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Survey of mathematical methods for engineers and scientists. Ordinary differential equations and Green's functions; partial differential equations and separation of variables; special functions, Fourier series. Applications to engineering and science. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics. Credit for this course and MAÌý401 is not allowed.

Prerequisite: MAÌý341; credit not allowed for both MAÌý501 and MAÌý401

Typically offered in Fall, Spring, and Summer

Determinants and matrices; line and surface integrals, integral theorems; complex integrals and residues; distribution functions of probability. Not for credit by mathematics majors. Any student receiving credit for MAÌý502 may receive credit for, atmost, one of the following: MAÌý405, MA 512, MAÌý513

Prerequisite: MAÌý341.

Typically offered in Spring only

Basic concepts of linear, nonlinear and dynamic programming theory. Not for majors in OR at Ph.D. level.

Prerequisite: MAÌý242, MAÌý405

Typically offered in Fall only

Introduction including: applications to economics and engineering; the simplex and interior-point methods; parametric programming and post-optimality analysis; duality matrix games, linear systems solvability theory and linear systems duality theory; polyhedral sets and cones, including their convexity and separation properties and dual representations; equilibrium prices, Lagrange multipliers, subgradients and sensitivity analysis.

Prerequisite: An introductory linear algebra course similar to MAÌý405

Typically offered in Fall only

A broad overview of topics in analysis. Historical development, logical refinement and applications of concepts such as limits, continuity, differentiation and integration. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics.

Prerequisite: Graduate standing

Typically offered in Fall, Spring, and Summer

This course is offered alternate years

A broad overview of topics in geometry. Various approaches to study of geometry, including vector geometry, transformational geometry and axiomatics. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics.

Prerequisite: Graduate standing

Typically offered in Fall, Spring, and Summer

This course is offered alternate years

A broad overview of topics in abstract algebra. Theory of equations, polynomial rings, rational functions and elementary number theory. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics.

Prerequisite: Graduate standing

Typically offered in Fall, Spring, and Summer

This course is offered alternate years

Coverage of various topics in mathematics of concern to secondary teachers. Topics selected from areas such as mathematics of finance, probability, statistics, linear programming and theory of games, intuitive topology, recreational math, computers and applications of mathematics. Course may be taken for graduate credit for certification renewal by secondary school teachers. Credit towards a graduate degree may be allowed only for students in mathematics education.

Prerequisite: Graduate standing

Typically offered in Spring and Summer

This course is offered alternate years

Fundamental theorems on continuous functions; convergence theory of sequences, series and integrals; the Riemann integral. Credit for both MAÌý425 and MAÌý511 is not allowed

Prerequisite: MAÌý341

Typically offered in Fall and Spring

Operations with complex numbers, derivatives, analytic functions, integrals, definitions and properties of elementary functions, multivalued functions, power series, residue theory and applications, conformal mapping.

Prerequisite: MAÌý242

Typically offered in Fall and Spring

Cryptography is the study of mathematical techniques for securing digital information, systems and distributed computation against adversarial attacks. In this class you will learn the concepts and the algorithms behind the most used cryptographic protocols: you will learn how to formally define security properties and how to formally prove/disprove that a cryptographic protocol achieves a certain security property. You will also discover that cryptography has a much broader range of applications. It solves absolutely paradoxical problems such as proving knowledge of a secret without ever revealing the secret (zero-knowledge proof), or computing the output of a function without ever knowing the input of the function (secure computation). Finally, we will look closely at one of the recent popular application of cryptography: the blockchain technology. Additionally, graduate students will study some of the topics in greater depth.

Prerequisite: (CSCÌý226 AND CSCÌý333) OR MAÌý225

Typically offered in Fall only

Metric spaces: contraction mapping principle, Tietze extension theorem, Ascoli-Arzela lemma, Baire category theorem, Stone-Weierstrass theorem, LP spaces. Banach spaces: linear operators, Hahn-Banach theorem, open mapping and closed graph theorems. Hilbert spaces: projection theorem, Riesz representation theorem, Lax-Milgram theorem, complete orthonormal sets.

Prerequisite: MAÌý426

Typically offered in Fall only

Geometry of curves and surfaces in space; Arclength, torsion, and curvature of curves; Tangent spaces, shape operators, and curvatures of surfaces; metrics, covariant derivatives, geodesics, and holonomy. Applications in the physical sciences and/or projects using computer algebra.

