MATH 630. Introduction to Complex Analysis. (3) I, II. Complex analytic functions and power series, complex integrals. Taylor and Laurent expansions, residues, Laplace transformations, and the inversion integral. Pr.: MATH 240.

MATH 632. Elementary Partial Differential Equations. (3) I. Orthogonal functions, Fourier series, boundary value problems in partial differential equations. Pr.: MATH 240.

MATH 633. Advanced Calculus I. (3) I. Functions of one variable; limits, continuity, differentiability, Riemann-Stieltjes integral, sequences, series, power series, improper integrals. Pr.: MATH 222.

MATH 634. Advanced Calculus II. (3) II. Functions of several variables; partial differentiation and implicit function theorems, curvilinear coordinates, differential geometry of curves and surfaces, vectors and vector fields, line and surface integrals, double and triple integrals, Green's Theorem, Stokes' Theorem, and Divergence Theorem. Pr.: MATH 633.

MATH 655. Elementary Numerical Analysis I. (3) I. Error analysis, root finding, interpolation, approximation of functions, numerical integration and differentiation, systems of linear equations. Pr.: MATH 221, a computer language, and either MATH 515 or 551.

MATH 656. Elementary Numerical Analysis II. (3) II. A continuation of MATH 655. Linear programming, numerical solutions of differential equations, and the use of standard packages for the solutions of applied problems. Pr.: MATH 655 and 240.

MATH 670. Mathematical Modeling. (3) Introduction of modeling procedures. Case studies in mathematical modeling projects from physical, biological, and social sciences. Pr.: Four mathematics courses numbered 500 or above.

Graduate credit

MATH 715. Applied Mathematics I. (3) I. Topics from vector calculus, higher-dimensional calculus, ordinary differential equations, matrix theory, linear algebra, and complex analysis. Pr.: MATH 222.

MATH 716. Applied Mathematics II. (3) II. Topics from Fourier series, Fourier and Laplace transforms, partial differential equations, calculus of variations and linear algebra. Pr.: MATH 715.

MATH 721. Introduction to Real Analysis. (3) I. Limits, continuity, uniform convergence, completeness, differentiation, Riemann integration. Pr.: MATH 634 or graduate standing.

MATH 722. Introduction to Functions of Several Variables. (3) II. Analysis of functions of several variables, including differentiability, partial differentiability, maxima and minima, inverse function theorem, implicit function theorem, integration, Fubini's theorem. Pr.: MATH 634 or graduate standing.

MATH 740. Calculus of Variation. (3) On sufficient demand. Necessary conditions and the Euler-Lagrange equations. Hamilton-Jacobi theory, Noether's theorems, direct methods, applications to geometry and physics. Pr.: MATH 722 or equivalent.

MATH 745. Ordinary Differential Equations. (3) I. First-order equations and applications, second-order equations and oscillation theorems, series solutions and special functions, Sturm Liouville problems, linear systems, autonomous systems and phase plane analysis, stability, Liapunov's method, periodic solutions, perturbation and asymptotic methods, existence and uniqueness theorems. Pr.: MATH 240

MATH 755. Dynamic Modeling Processes. (3) Topics to include equilibrium and stability, limit circles, reaction-diffusion, and shock phenomena, Hopf bifurcation and cusp catastrophes, chaos and strange attractors, bang-bang principle. Applications from physical and biological sciences and engineering. Pr.: MATH 240 and 551.

MATH 757. Mathematical Control Theory. (3) Mathematical analysis of dynamical systems governed by differential equations and their optimal processes, feedback, and filtering. Topics include: dynamical systems with controls, axioms of control systems, input-output behaviors, stability and instability, reachability and controllability, dynamic feedback and stabilization, optimal control processes, piecewise constant control and bang-bang principle, Pontryagin maximum principle, tracking, and filtering. Pr.: MATH 560 and MATH 615.

MATH 760. Probability Theory. (3) An introduction to the mathematical theory of probability. Material covered includes combinatorial probability, random variables, independence, expectations, limit theorems, Markov chains, random walks, and martingales. Pr.: MATH 633 and STAT 510.

