| MATH 514 | Advanced Mathematical Methods | (3-0-3) |
| Integral transforms: Fourier, Laplace, Hankel and Mellin transforms and their applications. Singular integral equations. Wiener-Hopf techniques. Applications of conformal mapping. Introduction to asymptotic expansion. Prerequisite: Graduate Standing. | ||
| MATH 521 | General Topology I | (3-0-3) |
| Basic set theory (countable and uncountable sets, Cartesian products). Topological spaces (basis for a topology, product topology, continuous functions, homeomorphisms, standard examples). Connected spaces, path connectedness. Compact spaces, compactness in metrizable spaces. Countability axioms, first countable and second countable spaces. Separation axioms, Urysohn's Lemma, Urysohn's Metrization Theorem. Complete metric spaces. Prerequisite: Graduate Standing. | ||
| MATH 523 | Algebraic Topology | (3-0-3) |
| The fundamental group. The fundamental group of the circle. Van Kampen's Theorem. Covering spaces and unique lifting property. Homology: definition, invariance under homotopies. Homology of a pair. The relationship between the fundamental group and the first homology group. Basic exact sequences in homology: Mayer-Vietoris and excision. Homology of CW-complexes. Homology of product spaces. Prerequisite: Graduate Standing. | ||
| MATH 525 | Graph Theory | (3-0-3) |
| A basic introduction to graph theory for advanced students in computer science, Mathematics, and related fields. Connectivity, matching, factorization and covering of graphs, embeddings, edge and vertex coloring. Line graphs. Reconstruction of graphs. Networks and algorithms. Topological subgraphs: Contractions. Prerequisite: Graduate Standing. | ||
| MATH 527 | Differential Geometry | (3-0-3) |
| Curves in Euclidean spaces: arclength, tangent, normal and binormal vectors, curvature and torsion. Frenet formulas. Isoperimetric inequality. Differential geometry and local theory of surfaces, the first and second fundamental forms. Local isometries. Geodesics. Gaussian and mean curvature of surfaces. The Gauss- Bonnet theorem. Manifolds and differential forms. Introduction to Riemannian geometry Prerequisite: Graduate Standing. | ||
| MATH 531 | Real Analysis | (3-0-3) |
| Lebesgue measure and outer measure. Measurable functions. The Lebesgue integral. Lebesgue Convergence Theorem. Differentiation and integration. spaces. Riesz Representation Theorem. Introduction to Banach and Hilbert spaces. Product spaces, Fubini's Theorem. Prerequisite: Graduate Standing. | ||
| MATH 533 | Complex Variables | (3-0-3) |
| Analytic functions. Cauchy's theorem and consequences. Singularities and expansion theorems. Maximum modulus principle. Residue theorem and its application. Compactness and convergence in spaces of analytic functions. Weierstrass and Mittag-Leffler representation theorems. Elementary conformal mappings. Prerequisite: Graduate Standing. | ||
| MATH 535 | Functional Analysis | (3-0-3) |
| Normed linear spaces. Banach spaces. Hilbert spaces. Banach algebras (definitions, examples, geometric properties). Bounded linear operators. Convex sets. Linear functionals. Duality. Reflexive spaces. Weak topology and weak convergence. Banach contraction principle and its applications. Hahn-Banach Theorem. Uniform boundedness principle. Open Mapping Theorem. Closed Graph Theorem. Representation of functionals on Hilbert spaces (Riesz Representation Theorem). Prerequisite: Graduate Standing. | ||
| MATH 537 | Topological Vector Spaces | (3-0-3) |
| Topological vector spaces. Locally convex spaces. Krein-Milman Theorem. Duality in locally convex spaces. Separation theorem for compact convex sets. Topological tensor products. Nuclear mappings and spaces. Prerequisite: MATH 535 | ||
| MATH 550 | Linear Algebra | (3-0-3) |
| Basic properties of vector spaces and linear transformations, characteristic values and diagonalizable operators, invariant subspaces and triangulable operators. The Primary Decomposition Theorem, cyclic decompositions and the Generalized Cayley-Hamilton Theorem. Rational and Jordan forms, inner product spaces. The Spectral Theorem, bilinear forms, symmetric and skew symmetric bilinear forms. Prerequisite: Graduate Standing. | ||