Prerequisite: MAÌý242 and MAÌý405

Typically offered in Spring only

Vector spaces. Bases and dimension. Changes of basis. Linear transformations and their matrices. Linear functionals. Simultaneous triangularization and diagonalization. Rational and Jordan canonical forms. Bilinear forms.

Prerequisite: MAÌý405

Typically offered in Fall and Spring

Groups, quotient groups, group actions, Sylow's Theorems. Rings, ideals and quotient rings, factorization, principal ideal domains. Fields, field extensions, Galois theory.

Prerequisite: MAÌý405 and MAÌý407

Typically offered in Fall only

Basic techniques and algorithms of computer algebra. Integer arithmetic, primality tests and factorization of integers, polynomial arithmetic, polynomial factorization, Groebner bases, integration in finite terms.

Prerequisite: MAÌý405 and MAÌý407

Typically offered in Fall only

Vector spaces, linear transformations and matrices, orthogonality, orthogonal transformations with emphasis on rotations and reflections, matrix norms, projectors, least squares, generalized inverses, definite matrices, singular values.

Prerequisite: MAÌý405

Typically offered in Fall and Spring

Enumerative combinatorics, including placements of balls in bins, the twelvefold way, inclusion/exclusion, sign-reversing involutions and lattice path enumeration. Partically ordered sets, lattices, distributive lattices, Moebius functions, and rational generating functions.

Prerequisite: MAÌý405 and MAÌý407

Typically offered in Fall only

The course covers (i) structure and operation of derivative markets, (ii) valuation of derivatives, (iii) hedging of derivatives, and (iv) applications of derivatives in areas of risk management and financial engineering. Models and pricing techniques include Black-Scholes model, binomial trees, Monte-Carlo simulation. Specific topics include simple no-arbitrage pricing relations for futures/forward contracts; put-call parity relationship; delta, gamma, and vega hedging; implied volatility and statistical properties; dynamic hedging strategies; interest-rate risk, pricing of fixed-income product; credit risk, pricing of defaultable securities.

Prerequisites: MAÌý341 and MAÌý405 and MAÌý421

Typically offered in Fall only

Introduction to modeling, analysis and control of linear discrete-time and continuous-time dynamical systems. State space representations and transfer methods. Controllability and observability. Realization. Applications to biological, chemical, economic, electrical, mechanical and sociological systems.

Prerequisite: MAÌý341, MAÌý405

Typically offered in Fall only

Existence and uniqueness theorems, systems of linear equations, fundamental matrices, matrix exponential, nonlinear systems, plane autonomous systems, stability theory.

Prerequisite: MAÌý341, 405, 425 or 511, Corequisite: MAÌý426 or 512

Typically offered in Fall only

Linear first order equations, method of characteristics. Classification of second order equations. Solution techniques for the heat equation, wave equation and Laplace's equation. Maximum principles. Green's functions and fundamental solutions.

Prerequisite: MAÌý425 or MAÌý511, MAÌý341, Corequisite: MAÌý426 or 512

Typically offered in Fall only

Usage of computer experiments for demonstration of nonlinear dynamics and chaos and motivation of mathematical definitions and concepts. Examples from finance and ecology as well as traditional science and engineering. Difference equations and iteration of functions as nonlinear dynamical systems. Fixed points, periodic points and general orbits. Bifurcations and transition to chaos. Symbolic dynamics, chaos, Sarkovskii's Theorem, Schwarzian derivative, Newton's method and fractals.

Prerequisite: MAÌý341 and MAÌý405

Typically offered in Spring only

Introduction to uncertainty quantification for physical and biological models. Parameter selection techniques, Bayesian model calibration, propagation of uncertainties, surrogate model construction, local and global sensitivity analysis.

Prerequisite: MAÌý341 and basic knowledge of probability, linear algebra, and scientific computation

Typically offered in Fall and Spring

This course is offered alternate even years

Convex optimization methods and their applications in various areas of data science including, but not limited to, signal and image processing, inverse problems, statistical data analysis, machine learning and classification. Basic theory, algorithm design and concrete applications.

Prerequisite: MAÌý141, 241, 242, or equivalent and MAÌý405 or equivalent; Some notions of elementary convex analysis are an asset but are neither required nor assumed known.

Typically offered in Fall only

Exposure of student to practice of performing mathematical experiments on computer, with emphasis on probability. Programming in an interactive language such as APL, MATLAB or Mathematica. Mathematical treatment of random number generation and application of these tools to mathematical topics in Monte Carlo method, limit theorems and stochastic processes for purpose of gaining mathematical insight.