MATH 772. Elementary Differential Geometry. (3) Curves and surfaces in Euclidean spaces, differential forms and exterior differentiation, differential invariants and frame fields, uniqueness theorems for curves and surfaces, geodesics, introduction to Riemannian geometry, some global theorems, minimal surfaces. Pr.: MATH 240.

MATH 801. Numerical Solution of Differential Equations I. (3) I. Single and multistep methods for initial-value problems for ordinary differential equations; discretization and round-off error; consistency, convergence, and stability of these methods; stiff equations and implicit methods; two-point boundary value problems; initial and boundary-value problems for partial differential equations; finite difference methods; marching schemes for parabolic and hyperbolic problems; consistency, stability, convergence, and the Lax equivalence theorem; treatment of boundary conditions; boundary-value problems for elliptic equations; relaxation, alternating direction, and strongly-implicit iterative methods; nonlinear problems; finite element methods. Pr.: MATH 655 and knowledge of a programming language.

MATH 802. Numerical Solution of Differential Equations II. (3) II. Continuation of MATH 801. Pr.: MATH 801.

MATH 821. Real Analysis. (3) I. Measure theory and integration, Lebesgue integration, L^p-spaces, Hilbert spaces, integration on product spaces, Fubini's theorem. Pr.: MATH 721 and 722.

MATH 822. Complex Analysis. (3) II. Analytic functions, the Cauchy integral theorem, power series, principle of the argument, conformal mapping. Pr.: MATH 821.

MATH 823. Geometric Function and Measure Theory I. (3) I. Topics include general measure theory, covering theorems, Hausdorff measure, area and coarea formulas, distributions, Sobolev spaces, Poincare' inequalities, embeddings theorem, changes of variables, extensions, and capacity. Applications include quasiconformal and quasiregular maps in IR^IN and analysis on fractals. Pr.: MATH 821.

MATH 824. Geometric Function and Measure Theory II. (3) II. Continuation of Geometric Function and Measure Theory I. Pr.: MATH 823.

MATH 825. Complex Analysis I. (3) I. Holomorphic functions, harmonic functions, the Cauchy integral theorem, normal families and the Riemann mapping theorem, and the Mittag-Leffler theorem. Pr.: MATH 822 or consent of department.

MATH 826. Complex Analysis II. (3) II. Analytic continuation, the Picard theorem, Hp-spaces, elementary theory of Banach algebra, the theory of Fourier transforms, and the Paley-Wiener theorems. Pr.: MATH 825.

MATH 827. Classical and Modern Fourier Analysis I. (3) I. Topics include Fourier analysis on the circle, singular integrals of convolution type, Littlewood-Paley theory and multipliers, BMO and Carleson Measures, and boundedness and convergence of Fourier integrals (or singular integrals of nonconvolution type). Pr.: MATH 821.

MATH 828. Classical and Modern Fourier Analysis II. (3) II. Continuation of Classical and Modern Fourier Analysis I. Pr.: MATH 827.

MATH 840. Differential Equations I. (3) I. Basic ordinary and partial differential equations. First-order ordinary differential equations: symmetries and solutions in quadratures; existence, uniqueness and dependence on parameters, systems of first order equations, analysis of equilibria. Second order equations: series solutions and special functions, initial- and boundary-value problems for second-order equations. Elements of integral equations. First-order partial differential equations. Basic second-order partial differential equations: wave equations, heat equations, Poisson equation, Schrodinger equation. Pr.: MATH 634 or MATH 745 or consent of instructor.

MATH 841. Differential Equations II. (3) II. Where PDEs come from. Initial and boundary-value problems. A crash course in distribution theory: different spaces of distributions, Fourier and Laplace transformations of distributions, Sobolev spaces. The Poisson equation in bounded and in exterior domains; properties of solutions of elliptic equations. The classical evolution equations revisited. Energy estimates, existence and uniqueness theorems, regularity and other properties of solutions. Simple examples of nonlinear PDEs. Pr.: MATH 840.