| MATH 551 | Abstract Algebra | (3-0-3) |
| Basic definitions of rings and modules. Homomorphisms. Sums and products. Exactness. Hom and tensor. Adjoint isomorphism. Free, projective and injective modules. Chain conditions. Primary decomposition. Noetherian rings and modules. Artinian rings. Structure theorems. Prerequisite: Graduate Standing. | ||
| MATH 552 | Fields and Galois Theory | (3-0-3) |
| Field extensions. The Fundamental Theorem. Splitting fields and algebraic closure. Finite fields. Separability. Cyclic, cyclotomic, and radical extensions. Structure of fields: transcendence bases. Prerequisite: Graduate Standing. | ||
| MATH 553 | Homological Algebra | (3-0-3) |
| Review of free, projective, and injective modules. Direct limits. Watts theorems. Flat modules. Localization. Noetherian, semisimple, Von Neumann regular, hereditary, and semi-hereditary rings. Homology, homology functors, derived functors, Ext. and Tor., homological dimensions, Hilbert Syzygy Theorem. Prerequisite: MATH 551. | ||
| MATH 555 | Commutative Algebra | (3-0-3) |
| Basics of rings and ideals. Rings of fractions. Integral dependence. Valuation rings, discrete valuation rings. Dedekind domains, fractional ideals. Topologies and completions. Filtrations. Graded rings and modules. Dimension theory. Prerequisite: MATH 551. | ||
| MATH 556 | Algebraic Geometry | (3-0-3) |
| Affine and projective algebraic varieties. Zariski topology. Morphisms of algebraic varieties, Veronese and Segre maps, coordinate rings. Hilbert's Nullstellensatz. Hilbert polynomial, dimension and degree. Quasi-projective varieties and their morphisms. Regular functions. Tangent space, smoothness. Birational maps and blow ups. Resolution of singularities. Prerequisite: MATH 551. | ||
| MATH 558 | Algebraic Categories | (3-0-3) |
| Categories, functors. Abstract algebraic categories, algebraic theories. Filtered colimits. Algebraic functors. Birkhoff's Variety Theorem. Concrete algebraic categories, equational categories. Monads, Abelian categories. Categories of modules over rings. Prerequisite: MATH 551. | ||
| MATH 559 | Numerical Linear Algebra | (3-0-3) |
| Concepts from linear algebra and numerical analysis. Direct methods for large, sparse linear systems. Cholesky, LU, QR and SVD factorizations. Sensitivity and conditioning of linear systems and least square problems. Iterative methods. Numerical methods for eigenvalues. Computations of SVD. Computer software and applications. Prerequisite: Graduate Standing. (Cannot be taken for credit with MATH 471.) | ||
| MATH 563 | Probability Theory | (3-0-3) |
| Foundations of probability theory. Measure-theoretic approach to definitions of probability space, random variables and distribution functions. Modes of convergence and relations between the various modes. Independence, Kolmogorov type inequalities. Tail events and the Kolmogorov 0-1 law. Borel-Cantelli lemma. Convergence of random series and laws of large numbers. Convergence in distribution. Characteristic functions. The Central Limit Theorem. Weak convergence of probability measures. Conditional expectations and martingales. Prerequisite: Graduate Standing. | ||
| MATH 565 | Advanced Ordinary Differential Equations I | (3-0-3) |
| Existence, uniqueness and continuity of solutions. Linear systems, solution space, linear systems with constant and periodic coefficients. Stability of linear systems and stability under nonlinear perturbations. Hyperbolicity and topological conjugacies. Existence of invariant manifolds. Index theory, Poincaré-Bendixson theory. Bifurcations and center manifolds. Hamiltonian Systems. Prerequisite: Graduate Standing. | ||
| MATH 566 | Fractional Differential Equations | (3-0-3) |
| Special functions (Gamma, Mittage-Leffler, and Wright). Riemann fractional integral. Riemann-Liouville and Caputo fractional derivatives. Composition rules. Embeddings. Equivalence with integral equations. Well posedness for Cauchy type problems. Successive approximation method. Laplace and Mellin transform methods. Prerequisite: Graduate standing. | ||
| MATH 568 | Advanced Partial Differential Equations I | (3-0-3) |
