Graduate Courses
MATH 401. Abstract Algebra I (3)
Basic properties of groups, rings, modules and fields. Isomorphism theorems for groups; Sylow theorem; nil potency and solvability of groups; Jordan-Holder theorem; Gauss lemma and Eisenstein's criterion; finitely generated modules over principal ideal domains with applications to abelian groups and canonical forms for matrices; categories and functors; tensor product of modules, bilinear and quadratic forms; field extensions; fundamental theorem of Galois theory, solving equations by radicals. Prereq: MATH 308.
MATH 402. Abstract Algebra II (3)
A continuation of MATH 401. Prereq: MATH 401.
MATH 406. Mathematical Logic and Model Theory (3)
A study of formal logical systems and their models. Propositional logic and quantification. First order theories; consistency, compactness, and the Lowenheim Skolem theorem. Cross-listed as PHIL 406.
MATH 408. Introduction to Cryptology (3)
Introduction to the mathematical theory of secure communication. Topics include: classical cryptographic systems; one-way and trapdoor functions; RSA, DSA, and other public key systems; Primality and Factorization algorithms; birthday problem and other attack methods; elliptic curve cryptosystems; introduction to complexity theory; other topics as time permits. Prereq: MATH 303.
MATH 410. Automata and Formal Languages (3)
Finite automata, Turing and Post machines, and pushdown automata. The languages generated, accepted, and decided by these machines. Closure properties. Decidability and undecidability. Regular expressions. Right linear, unrestricted, and context-free grammars. MATH 410 and MATH/EECS 343 cannot both be taken for credit. Prereq: MATH 304. Cross-listed as EECS 440.
MATH 413. Graph Theory (3)
Building blocks of a graph, trees, connectedness, transversability connectedness, transversability, matching, coverings, planarity, and NP-complete problems; various applications and algorithms. Prereq: MATH 201 or MATH 308.
MATH 415. Group Representation Theory (3)
Representation and character theory of finite groups and certain (infinite) compact groups. Fundamental concepts and methods of the theory together with examples which are useful, particularly in quantum chemistry or physics. Suitable for undergraduates and graduates who have some acquaintance with linear algebra and group theory. Prereq: MATH 308.
MATH 421. Fundamentals of Analysis I (3)
( See MATH 321.) Additional work required. (May not be taken for credit by graduate students in the Department of Mathematics.) Coreq: MATH 223.
MATH 422. Fundamentals of Analysis II (3)
(See MATH 322.) Additional work required. (May not be taken for credit by graduate students in the Department of Mathematics.) Prereq: MATH 321.
MATH 423. Introduction to Real Analysis I (3)
General theory of measure and integration. Measures and outer measures. Lebesgue measure on-N-space. Integration. Convergence theorems. Product measures and Fubini's theorem. Signed measures. Hahn-Jordan decomposition, Radon-Nikodym theorem, and Lebesgue decomposition. Lp spaces. Lebesgue differentiation theorem in N-space. Prereq: MATH 322.
MATH 424. Introduction to Real Analysis II (3)
Measures on locally compact spaces. Riesz representation theorem. Elements of functional analysis. Normed linear space. Hahn-Banach, Banach-Steinhaus, open mapping, closed graph theorems. Weak topologies. Banach-Alaoglu theorem. Function spaces. Stone-Weierstrass and Ascoli theorems. Basic Hilbert space theory. Application to Fourier series. Additional topics: Haar measure on locally compact groups. Prereq: MATH 423.
MATH 425. Complex Analysis I (3)
Analytic functions. Integration over paths in the complex plane. Index of a point with respect to a closed path; Cauchy's theorem and Cauchy's integral formula; power series representation; open mapping theorem; singularities; Laurent expansion; residue calculus; harmonic functions; Poisson's formula; Riemann mapping theorem. More theoretical and at a higher level than MATH 324. Prereq: MATH 322.
MATH 427. Convexity and Optimization (3)
(See MATH 327.) Cross-listed as OPRE 427.
MATH 428. Fourier Analysis (3)
Introduction to the mathematical aspects of Fourier analysis and synthesis. Accessible to upper level undergraduates and graduate students in the sciences and engineering. Periodic functions. Fourier series. Convergence theorems. The classical sine and cosine series. General orthogonal systems. Multiple Fourier series. Applications. Fourier integrals and Fourier Transforms. L^1 and L^2 theory. Inversion theorems. Classical sine and cosine transforms. Multiple Fourier Transform. Spherical symmetry. Other important transforms. Applications. In addition to the prerequisite listed below, MATH 321 and MATH 201 are recommended. Prereq: MATH 224.
MATH 431. Introduction to Numerical Analysis I (3)
Numerical linear algebra for scientists and engineers. Matrix and vector norms, computer arithmetic, conditioning and stability, orthogonality. Least squares problems: QR factorization, normal equations and Singular Value Decomposition. Direct solution of linear system: Gaussian elimination and Cholesky factorization. Eigenvalues and eigenvectors: the QR algorithm, Rayleigh quotient, inverse iteration. Introduction to iterative methods. Students will be introduced to MATLAB. Prereq: MATH 201 or MATH 308.