MATH 842. Differential Equations III. (3) II. Continuation of MATH 841. Pr.: MATH 841.

MATH 852. Functional Analysis I. (3) I, in alternate years. Topological vector spaces; locally convex spaces (Hahn-Banach Theorem, weak topology, dual pairs, Krein-Milman Theorem, theory of distributions); Banach spaces (Uniform Boundedness Principle, Open Mapping Theorem and applications, Alaoglu's Theorem, analytic vector-valued functions, Krein-Smulian Theorem); C(X) as a Banach space (Stone-Weierstrass Theorem, Riesz Theorem); L^p spaces. Pr.: MATH 821

MATH 853. Functional Analysis II. (3) II, in alternate years. Banach Algebras (spectrum, Gelfand Fourier transform, holomorphic functional calculus); Hilbert spaces (geometric properties, Riesz's Theorem, projections, the adjoint); functional calculus for normal operators; compact operators (spectral properties, Min-Max Theorem, Schatten-vol Neumann classes); Fredholm operators; other operator topologies on B(H); unbounded self-adjoint operators. Other topics. Pr.: MATH 852.

MATH 855. Methods of Applied Mathematics I. (3) An introduction to the mathematical techniques of problem solving in the sciences and engineering. Construction of mathematical models; problem formulation, dimensional analysis and scaling; solution methods for differential equations and difference equations; methods for obtaining approximate solutions; regular and singular perturbations methods, asymptotic series, applications to specific equations and scientific problems. Pr.: MATH 630, 633 and 551.

MATH 856. Methods of Applied Mathematics II. (3) A continuation of MATH 855. Asymptotic expansion of integrals; the methods of stationary phase and steepest descent; summations of series, the Shanks transformation and the Pade fractions; boundary layer theory; the WKB and Langer approximations; the method of averaging and the method of multiple scales. Pr.: MATH 855.

MATH 857. Nonlinear Analysis I. (3) I. The course deals with studies of nonlinear operator equations, existence of their solutions, uniqueness of the solutions, numerical methods for finding solutions. Fixed point theorems, topological principles, nonlinear elliptic equations and evolution equations, ill-posed and inverse problems are discussed. Pr. MATH 821, 822, 852, 853 and MATH 840 is recommended.

MATH 858. Nonlinear Analysis II. (3) II. Continuation of Nonlinear Analysis I. Pr.: MATH 857.

MATH 864. Theory of Ordinary Differential Equations I. (3) I. The modern theory of ordinary differential equations including general theory and the theory of linear differential equations. Pr.: MATH 641, 722 and 731.

MATH 865. Theory of Ordinary Differential Equations II. (3) II. Continuation of MATH 864 to include nonlinear equations and differential equations in Banach spaces. Pr.: MATH 864.

MATH 871. General Topology I. (3) I. Topological spaces and topological invariants; continuous mappings and their invariants; perfect mappings; topological constructions (product, quotient, direct and inverse limit spaces). Pr.: MATH 700 and 701.

MATH 872. General Topology II. (3) II. Compact spaces and compactification, uniform and proximity spaces, metric spaces and metrization, topology of Rn, function spaces, complete spaces, introduction to homotopy theory. Pr.: MATH 871.

MATH 924. Several Complex Variables. (3) An introduction to the theory and analytic functions of several variables, domains of homomorphy and pseudoconvexity, the Levi problem, delta bar equations, Cousin problems, zeros of analytic functions, integral formulas, holomorphic mappings. Pr.: MATH 826.

MATH 992. Topics in Analysis. (3) On sufficient demand. Selected topics in modern analysis. May be repeated for credit. Pr.: Consent of instructor.

MATH 993. Topics in Harmonic Analysis. (3) On sufficient demand. Selected topics in harmonic analysis. May be repeated for credit. Pr.: Consent of instructor.

MATH 994. Topics in Applied Mathematics. (3) On sufficient demand. Selected topics in applied mathematics. May be repeated for credit. Pr.: Consent of instructor.