| First order linear and nonlinear equations. Classification of Second order equations. The wave equation, heat equation and Laplace's equation. Green's functions, conformal mapping. Separation of variables. Sturm-Liouville theory. Maximum principles and regularity theorems. Prerequisite: Graduate Standing. | ||
| MATH 569 | Linear Elliptic Partial Differential Equations | (3-0-3) |
| Sobolev spaces. Mollifiers. Dual spaces. Poincaré's inequality. Lax-Milgram Theorem. Linear elliptic problems. Weak formulation. Weak derivatives. Weak solutions. Existence uniqueness and regularity. Maximum principle. Prerequisite: Graduate Standing. | ||
| MATH 571 | Numerical Analysis of Ordinary Differential Equations | (3-0-3) |
| Theory and implementation of numerical methods for initial and boundary value problems in ordinary differential equations. One-step, linear multi-step, Runge-Kutta, and extrapolation methods: convergence, stability, error estimates, and practical implementation. Study and analysis of shooting, finite difference and projection methods for boundary value problems for ordinary differential equations. Prerequisite: Graduate Standing. | ||
| MATH 572 | Numerical Analysis of Partial Differential Equations | (3-0-3) |
| Theory and implementation of numerical methods for boundary value problems in partial differential equations (elliptic, parabolic, and hyperbolic). Finite difference and finite element methods: convergence, stability, and error estimates. Projection methods and fundamentals of variational methods. Ritz-Galerkin and weighted residual methods. Prerequisite: Graduate Standing. | ||
| MATH 574 | Numerical Methods of Partial Differential Equations | (3-0-3) |
| Concepts of consistency, stability, and convergence of numerical schemes. Initial and boundary value problems for ordinary differential equations. Various finite difference and finite element methods and their applications to fundamental partial differential equations in engineering and applied sciences. Case studies selected from computational fluid mechanics, solid mechanics, structural analysis, and plasma dynamics. Prerequisite: Graduate Standing. (Not Open to Mathematics Majors.) | ||
| MATH 575 | Introduction to Approximation Theory | (3-0-3) |
| Best approximation in normed linear spaces: basic concepts. Lagrange and Hermite interpolation. Approximate solution of over-determined system of linear equations. Linear approximation of continuous functions in Chebyshev and least squares norms. Rational approximation. Piecewise polynomial approximation. Cubic and B-splines. Prerequisite: Graduate Standing. | ||
| MATH 580 | Convex Analysis | (3-0-3) |
| Basic properties of convex sets and convex functions. Analytical and geometrical aspects of convexity and duality. Convexity connections with optimization.Lagrangian and Fenchel duality.Non-smooth convex optimization and subdifferential calculus. Prerequisite: Graduate Standing. | ||
| MATH 581 | Advanced Linear Programming | (3-0-3) |
| Theory of linear programming and its duality. Simplex method, revised simplex method, dual simplex method. Interior point method. Sensitivity analysis. Integer and mixed integer programming (cutting plane, and branch and bound methods). Implementation and current solvers. Applications selected from transportation problems, assignment problems, game theory, goal programming, multiobjective programming, linear fractional programming, regression, classification and finance. Prerequisite: Graduate Standing. (Cannot be taken for credit with ISE 503.) | ||
| MATH 582 | Nonlinear Programming | (3-0-3) |
| An advanced introduction to theory of nonlinear programming, with emphasis on convex programs. First and second order optimality conditions, constraint qualifications, Lagrangian convexity and duality. Penalty function methods. Theory and algorithms of main computational methods of nonlinear programming. Representative applications of nonlinear programming in Economics, Operations Research and Mathematics. Prerequisite: Graduate Standing. | ||
| MATH 587 | Advanced Applied Regression | (3-0-3) |
