    Mathematics Track: A student in the traditional mathematics program must demonstrate knowledge of the basic concepts and techniques of algebra, analysis (real and complex), and topology. This must be done by taking all courses in the three basic areas: abstract algebra (MATH 401-MATH 402), analysis (MATH 423-MATH 424 and MATH 425), and topology (MATH 461). In addition, the student is required to take a minimum of 18 credit hours of approved course work.


    A student with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, must take 18 credit hours of approved courses. The graduate committee will determine which of the specific course work requirements stated above have been satisfied by the master’s course work.


    Applied Mathematics Track: A student in the applied mathematics track must demonstrate knowledge of scientific computing, mathematical modeling, and differential equations. This may be done by taking:

    In addition, a student in this track must take at least 24 credit hours of approved courses, which must include at least 9 credit hours of courses offered outside the Department of Mathematics, and at least 9 credit hours offered by the Department of Mathematics.


    A student with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, must take 18 credit hours of approved courses, which must include at least 6 credit hours of courses offered outside the Department of Mathematics and at least 9 credit hours offered by the Department of Mathematics. The graduate committee will determine which of the specific course work requirements stated above have been satisfied by the master’s course work.


    Sample study plans for students with concentrations in scientific computing, imaging, mathematical biology, and stochastics follow. The graduate committee will entertain ideas for other serious study plans or qualifying exam subjects in addition to the most common variants specifically suggested.


    Scientific Computing Concentrations

    MATH 431

    Application area

    MATH 432

    Application area

    MATH 433

    Application area

    MATH 441

    Application area

    MATH 445

    Application area

    MATH 439/440

    Application area

    MATH 448

    Application area

    MATH 487

    Application area

    MATH 449/469/478

    Application area

    Imaging Concentrations

    MATH 431

    PHYS 431

    MATH 432

    PHYS 460

    MATH 433

    EBME 410

    MATH 441

    PHYS 431

    MATH 428

    PHYS 460

    MATH 475

    EBME 410

    MATH 445

    PHYS 431

    MATH 439/440

    PHYS 460

    MATH 444

    EBME 410

    Life Science Concentrations

    MATH 431

    Application area

    MATH 432

    Application area

    MATH 433

    Application area

    MATH 441

    Application area

    MATH 449

    Application area

    MATH 478

    Application area

    MATH 445

    Application area

    MATH 439/487

    Application area

    MATH 440

    Application area

    Stochastics Concentrations

    MATH 431

    Application area

    MATH 423

    Application area

    MATH 441

    Application area

    MATH 481

    Application area

    MATH 424

    Application area

    MATH 491

    Application area

    MATH 487

    Application area

    MATH 469

    Application area

    MATH 492

    Application area

     

    Ph.D. students entering with a bachelor’s degree are also subject to the breadth requirements for students in the program for the M.S. degree in applied mathematics.


    Petitions


    Any exceptions to departmental regulations or requirements must have the formal approval of the graduate committee of the department. Such exceptions are to be sought by a written petition, approved by the student’s advisory committee or thesis advisor, to the graduate committee.


    Any exception to university rules and regulations must be approved by the dean of graduate studies. Such exceptions are to be sought by presenting a written petition to the graduate committee for departmental endorsement and approval prior to forwarding the petition to the dean.


    Course Descriptions


    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 or MATH 123 or MATH 126.


    MATH 123. Calculus I (4)
    Limits, continuity, derivatives of algebraic and transcendental functions, including applications, basic properties of integration. Techniques of integration and applications. Students must have 31/2 years of high school mathematics.


    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 or placement by department.


    MATH 125. Math and Calculus Applications for Life, Managerial, and Social Sci 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. Math and Calculus Applications for Life, Managerial, and Social Sci 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 121 or MATH 123 or 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 308. May not be taken for credit by mathematics majors. Only one of MATH 201 or MATH 308 may be taken for credit.
    Prereq: MATH 122 or MATH 124 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 or MATH 124.


    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 or MATH 227.


    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 the department.