Prerequisite: MAÌý421

Typically offered in Spring only

Mathematical foundations of probability theory. Probabilistic measure theory, random variables and their distributions, construction of expectation. Notions of convergence: almost sure, in probability, in L^p, weak convergence, vague convergence. Conditioning, independence, Borel-Cantelli lemmas, weak and strong laws of large numbers, characteristic functions, central limit theorem, and related concentration inequalities.

Prerequisite: MAÌý421 and MAÌý425 or MAÌý511

Typically offered in Fall only

This course explores stochastics calculus with its applications in pricing and hedging problems for financial derivatives such as options. Topics to be covered in the course include 1) discrete and continuous martingales, 2) Brownian motions and Ito's stochastic calculus, and 3) Black-Scholas framework for financial derivatives pricing and hedging.

Prerequisite: FIMÌý528 and MA(ST) 546

Typically offered in Spring only

Monte Carlo (MC) methods for accurate option pricing, hedging and risk management. Modeling using stochastic asset models (e.g. geometric Brownian motion) and parameter estimation. Stochastic models, including use of random number generators, random paths and discretization methods (e.g. Euler-Maruyama method), and variance reduction. Implementation using Matlab. Incorporation of the latest developments regarding MC methods and their uses in Finance.

Prerequisites: (MAÌý421 or STÌý421), MAÌý341, and MAÌý405

Typically offered in Spring only

This course focuses on mathematical methods to analyze and manage risks associated with financial derivatives. Topics covered include aggregate loss distributions, extreme value theory, default probabilities, Value-at-Risk and expected shortfall, coherent risk measures, correlation and copula, applications of principle component analysis and Monte Carlo simulations in financial risk management, how to use stochastic differential equations to price financial risk derivatives, and how to back-test and stress-test models.

Prerequisites: MAÌý405 and (MAÌý421 or STÌý421) and (MA/STÌý412 or MA/STÌý413)

Typically offered in Spring only

Set theory, topological spaces, metric spaces, continuous functions, separation, cardinality properties, product and quotient topologies, compactness, connectedness.

Prerequisite: MAÌý426

Typically offered in Fall only

An introduction to smooth manifolds. Topics include: topological and smooth manifolds, smooth maps and differentials, vector fields and flows, Lie derivatives, vector bundles, tensors, differential forms, exterior calculus, and integration on manifolds.

Prerequisite: MAÌý405 and MAÌý426

Typically offered in Fall only

Logic and axiomatic approach, the Zermelo-Fraenkel axioms and other systems, algebra of sets and order relations, equivalents of the Axiom of Choice, one-to-one correspondences, cardinal and ordinal numbers, the Continuum Hypothesis.

Prerequisite: MAÌý407

Typically offered in Spring only

Basic concepts of graph theory. Trees and forests. Vector spaces associated with a graph. Representation of graphs by binary matrices and list structures. Traversability. Connectivity. Matchings and assignment problems. Planar graphs. Colorability. Directed graphs. Applications of graph theory with emphasis on organizing problems in a form suitable for computer solution.

Prerequisite: CSCÌý226 or MAÌý351.

Typically offered in Spring only

This course is offered alternate even years

Introduction to model development for physical and biological applications. Mathematical and statistical aspects of parameter estimation. Compartmental analysis and conservation laws, heat transfer, and population and disease models. Analytic and numerical solution techniques and experimental validation of models. Knowledge of high-level programming languages required.

Prerequisite: MAÌý341 and knowledge of high-level programming language.

Typically offered in Fall only

Model development, using Newtonian and Hamiltonian principles, for acoustic and fluid applications, and structural systems including membranes, rods, beams, and shells. Fundamental aspects of electromagnetic theory. Analytic and numerical solution techniques and experimental validation of models.

Prerequisite: MA/BMAÌý573

Typically offered in Spring only

Algorithm behavior and applicability. Effect of roundoff errors, systems of linear equations and direct methods, least squares via Givens and Householder transformations, stationary and Krylov iterative methods, the conjugate gradient and GMRES methods, convergence of method.

Prerequisite: MAÌý405; MAÌý425 or MAÌý511; high-level computer language

Typically offered in Fall and Spring

Introduction to basic parallel architectures, algorithms and programming paradigms; message passing collectives and communicators; parallel matrix products, domain decomposition with direct and iterative methods for linear systems; analysis of efficiency, complexity and errors; applications such as 2D heat and mass transfer.