MATH 432. Numerical Differential Equations (3)
Numerical solution of differential equations for scientists and engineers. Solution of ordinary differential equations by multistep and single step methods. Stability, consistency, and convergence. Stiff equations. Finite difference schemes. Introduction to the finite element method. Introduction to multigrid techniques. The diffusion equation: numerical schemes and stability analysis. Introduction to hyperbolic equations. MATLAB will be used in this course.
MATH 433. Numerical Solutions of Nonlinear Systems and Optimization (3)
The course provides an introduction to numerical solution methods for systems of nonlinear equations and optimization problems. The course is suitable for upper-undergraduate and graduate students with some background in calculus and linear algebra. Knowledge of numerical linear algebra is helpful. Among the topics which will be covered in the course are Nonlinear systems in one variables; Newton's method for nonlinear equations and unconstrained minimization; Quasi-Newton methods; Global convergence of Newton's methods and line searches; Trust region approach; Secant methods; Nonlinear least squares. Prereq: MATH 223, MATH 201, MATH 431 or permission.
MATH 434. Optimization of Dynamic Systems (3)
Fundamentals of dynamic optimization with applications to control. Variational treatment of control problems and the Maximum Principle. Structures of optimal systems; regulators, terminal controllers, time-optimal controllers. Sufficient conditions for optimality. Singular controls. Computational aspects. Selected applications. Cross-listed as EECS 421.
MATH 439. Integrated Numerical and Statistical Computation (3)
This course will embed numerical methods into a Bayesian framework. The statistical framework will make it possible to integrate a priori 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 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.
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 then see how to 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 445. Introduction to Partial Differential Equations (3)
Method of characteristics for linear and quasilinear equations. Second order equations of elliptic, parabolic, type; initial and boundary value problems. Method of separation of variables, eigenfunction expansions, Sturm-Liouville theory. Fourier, Laplace, Hankel transforms; Bessel functions, Legendre polynomials. Green's functions. Examples include: heat diffusion, Laplace's equation, wave equations, one dimensional gas dynamics and others. Appropriate for seniors and graduate students in science, engineering, and mathematics. Prereq: MATH 201 and MATH 224.
MATH 448. Applied Partial Differential Equations (3)
Continuation of MATH 445. Linear and nonlinear partial differential equations, with emphasis on applications. Variational methods; asymptotic and perturbation methods: regular and singular perturbations; boundary layer, multiple scales, method of geometric optics and stationary phase. Applications to fluid dynamics, elasticity; optics; wave propagation. Topics depend upon instructor and may vary from year to year. Appropriate for seniors and graduate students in science, engineering and mathematics. Prereq: MATH 445.
MATH 450. Domain Theoretic Methods for Artificial Intelligence (3)
Resolution for propositional logic and completeness via Zorn's Lemmq, Domain theory and topology through three-value logic. Default reasoning and extensions. Clausal logic for Scott domains and Smyth power domains. Power defaults theory and the semantics of nonmonotonic reasoning and disjunctive logic programming. Prereq: EECS 343, EECS 391, MATH 307, or PHIL 306. Cross-listed as EECS 459X.
MATH 452. Continuum Mechanics (3)
Kinematics of deformation. Tensors. Mathematical and physical formulation of continuum mechanics. Thermo-dynamical notions. Constitutive relations. Simple nonlinear materials with memory. Role of the classical theories of solids and fluids. Modern developments. MATH 201 recommended. Prereq: MATH 224.
MATH 460. Mathematics and the Imagination (3)
This course explores mathematical ideas in geometry, algebra, and combinatorics relating to content areas in the secondary school curriculum. The course is structured around a series of problems and projects not generally covered in the undergraduate curriculum. This course is designed for present and future mathematics teachers in secondary schools. It is offered as an intensive, three-week seminar. Requirements for the class include daily reading assignments and problems taken from the readings. Considerable time will be devoted to group work. Each student will be required to prepare a report and make a 30-minute presentation to the class on a topic relevant to the materials developed in the course.
MATH 461. Introduction to Topology (3)
Metric spaces, topological spaces, and continuous functions. Compactness, connectedness, path connectedness. Topological manifolds; topological groups. Polyhedra, simplical complexes. Fundamental groups. Prereq: MATH 224.
MATH 462. Algebraic Topology (3)
The fundamental group and covering spaces; van Kampen's theorem. Higher homotopy groups; long-exact sequence of a pair. Homology theory; chain complexes; short and long exact sequences; Mayer-Vietoris sequence. Homology of surfaces and complexes; applications. Prereq: MATH 461.