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


    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. Departmental Seminar (3)
    A seminar devoted to understanding the formulation and solution of mathematical problems. SAGES Department Seminar. Students will investigate, from different possible viewpoints, via case studies, how mathematics advances as a discipline--what mathematicians do. The course will largely be in a seminar format. There will be two assignments involving writing in the style of the discipline. Enrollment by permission (limited to majors depending on demand).
    SAGES Dept Seminar


    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 or MATH 124.


    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.
    Offered as EECS 302 and MATH 304.
    Prereq: MATH 122 or MATH 124 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 or MATH 124.


    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.
    Offered as BIOL 319, EECS 319, MATH 319, BIOL 419, EBME 419, and PHOL 419.
    Prereq: MATH 224 or MATH 228 or BIOL 300 or BIOL 306.


    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. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
    Offered as MATH 321 and MATH 421.
    Prereq: MATH 223 or MATH 227.


    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. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
    Offered as MATH 322 and MATH 422.
    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 or MATH 228.


    MATH 326. Geometry and Complex Analysis (3)
    The theme of this course will be the interplay between geometry and complex analysis, algebra and other fields of mathematics. An effort will be made to highlight significant, unexpected connections between major fields, illustrating the unity of mathematics. The choice of text(s) and syllabus itself will be flexible, to be adapted to the range of interests and backgrounds of pre-enrolled students. Possible topics include: the Mobius group and its subgroups, hyperbolic geometry, elliptic functions, Riemann surfaces, applications of conformal mapping, and potential theory in classical physical models.
    Offered as MATH 326 and MATH 426.
    Prereq: MATH 324.


    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.
    Offered as MATH 327, MATH 427, and OPRE 427.
    Prereq: MATH 223 or MATH 227.


    MATH 330. Introduction of Scientific Computing (3)
    Among the topics which will be covered in the course are solutions of linear systems and least squares, approximation and interpolation, solution of nonlinear systems, numerical integration and differentiation, and numerical solution of differential equations. Projects where the numerical methods are used to solve problems from various application areas will be assigned throughout the semester.
    Prereq or Coreq: MATH 224 or MATH 228.


    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 or MATH 227.


    MATH 342. Introduction to Research in Mathematical Biology (1)
    The purpose of this seminar is to introduce students to some of the research being done at Case Western Reserve that explores questions at the intersection of mathematics and biology. Students will explore roughly five research collaborations, spending two weeks with each research group. In the first three classes of each two-week block, students will read and discuss relevant papers, guided by members of that research group, and the two-week period will culminate in a talk in which a member of the research group will present a potential undergraduate project in that area. After the final group’s talk, students will divide themselves into groups of two to four people and choose one project for further exploration. Together, they will write up this project as a research proposal, introducing the problem, explaining how it connects to broader scientific questions, and outlining the proposed work. It is expected that students will use the associated research group as a resource, but the proposal should be their own work. Students will submit a first draft, receive feedback, and then submit a revised draft.
    Offered as BIOL 309 and MATH 342.


    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.
    Offered as EECS 343 and MATH 343.
    Prereq: MATH 304 and EECS 340.


    MATH 351. Senior Project for the Mathematics and Physics Program (2)
    A two-semester course (2 credits per semester) in the joint B.S. in Mathematics and Physics program. Project based on numerical and/or theoretical research under the supervision of a mathematics faculty member, possibly jointly with a faculty member from physics. Study of the techniques utilized in a specific research area and of recent literature associated with the project. Work leading to meaningful results which are to be presented as a term paper and an oral report at the end of the second semester. Supervising faculty will review progress with the student on a regular basis, including detailed progress reports made twice each semester, to ensure successful completion of the work.
    SAGES Senior Cap


    MATH 352. Mathematics Capstone (3)
    Mathematics Senior Project. Students pursue a project based on experimental, theoretical or teaching research under the supervision of a mathematics faculty member, a faculty member from another Case Western Reserve department or a research scientist or engineer from another institution. A departmental Senior Project Coordinator must approve all project proposals and this same person will receive regular oral and written progress reports. Final results are presented at the end of the second semester as a paper in a style suitable for publication in a professional journal as well as an oral report in a public Mathematics Capstone symposium.
    SAGES Senior Cap


    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, isometrics 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.
    Prereq: MATH 224.
    SAGES Dept Seminar


    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 or MATH 227.