| Least square method and properties. Simple and multiple linear regression with matrix approach. Development of liner models. Residual analysis. Polynomial models. Use of dummy variables in multiple linear regression. Analysis of variance approach. Selection of 'best' regression equation. Concepts of mathematical model building. Non-linear regression and estimation. Extensive use of computer packages. Prerequisites: Graduate Standing. | ||
| MATH 601 | Introduction to Stochastic Differential Equations and Applications | (3-0-3) |
| Probability spaces. Characteristic functions. Stochastic processes. Martingales. Markov Chains. Brownian motion. Itô calculus. Itô formula. Stochastic differential equations, applications of stochastic differential equations. Prerequisite: Graduate Standing. | ||
| MATH 605 | Asymptotic Expansions and Perturbation Methods | (3-0-3) |
| Asymptotic sequences and series. Asymptotic expansions of integrals. Solutions of differential equations at regular and irregular singular points. Nonlinear differential equations. Perturbation methods. Regular and singular perturbations. Matched asymptotic expansions and boundary layer theory. Multiple scales. WKB theory. Prerequisite: Graduate Standing. | ||
| MATH 608 | Inverse and Ill-Posed Problems | (3-0-3) |
| Mathematical and numerical analysis of linear inverse and/or ill-posed problems for partial differential, integral and operator equations. Tikhonov regularization. Constraints and a priori bounds. Methodologies for achieving «optimal» compromise between accuracy and stability. Applications to practical problems in remote sensing, profile inversion, geophysics, inverse scattering and tomography. Prerequisite: Graduate Standing. | ||
| MATH 611 | Hilbert Space Methods in Applied Mathematics I | (3-0-3) |
| Review of normed and product spaces. Theory of distributions, weak solution. Complete orthonormal sets and generalized Fourier expansions. Green's functions and boundary-value problems, modified Green's functions. Operator theory, invertibility, adjoint operators, solvability conditions. Fredholm alternative. Spectrum of an operator. Extremal principles for eigenvalues and perturbation of eigenvalue problems. Applications. Prerequisite: MATH 535. | ||
| MATH 612 | Hilbert Space Methods in Applied Mathematics II | (3-0-3) |
| Integral equations. Fredholm integral equation. Spectrum of a self-adjoint compact operator. Inhomogeneous equation. Variational principles and related approximation methods. Spectral theory of second-order differential operator. Weyl's classification of singular problems. Continuous spectrum. Applications. Introduction to nonlinear problems. Perturbation theory. Techniques for nonlinear problems. Prerequisite: MATH 611. | ||
| MATH 621 | General Topology II | (3-0-3) |
| The Tychonoff theorem, one-point compactification, the Stone-Cech compactification. Paracompactness, Lindelöf spaces, Stone's Theorem. Metrizability, the Nagata-Smirnov Metrization Theorem. Homotopy paths, fundamental group, simply-connected spaces, retracts and deformation retracts; the fundamental groups of the circle, the punctured plane and the n-sphere; Van Kampen's Theorem. Prerequisite: MATH 521. | ||
| MATH 627 | Differentiable Manifolds and Global Analysis | (3-0-3) |
| Calculus on manifolds. Differentiable manifolds, mappings, and embeddings. Implicit Functions Theorem, exterior differential forms, and affine connections. Tangent bundles. Stoke's Theorem. Critical points. Sard's Theorem. Whitney's Embedding Theorem. Introduction to Lie groups and Lie algebras. Applications. Prerequisite: MATH 527. | ||
| MATH 636 | Operators and Spectral Theory | (3-0-3) |
| Algebra of bounded operators, self-adjoint operators in Hilbert Spaces, Normal operators, compact operators. Projections. Spectral theory of linear operators in normed spaces and Hilbert spaces. Spectral Mapping Theorem. Banach-Alaoglu Theorem. Prerequisite: MATH 535. | ||
| MATH 637 | Non linear Functional Analysis and Applications | (3-0-3) |
| Fixed points methods. Nonexpansive mappings. Differential and integral calculus in Banach spaces. Implicit and inverse function theorems. Potential operators and variational methods for linear and nonlinear operator equations. Extrema of functional. Monotone operators and monotonicity methods for nonlinear operator equations. Applications to differential and integral equations and physical problems. Prerequisite: MATH 535. | ||