MATH 465. Differential Geometry (3)
Manifolds and differential geometry. Vector fields; Riemannian metrics; curvature; intrinsic and extrinsic geometry of surfaces and curves; structural equations of Riemannian geometry; the Gauss-Bonnet theorem. Prereq: MATH 321.
MATH 467. Differentiable Manifolds (3)
Differentiable manifolds and structures on manifolds. Tangent and cotangent bundle; vector fields; differential forms; tensor calculus; integration and Stokes' theorem. May include Hamiltonian systems and their formulation on manifolds; symplectic structures; connections and curvature; foliations and integrability. Prereq: MATH 322.
MATH 469. Calculus of Variations (3)
Examples of variational problems; variation of a functional; linear spaces; Frechet derivative; Euler Lagrange equations; Lagrange multipliers; Hamiltonian formulation; canonical coordinates; Noether's theorem; second variation; conjugate points; direct methods. Other topics such as existence and regularity of solutions; Sobolev spaces; depending on audience. Prereq: MATH 224.
MATH 471. Advanced Engineering Mathematics (3)
Vector analysis, Fourier series and integrals. Laplace transforms, separable partial differential equations, and boundary value problems. Bessel and Legendre functions. Emphasis on techniques and applications. Students may not take both MATH 345 and MATH 471 for credit. Prereq: MATH 224.
MATH 475. Mathematics of Imaging in Industry and Medicine (3)
The mathematics of image reconstruction; properties of radon transform, relation to Fourier transform; inversion methods, including convolution, backprojection, rho-filtered layergram, algebraic reconstruction technique (ART), and orthogonal polynomial expansions. Reconstruction from fan beam geometry limited angle techniques used in NMR; survey of applications. Prereq: PHYS 431 and MATH 345 or MATH 471.
MATH 478: Computational Neuroscience (3)
Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems. Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural circuits, as well as a cable theory, passive and active compartmental modeling, numerical integration methods, models of plasticity and learning, models of brain systems, and their relationship to artificial and neural networks. Term project
required. Students enrolled in MATH 478 will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Prereq: MATH 223 & MATH 224 or BIOL 300 & BIOL 306, or consent of department. Cross-listed as EBME 478, EECS 478, MATH 478, NEUR 478
MATH 487. Stochastic Processes in Engineering and Science (3)
Review of basic probability concepts. Discrete-time Markov chains. Transition probability matrices. Classification of states. Stationary distributions. Limiting behavior. Random walk; application to the gambler's ruin problem. Branching processes; application to population growth models. Examples of continuous time Markov chains. Poisson and compound Poisson processes. Birth and death processes. Limiting behavior. Renewal processes. Examples are drawn from queuing theory, reliability theory, population growth processes and other biological models. Prereq: MATH 380.
MATH 491. Probability I (3)
Probabilistic concepts. Discrete probability, elementary distributions. Measure theoretic framework of probability theory. Probability spaces, sigma algebras, expectations, distributions. Independence. Classical results on almost sure convergence of sums of independent random variables. Kolmogorov's law of large numbers. Recurrence of sums. Weak convergence of probability measures. Inversion, Levy's continuity theorem. Central limit theorem. Introduction to the central limit problem. Prereq: MATH 423.
MATH 492. Probability II (3)
Conditional expectations. Discrete parameter martingales. Stopping times, optional stopping. Discrete parameter stationary processes and ergodic theory. Discrete time Markov processes. Introduction to continuous parameter stochastic processes. Kolmogorov's consistency theorem. Gaussian processes. Brownian motion theory (sample path properties, strong Markov property, Martingales associated to Brownian motion, functional central limit theorem). Prereq: MATH 491.
MATH 495. Combinatorics (3)
Permutations, combinations and variations. Principle of inclusion and exclusion. Generating functions. Difference equations. Partitions. Stirling numbers. Eulerian numbers. Ballot problems. Ramsey's theorem. Finite groups. Polya's theorem. Debruijn's theorem. Graphs. Trees. Finite fields. Finite geometries. Orthogonal Latin squares. Hadamard matrices. Block designs. Coding theory. Prereq: MATH 321 or consent.
MATH 499. Special Topics (3)
Special topics in mathematics.
MATH 501. Topics in Algebra (3)
Selected topics from fields, rings, and modules. Prereq: MATH 402.
MATH 527. Functional Analysis (3)
Selected topics in Functional Analysis. Prereq: MATH 424 and MATH 425.
MATH 563. Topology Seminar (1-3)
Continuing seminar on areas of current interest in topology and geometry. Topics may include: minimal submanifolds; hyperbolic geometry and diffeomorphisms of surfaces; global analysis; discrete dynamical systems; gauge theory; symplectic geometry; closed geodesics. May be taken more than once for credit.
MATH 601. Reading and Research Problems (1-18)
Presentation of individual research, discussion, and investigation of research papers in a specialized field of mathematics.
MATH 651. Thesis (M.S.) (1-18)
MATH 701. Dissertation (Ph.D.) (1-18)
MATH 702. Appointed Dissertation Fellow (9)
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