    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 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. Recommended preparation: MATH 223 and MATH 224 or BIOL 300 and BIOL 306.
    Offered as BIOL 378, COGS 378, MATH 378, BIOL 478, EBME 478, EECS 478, MATH 478 and 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 223 or MATH 227.


    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.
    Offered as MATH 381 and MATH 481.
    Prereq: MATH 380.


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


    MATH 400. Mathematics Teaching Practicum (1)
    Practicum for teaching college mathematics. Includes preparation of syllabi, exams, lectures. Grading, alternative teaching styles, use of technology, interpersonal relations and motivation. Handling common problems and conflicts.


    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.
    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)
    Propositional calculus and quantification theory; consistency and completeness theorems; godal incompleteness results and their philosophical significance; introduction to basic concepts of model theory; problems of formulation of arguments in philosophy and the sciences.
    Offered as PHIL 306, MATH 406 and 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. Recommended preparation: MATH 303.


    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 421. 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. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
    Offered as MATH 321 and MATH 421.


    MATH 422. 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. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
    Offered as MATH 322 and MATH 422.
    Prereq: 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. SpaceP-integrable function. Lebesgue differentiation theorem in n-space.
    Prereq: MATH 322 or MATH 422.


    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. 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 or MATH 422.


    MATH 426. Geometry and Complex Analysis (3)
    The theme of this course will be the interplay between geometry and complex analysis, algebra and other fields of mathematics. An effort will be made to highlight significant, unexpected connections between major fields, illustrating the unity of mathematics. The choice of text(s) and syllabus itself will be flexible, to be adapted to the range of interests and backgrounds of pre-enrolled students. Possible topics include: the Mobius group and its subgroups, hyperbolic geometry, elliptic functions, Riemann surfaces, applications of conformal mapping, and potential theory in classical physical models.
    Offered as MATH 326 and MATH 426.


    MATH 427. 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.
    Offered as MATH 327, MATH 427, and 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. Integrable and square integrable function theory. Inversion theorems. Classical sine and cosine transforms. Multiple Fourier Transform. Spherical symmetry. Other important transforms. Applications.
    Prereq: MATH 224 or MATH 228.


    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 307.


    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.
    Prereq: MATH 224 or MATH 228.


    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 or MATH 227, and 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. Recommended preparation: EECS 408.
    Offered as EECS 421 and MATH 434.


    MATH 435. Ordinary Differential Equations (3)
    MATH 435 is a second course in ordinary equations. The course will begin with a quick review of one-dimensional o.d.e.’s with the idea of introducing tools for studying general systems that will be developed later in the course (this is the get-acquainted part of the course; topics will be selected from Parts I and II of the text). After this introduction we will turn to general features of abstract o.d.e.’s existence and uniqueness of solutions, dependence of solutions on parameters, and the idea of a flow. Next we will consider linear systems. The remainder of the semester will be spent on nonlinear systems: topics that will be discussed are linearization, stability, the Poncare-Bendixson theory and bifurcations.
    Prereq: MATH 224 and either MATH 201 or MATH 307.


    MATH 439. Integrated Numerical and Statistical Computations (3)
    This course will embed numerical methods into a Bayesian framework. The statistical framework will make it possible to integrate a prori information about the unknowns and the error in the data directly into the most efficient numerical methods. A lot of emphasis will be put on understanding the role of the priors, their encoding into fast numerical solvers, and how to translate qualitative or sample-based information--or lack thereof--into a numerical scheme. Confidence on computed results will also be discussed from a Bayesian perspective, at the light of the given data and a priori information. The course should be of interest to anyone working on signal and image processing statistics, numerical analysis and modeling.
    Recommended Preparation: Math 431.
    Offered as MATH 439 and STAT 439.