Prerequisite: CSCÌý302 or MAÌý402 or MA/CSCÌý428 or MA/CSCÌý580

Typically offered in Spring only

Survey of finite difference methods for partial differential equations including elliptic, parabolic and hyperbolic PDE's. Consideration of both linear and nonlinear problems. Theoretical foundations described; however, emphasis on algorithm design and implementation.

Prerequisite: MAÌý501; knowledge of a high level programming language

Typically offered in Fall only

Introduction to finite element method. Applications to both linear and nonlinear elliptic and parabolic partial differential equations. Theoretical foundations described; however, emphasis on algorithm design and implementation.

Prerequisite: MAÌý501; knowledge of a high level programming language

Typically offered in Spring only

Typically offered in Fall and Spring

Review and discussion of scientific articles, progress reports on research and special problems of interest to mathematicians.

P: Graduate Standing

Typically offered in Fall and Spring

Independent study of an advanced mathematics topic under the direction of mathematics faculty member on tutorial basis. Requires a faculty sponsor and departmental approval.

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

Readings in advanced topics in mathematics

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

Investigation of some topic in mathematics to a deeper and broader extent than typically done in a classroom situation. For the applied mathematics student the topic usually consists of a realistic application of mathematics to student's minor area.A written and oral report on the project required.

Typically offered in Fall, Spring, and Summer

Teaching experience under the mentorship of faculty who assist the student in planning for the teaching assignment, observe and provide feedback to the student during the teaching assignment, and evaluate the student upon completion of the assignment.

Prerequisite: Master's student

Typically offered in Fall and Spring

For students in non-thesis master's programs who have completed all credit hour requirements for their degree but need to maintain full-time continuous registration to complete incomplete grades, projects, final master's exam, etc. Students may register for this course a maximum of one semester.

Prerequisite: Master's student

Typically offered in Fall, Spring, and Summer

For students in non thesis master's programs who have completed all other requirements of the degree except preparing for and taking the final master's exam.

Prerequisite: Master's student

Typically offered in Fall and Spring

Instruction in research and research under the mentorship of a member of the Graduate Faculty.

Prerequisite: Master's student

Typically offered in Fall, Spring, and Summer

Thesis Research

Prerequisite: Master's student

Typically offered in Fall and Spring

For graduate students whose programs of work specify no formal course work during a summer session and who will be devoting full time to thesis research.

Prerequisite: Master's student

Typically offered in Summer only

For students who have completed all credit hour requirements and full-time enrollment for the master's degree and are writing and defending their thesis. Credits Arranged

Prerequisite: Master's student

Typically offered in Fall, Spring, and Summer

An advanced mathematical treatment of analytical and algorithmic aspects of finite dimensional nonlinear programming. Including an examination of structure and effectiveness of computational methods for unconstrained and constrained minimization. Special attention directed toward current research and recent developments in the field.

Prerequisite: OR(IE,MA) 505 and MAÌý425

Typically offered in Spring only

General integer programming problems and principal methods of solving them. Emphasis on intuitive presentation of ideas underlying various algorithms rather than detailed description of computer codes. Students have some "hands on" computing experience that should enable them to adapt ideas presented in course to integer programming problems they may encounter.

Prerequisite: MAÌý405, OR (MA,IE) 505, Corequisite: Some familiarity with computers (e.g., CSCÌý112)

Typically offered in Spring only

This course is offered alternate years

Integration: Legesgue measure and integration, Lebesgue-dominated convergence and monotone convergence theorems, Fubini's theorem, extension of the fundamental theorem of calculus. Banach spaces: Lp spaces, weak convergence, adjoint operators, compact linear operators, Fredholm-Fiesz Schauder theory and spectral theorem.

Prerequisite: MAÌý515

Typically offered in Spring only

Advanced topics in functional analysis such as linear topological spaces; Banach algebra, spectral theory and abstract measure theory and integration.

Prerequisite: MAÌý715

Typically offered in Fall only

This course is offered alternate years

Introduction to algebraic and function-analytic concepts used in system modeling and optimization: vector space, linear mappings, spectral decomposition, adjoints, orthogonal projection, quality, fixed points and differentials. Emphasis on geometricinsight. Topics include least square optimization of linear systems, minimum norm problems in Banach space, linearization in Hilbert space, iterative solution of system equations and optimization problems. Broad range of applications in operations research and system engineering including control theory, mathematical programming, econometrics, statistical estimation, circuit theory and numerical analysis.