| MATH 640 | Calculus of Variations | (3-0-3) |
| Gateaux and Fréchet differentials. Classical calculus of variations. Necessary conditions. Sufficient conditions for extrema. Jacobi and Legendre conditions. Natural boundary conditions. Broken extrema, Erdmann-Weierstrass condition. Multiple integral problems. Constrained extrema. Hamilton principle with applications to mechanics and theory of small oscillations. Problems of optimal control. Direct methods including the Galerkin and the Ritz-Kantorovich methods. Variational methods for eigenvalue problems. Prerequisite: Graduate Standing. | ||
| MATH 642 | Control and Stability of Linear Systems | (3-0-3) |
| Review of systems of linear differential equations to include existence and uniqueness, contraction mappings, fixed points, transition matrix, matrix exponentials, the Laplace transform and stability. Linear control systems. Controllability, observability and duality. Weighting patterns and minimal realizations. Feedback. Linear regulator problem and matrix Riccati equations. Fixed-end point problems. Minimum cost and final-value problems in control theory. Stability of linear systems. Uniform stability. Exponential stability. Prerequisite: Graduate Standing. | ||
| MATH 645 | Combinatorics | (3-0-3) |
Basic Counting Principles. Arrangements and Derangements. Generating Functions. Exponential generating functions. Recurrence Relations. The Sieve Formula. The Möbius inversion formula in partially ordered sets. Group Actions: Burnside and Polya-Redfield Counting theorems. Prerequisite: Graduate Standing. | ||
| MATH 652 | Advanced Topics in Group Theory | (3-0-3) |
| Advanced theory of solvable and nilpotent groups. General free groups. Krull- Schmidt theorem. Extensions. The general linear group. Group rings and group algebras. Representation theory of groups. Prerequisite: Graduate Standing. | ||
| MATH 653 | Advanced Topics in Commutative Algebra | (3-0-3) |
| Selected topics from: Prime spectra and dimension theory. Class groups. Ideal systems and star operations. Multiplicative ideal theory. Generator property. Homological aspects of commutative rings. Pullbacks of commutative rings. Prerequisite: MATH 555. | ||
| MATH 654 | Advanced Topics in Algebra | (3-0-3) |
| Selected topics from: Groups, rings, modules, and general algebraic systems. Prerequisite: MATH 551. | ||
| MATH 655 | Applied & Computational Algebra | (3-0-3) |
| Contents vary. Concepts and methods in algebra which have wide applications in mathematics as well as in computer science, systems theory, information theory, physical sciences, and other areas. Topics may be chosen from fields of advanced matrix theory; algebraic coding theory; group theory; Gröbner bases; or other topics of computational and applied algebra. Prerequisite: Graduate Standing. | ||
| MATH 663 | Advanced Probability | (3-0-3) |
| Measurable functions and integration. Radon-Nikodym Theorem. Probability space. Random vectors and their distributions. Independent and conditional probabilities. Expectation. Strong laws of large numbers. The Weak Compactness Theorem. Basic concepts of martingales. Invariance principles. The Law of the Iterated Logarithm. Stable distributions and infinitely divisible distributions. Prerequisites: MATH 531, MATH 563. | ||
| MATH 665 | Advanced Ordinary Differential Equations II | (3-0-3) |
| Self-adjoint boundary-value problems, Sturm-Liouville theory. Oscillation and comparison theorems. Asymptotic behavior of solutions. Singular Sturm-Liouville problems and non-self-adjoint problems. Hypergeometric functions and related special functions. Bifurcation phenomena. Prerequisite: MATH 565. | ||
| MATH 667 | Advanced Partial Differential Equations II | (3-0-3) |
| Classification of first order systems. Hyperbolic systems, method of characteristics. Applications to gas dynamics. Dispersive waves; application to water waves. Potential theory, single and double layers, existence theory for Dirichlet and Neumann problems. Prerequisite: MATH 568. | ||