    MATH 440. Computational Inverse Problems (3)
    This course will introduce various computational methods for solving inverse problems under different conditions. First the classical regularization methods will be introduced, and the computational challenges which they pose, will be addressed. Following this, the statistical methods for solving inverse problems will be studied and their computer implementation discussed. We will combine the two approaches to best exploit their potentials. Applications arising from various areas of science, engineering, and medicine will be discussed throughout the course.


    MATH 441. Mathematical Modeling (3)
    Mathematics is a powerful language for describing real world phenomena and providing predictions that otherwise are hard or impossible to obtain. The course gives the students prerequisites for translating qualitative descriptions given in the professional non-mathematical language into the quantitative language for mathematics. While the variety in the subject matter is wide, some general principles and methodologies that a modeler can pursue are similar in many applications. The course focuses on these similarities. The course is based on representative case studies that are discussed and analyzed in the classroom, the emphasis being on general principles of developing and analyzing mathematical models. The examples will be taken from different fields of science and engineering, including life sciences, environmental sciences, biomedical engineering and physical sciences. Modeling relies increasingly on computation, so the students should have basic skills for using computers and programs like Matlab or Mathematica.
    Prereq: MATH 224 or MATH 228.


    MATH 444. Mathematics of Data Mining and Pattern Recognition (3)
    This course will give an introduction to a class of mathematical and computational methods for the solution of data mining and pattern recognition problems. By understanding the mathematical concepts behind algorithms designed for mining data and identifying patterns, students will be able to modify to make them suitable for specific applications. Particular emphasis will be given to matrix factorization techniques. The course requirements will include the implementations of the methods in MATLAB and their application to practical problems.
    Prereq: MATH 201 or MATH 307.


    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 or MATH 308 and MATH 224 or MATH 228.


    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 optic 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 449. Dynamical Models for Biology and Medicine (3)
    Introduction to discrete and continuous dynamical models with applications to biology and medicine. Topics include: population dynamics and ecology; models of infectious diseases; population genetics and evolution; biological motion (reaction-diffusion and chemotaxis); Molecular and cellular biology (biochemical kinetics, metabolic pathways, immunology). The course will introduce students to the basic mathematical concepts and techniques of dynamical systems theory (equilibria, stability, bifurcations, discrete and continuous dynamics, diffusion and wave propagation, elements of system theory and control). Mathematical exposition is supplemented with introduction to computer tools and techniques (Mathematica, Matlab).
    Prereq: MATH 224 or MATH 228, or BIOL/EBME 300, and MATH 201.


    MATH 461. Introduction to Topology
    Metric spaces, topological spaces, and continuous functions. Compactness, connectedness, path connectedness. Topological manifolds; topological groups. Polyhedra, simplical complexes. Fundamental groups.
    Prereq: MATH 224 or MATH 228.


    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 or MATH 228.


    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.
    Prereq: MATH 224 or MATH 228.


    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 MRI; survey of applications. Recommended preparation: PHYS 431 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 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. Recommended preparation: MATH 223 and MATH 224 or BIOL 300 and BIOL 306.
    Offered as BIOL 378, COGS 378, MATH 378, BIOL 478, EBME 478, EECS 478, MATH 478 and NEUR 478.


    MATH 481. 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.
    Offered as MATH 381 and MATH 481.


    MATH 487. Stochastic Processes in Engineering and Sciences (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 322.


    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 307.


    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. Recommended preparation: MATH 425.
    Prereq: MATH 424.


    MATH 528. Analysis Seminar (1-3)
    Continuing seminar on areas of current interest in analysis. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include functional analysis, convexity theory, and their applications. May be taken more than once for credit. Consent of department required.


    MATH 535. Applied Mathematics Seminar (1-3)
    Continuing seminar on areas of current interest in applied mathematics. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include topics in applied probability and stochastic processes, continuum mechanics, numerical analysis, mathematical physics or mathematical biology. May be taken more that once for credit.


    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.


    MATH 651. Thesis (M.S.) (1-18)


    MATH 701. Dissertation (Ph.D.) (1-18)

    Prereq: Predoctoral research consent or advanced to Ph.D. candidacy milestone.