Prerequisite: MAÌý405, 511

Typically offered in Fall only

Definition of Lie algebras and examples. Nilpotent, solvable and semisimple Lie algebras. Engel's theorem, Lie's Theorem, Killing form and Cartan's criterion. Weyl's theorem on complete reducibility. Representations of s1(2,C). Root space decomposition of semisimple Lie algebras. Root system and Weyl group.

Prerequisite: MAÌý520

Typically offered in Spring only

This course covers: Module theory including the structure theory of modules over a PID and primary decomposition; Tensor, exterior, and symmetric algebras; introductory homological algebra including: complexes, derived functors, Ext and Tor; and the representation theory of groups. Further topics will be covered as time permits.

Prerequisite: MAÌý521

Typically offered in Spring only

Effective algorithms for symbolic matrices, commutative algebra, real and complex algebraic geometry, and differential and difference equations. The emphasis is on the algorithmic aspects.

Prerequisite: MAÌý522

Typically offered in Spring only

Canonical forms, functions of matrices, variational methods, perturbation theory, numerical methods, nonnegative matrices, applications to differential equations, Markov chains.

Prerequisite: MAÌý520 or 523

Typically offered in Spring only

Polytopes(V-polytopes and H-polytopes). Fourier-Motzkin elimination, Farkas Lemma, face numbers of polytopes, graphs of polytopes, linear programming for geometers, Balinski's Theorem, Steinitz' Theorem, Schlegel diagrams, polyhedral complexes, shellability, and face rings.

Prerequisite: MAÌý524

Typically offered in Spring only

Semisimple Lie algebras, root systems, Weyl groups, Cartan matrices and Dynkin diagrams, universal enveloping algebras, Serre's Theorem, Kac-Moody algebras, highest weight representations of finite dimensional semisimple algebras and affine Lie algebras, Kac-Weyl character formula.

Prerequisite: MAÌý720

Typically offered in Fall only

This course is offered alternate odd years

Abstract theory of solutions of systems of polynomial equations. Topics covered: ideals and affine varieties, the Nullstellensatz, irreducible varieties and primary decomposition, morphisms and rational maps, computational aspects including Groebner bases and elimination theory, projective varieties and homogeneous ideals, Grassmannians, graded modules, the Hilbert function, Bezout's theorem.

Prerequisite: MAÌý521

Typically offered in Spring only

Stability of equilibrium points for nonlinear systems. Liapunov functions. Unconstrained and constrained optimal control problems. Pontryagin's maximum principle and dynamic programming. Computation with gradient methods and Newton methods. Multidisciplinary applications.

Prerequisite: OR(E,MA) 531

Typically offered in Spring only

This course is offered alternate years

Existence-uniqueness theory, periodic solutions, invariant manifolds, bifurcations, Fredholm's alternative.

Prerequisite: MAÌý532, Corequisite: MAÌý515

Typically offered in Spring only

Linear second order parabolic, elliptic and hyperbolic equations. Initial value problems and boundary value problems. Iterative and variational methods. Existence, uniqueness and regularity. Nonlinear equations and systems.

Prerequisite: MAÌý515 and MAÌý534

Typically offered in Spring only

Advanced development of stochastic processes. Conditional expectation, filtrations of sigma-algebras, stopping times. Martingales, associated convergence theorems and inequalities, martingale decomposition, optional stopping. Markov chains including random walks, recurrence versus transience, asymptotic behavior. General Markov processes and the related semigroup operators. Construction and properties of Brownian motion, Donsker's invariance principle. Other potential topics include stationary processes, Birkhoff's ergodic theorem, branching processes, Poisson processes.

Prerequisite: MA(ST) 546

Typically offered in Spring only

Theory of stochastic differential equations driven by Brownian motions. Current techniques in filtering and financial mathematics. Construction and properties of Brownian motion, wiener measure, Ito's integrals, martingale representation theorem, stochastic differential equations and diffusion processes, Girsanov's theorem, relation to partial differential equations, the Feynman-Kac formula.

Prerequisite: MA(ST) 747

Typically offered in Fall only

Homotopy, fundamental group, covering spaces, classification of surfaces, homology and cohomology.

Prerequisite: MAÌý551 or MAÌý555

Typically offered in Spring only

Properties of cohomology, homotopy groups, fiber bundles, characteristic classes, and homological algebra. Additional topics may include spectra, spectral sequences, K-theory, group cohomology, and connections with smooth manifold topology.