| MATH 668 | Evolution Equations | (3-0-3) |
| Maximum Monotone Operators. Bounded and unbounded operators. Pseudo monotone operators. Self-adjoint. Evolution Equations in Hilbert and Banach spaces. Hille-Yosida Theorem, application to linear heat and wave equations. Nonlinear Evolution equations. The Galerkin Method. Prerequisite: MATH 569. | ||
| MATH 673 | Numerical Solution of Integral Equations | (3-0-3) |
| Numerical methods and approximate solutions of Fredholm integral equations of the second kind (both linear and nonlinear). Approximation of integral operators and quadrature methods. Nystrom method. Method of degenerate kernels. Collectively compact operator approximations. Numerical methods for Volterra integral equations. Methods of collocation, Galerkin, moments, and spline approximations for integral equations. Iterative methods for linear and nonlinear integral equations. Eigenvalue problems. Prerequisite: Graduate Standing. | ||
| MATH 674 | Numerical Functional Analysis | (3-0-3) |
| Theoretical topics in numerical analysis based on functional analysis methods. Operator approximation theory. Iterative and projection methods for linear and nonlinear operator equations. Methods of steepest descent, conjugate gradient, averaged successive approximations, and splittings. Stability and convergence. Abstract variational methods and theoretical aspects of spline and finite element analysis. Minimization of functionals. Vector space methods of optimization. Newton and quasi-Newton methods for operator equations and minimization. Prerequisite: MATH 535. | ||
| MATH 680 | Dynamic Programming | (3-0-3) |
| Development of the dynamic programming algorithm. Optimality principle and characterizations of optimal policies based on dynamic programming. Shortest route problems and maximum flow problems. Adaptive process. One-dimensional allocation processes. Reduction of dimensionality. Additional topics include imperfect state information models, the relation of dynamic programming to the calculus of variations, and network programming. Computational experience will be acquired by working on individual projects of applying dynamic programming to case study problems. Prerequisite: MATH 640. | ||
| MATH 681 | Topics in Mathematical Programming | (3-0-3) |
| Contents vary. Topics selected from: Nonconvex optimization, geometric programming, Lagrangian algorithms, sensitivity analysis, large-scale programming, nonsmooth optimization problems and optimality conditions in infinite-dimensional spaces, combinatorial optimization, computation of fixed points, complementarity problems, multiple-criteria optimization, and semi-infinite programming. Prerequisite: MATH 582. | ||
| MATH 690 | Special Topics in Mathematics | (3-0-3) |
| Variable Content. Advanced topics not covered in regular courses. Prerequisite: Admission to Ph.D. Program. | ||
| MATH 699 | Seminar | (1-0-0) |
| Ph.D. students are required to attend Departmental seminars delivered by faculty, visiting scholars and graduate students. Additionally, each Ph.D. student should present at least one seminar on a timely research topic. Ph.D. students should pass the comprehensive examination as part of this course. This course is a prerequisite to registering the Ph.D. Pre-dissertation MATH 711. The course is graded on Pass or Fail basis. IC grade is awarded if the Ph.D. Comprehensive exam is not yet passed. Prerequisite: Admission to Ph.D. Program. | ||
| MATH 701 | Directed Research I | (3-0-3) |
| Variable Content. Advanced mathematical topics not covered in regular courses. Prerequisite: Admission to Ph.D. Program. | ||
| MATH 702 | Directed Research II | (3-0-3) |
| Variable Content. Advanced mathematical topics not covered in regular courses. Prerequisite: Admission to Ph.D. Program. | ||
| MATH 711 | Pre-Ph.D. Dissertation | (0-0-3) |
| Preparation and defense of the PhD dissertation proposal. This is an NP/NF course. Prerequisite: PhD Candidacy. Co-requisite: MATH 699. | ||
| MATH 712 | Ph.D. Dissertation | (0-0-9) |
| Preparation and defense of the PhD dissertation. This is an NP/NF/IP course. Prerequisite: MATH 711. | ||