112 / FIELDS OF INSTRUCTION The requirements for a doctoral degree include 36 hours of 6000-level course work in mathematics. No hours of teaching, colloquium, dissertation, or indi- vidual work will count toward this requirement. The doctoral student must pass a written and oral comprehensive preliminary examination adminis- tered by the department to become a candidate for the degree. The doctoral student must pass reading knowledge examinations in one of the following for- eign languages: French, German, or Russian. The dissertation is an important requirement fordjuthedoctoral degree in mathematics. Thetopics for the dissertation may be chosen from a number of areas of current research in pure and applied mathematics. Every graduate student is expected to attend the regular colloquium. Details concerning all requirements for graduate degrees in mathematics can be obtained by writing the Mathematics Department Graduate Selection Committee. MAA 5102-Advanced Calculus for Engineers and Physical Scientists I (3) MAA 5103-Advanced Calculus for Engineers and Physical Scientists II (3) Prereq; maa 5102. MAA 5226-Modern Analysis I (3) Prereq: MAC 3313. Topol- ogy of metric spaces, numerical sequences and series, con- tinuity, differentiation, the Riemann-Stieltjes integral, se- quences and series of functions, the Stone-Weierstrass the- orem, functions of several variables. Stokes' theorem, the Lebesque theory. MAA 5227-Modern Analysis II (3) Prereq: MAA 5226. MAA 5402-Introduction to Complex Variables for Engi- neers and Physical Scientists (3) MAA 5506-Introduction to Functional Analysis I (3) MAA 5507-Introduction to Functional Analysis 11 (3) Pre- req: MAA 5506. MAA 6236-Mathematical Analysis for Statisticians (4) Coreq: STA 6326. Numerical sequences and series, limits, continuity, differentiation, -integration, series of functions. Applications to probability and statistics stressed. MAA 6406-Complex Analysis I (3) Prereq: MAA 5227. Rapid survey of properties of complex numbers, linear trans- formations, geometric forms and necessary concepts from topology. Complex integration, Cauchy's theorem and its corollaries. Taylor's series and the implicit function theorem in complex form. Conformality and the Riemann-Caratheo- dory mapping theorem. Theorems of Bloch, Schottky, and the big and little theorems of Picard. Harmonicity- and Dirichlet's problems. MAA 6407-Complex Analysis II (3) Prereq: MAA 6406. MAA 6616-General Theory of Measure and Integration I (3) Prereq: MAA 5227. The Daniell approach to integration, ab- stract measure theory, abstract Lebesque integral, con- vergence theorems for integrals, Riesz representation the- orem. LP spaces, various modes of convergence. Banach spaces, Hahn-Banach theorem, open-mapping theorem, uniform-boundedness principle. Hilbert spaces, differentia- tion of functions of one real variable. Vitali's covering the- orem, differentiability of monotone functions. Absolutely continuous functions and indefinite integrals, the Lebesque- Radon-Nikodym theorem. MAA 6617-General Theory of Measure and Integration II (3) Prereq: MAA 6616. MAA 6626-Advanced Topics in Integration I (3; max: 6) Pre- req: MAA 6617 and MTG 6347. Current topics in integration, reference to function spaces and to spaces provided with al- gebraic or topological structures. MAA 6627-Advanced Topics in Integration II (3; max: 6) Prereq: MAA 6626. MAA 7526-Advanced Topics in Functional Analysis I (3; max: 6) Prereq: MAA 6617, MAS 6347. Algebraic and topological approach to current material and methods in analysis. MAA 7527-Advanced Topics in Functional Analysis 11 (3; max: 6) Prereq: MAA 7526. MAD 5405-Numerical Methods of Differential Equations (3) Numerical methods for the solution of ordinary and par- tial differential equations. MAD 6206-Combinatorial Theory I (3) Matching theory, Ramsey's theorem, lattice theory, Mobius inversion, gener- ating functions. Polya's theorem, matroids,. applications, block designs, graph theory. MAD 6207-Combinatorial Theory II (3) Prereq: MAD 6206. MAD 6407-Numerical Analysis (3) Prereq: MAA 4212, 5227 or MAD 4401. Designed to acquaint research student with numerical analysis. Topics may include differential equa- tions, integral equations, eigenvalues, systems of linear and nonlinear equations, approximation theory. MAD 7396-Topics in Combinatorial Theory I (3; max: 6) Prereq: MAS 5332. Topics chosen from among graph theory, coding theory, matroid theory, finite geometries, projective geometry, difference methods, and Latin squares. MAD 7397-Topics in Combinatorial Theory II (3; max: 6) Prereq: MAD 7396. MAE 6940-Supervised Teaching (1-5; max: 5) S/U. MAP 5304-Intermediate Differential Equations for Engi- neers and Physical Scientists (3) MAP 5345-Introduction to Partial Differential Equations (3) MAP 5484-Mathematics for Biological Sciences I (3) Prereq: MAC 3311. Difference equations, differential equations, probability theory, matrix theory, and Markov chains with applications in the biological sciences. Among the applica- tions covered are population dynamics, epidemiology, and genetics. MAP 5485-Mathematics for Biological Sciences II (3) Pre- req: MAP 5484. Continuation of MAP 5484. MAP 6216-Introduction to Calculus of Variations for Engi- neers and Physical Scientists (3) Prereq: MAP 5304, MAS 5156 or equivalent. Extremum problems, first variation. Euler equation problems with fixed and movable boundaries. La- grange multiplier methods for problems with constraints, canonical form, second variation, applications to physics and engineering. MAP 6327-Applied Differential Equations I (3) Prereq: MAA 5227. Theory and methods for solving linear and non- linear systems of differential equations and partial differen- tial equations. Applications and computer techniques in- cluded. MAP 6328-Applied Differential Equations II (3) Prereq: MAP 6327. MAP 6356-Partial Differential Equations I (3) Prereq: MAA 5227, MAP 5341 or 6506. Cauchy-Kowalewski theorem, first order equations, classification of equations, hyperbolic equations, elliptic equations, parabolic equations, hyper- bolic systems, nonlinear hyperbolic systems, existence theo- ry based on functional analysis. Applications to physical sci- ences. MAP 6357-Partial Differential Equations II (3) Prereq: MAP 6356. MAP 6417-Fourier Series I (3) Prereq: MAP 6505. Funda- mental theorems on convergence, differentiation, and inte- gration. Applications to boundary value problems. MAP 6418--Fourier Series II (3) Prereq: MAP 6417. MAP 6467-Stochastic Differential Equations and Filtering Theory 1 (3) Introduction to random functions; the Brownian motion process. Ito's stochastic integral; Ito's stochastic cal- culus; stochastic differential equations. Linear filtering; Kalman filtering; nonlinear filtering theory. MAP 6468-Stochastic Differential Equations and Filtering Theory II (3) Prereq: MAP 6467. MAP 6472-Probability and Potential Theory 1 (3) Prereq: MAA 5227, STA 6326. Probability laws. Uniformly integrable random variables. Independence and conditioning. General properties of stochastic processes. First and second canonical process. Separability of stochastic processes. De- compositions. Potentials. Excessive functions. Connections with Martingale theory. MAP 6473-Probability and Potential Theory II (3) Prereq: MAP 6472. MAP 6487-Biomathematics Seminar 1 (3) Prereq: MAC 3312, MAP 3302, STA 6327. Stochastic processes, differential equa- tions, and reaction-diffusion equations used to model vari- ous biological processes. Among the applications covered