Prerequisite: MAÌý753

Typically offered in Fall only

This course is offered alternate odd years

An introduction to smooth manifolds with metric. Topics include: Riemannian metric and generalizations, connections, covariant derivatives, parallel translation, Riemannian (or Levi-Civita) connection, geodesics and distance, curvature tensor, Bianchi identities, Ricci and scalar curvatures, isometric embeddings, Riemannian submanifolds, hypersurfaces, Gauss Bonnet Theorem; applications and connections to other fields.

Prerequisite: MAÌý555

Typically offered in Spring only

This course is offered alternate years

Study of problems of flows in networks. These problems include the determination of shortest chain, maximal flow and minimal cost flow in networks. Relationship between network flows and linear programming developed as well as problems with nonlinear cost functions, multi-commodity flows and problem of network synthesis.

Prerequisite: OR(IE,MA) 505

Typically offered in Spring only

This course is offered alternate years

Role of theory construction and model building in development of experimental science. Historical development of mathematical theories and models for growth of one-species populations (logistic and off-shoots), including considerations of age distributions (matrix models, Leslie and Lopez; continuous theory, renewal equation). Some of the more elementary theories on the growth of organisms (von Bertalanffy and others; allometric theories; cultures grown in a chemostat). Mathematical theories oftwo and more species systems (predator-prey, competition, symbosis; leading up to present-day research) and discussion of some similar models for chemical kinetics. Much emphasis on scrutiny of biological concepts as well as of mathematical structureof models in order to uncover both weak and strong points of models discussed. Mathematical treatment of differential equations in models stressing qualitative and graphical aspects, as well as certain aspects of discretization. Difference equation models.

Prerequisite: Advanced calculus, reasonable background in biology

Typically offered in Fall only

Theory of stochastic processes and its application to contemporary problems in biology. Discrete- and continuous-time Markov chains, branching processes, birth-and-death processes, diffusion approximations, and elementary stochastic differential equations. Survey of applications in areas such as population genetics, infectious disease dynamics, neurobiology, and community ecology.

Prerequisite: Elementary Probability Theory

Typically offered in Spring only

Survey of modeling approaches and analysis methods for data from continuous state random processes. Emphasis on differential and difference equations with noisy input. Doob-Meyer decomposition of process into its signal and noise components. Examples from biological and physical sciences, and engineering. Student project.

Prerequisite: BMAÌý772 or ST (MA) 746

Typically offered in Spring only

This course is offered alternate years

Modeling with and analysis of partial differential equations as applied to real problems in biology. Review of diffusion and conservation laws. Waves and pattern formation. Chemotaxis and other forms of cell and organism movement. Introduction to solid and fluid mechanics/dynamics. Introductory numerical methods. Scaling. Perturbations, Asymptotics, Cartesian, polar and spherical geometries. Case studies.

Prerequisite: BMAÌý771 or MA/ORÌý731; BMAÌý772 or MAÌý401 or MAÌý501

Typically offered in Spring only

Approximation and interpolation, Fast Fourier Transform, numerical differentiation and integration, numerical solution of initial value problems for ordinary differential equations.

Prerequisite: MAÌý580

Typically offered in Spring only

Computational methods for inverse problems that are governed by partial differential equations. Topics will include variational formulations, ill-posedness, regularization, discretization methods, and optimization algorithms, statistical formulations, and numerical implementations.

P: MAÌý401 and MAÌý580 or equivalent.

Typically offered in Spring only

The course provides a graduate-level introduction to the numerical methods of solving linear and nonlinear optimization problems and nonlinear equations, along with the fundamental mathematical theory necessary to develop these algorithms. Topics selected from: Newton's method and Quasi-Newton methods for nonlinear equations and optimization problems, globally convergent extensions, methods for sparse problems, applications to differential equations, integral equations and general minimization problems, methods appropriate for boundary value problems, conic programming, first-order methods for large-scale optimization problems.

P: MAÌý580 or MAÌý523

Typically offered in Spring only

A review of modern numerical techniques for time-dependent nonlinear partial differential equations. Topics include Finite Difference, Finite Volume, Particle and Hybrid Eulerian- Lagrangian Methods; Splitting Methods and Implicit-Explicit Discretization; Spectral and Pseudo-Spectral Methods including Stochastic Galerkin and Stochastic Collocation Methods, and others. Applications including problems in fluid and gas dynamics, geophysics, meteorology, astrophysics, biology, and other fields.

Prerequisite: MAÌý401 or MAÌý427 or MAÌý428; knowledge of a high level programming language

Typically offered in Spring only

This course is offered alternate years

Special advanced topics in mathematics.

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

Typically offered in Fall and Spring

Typically offered in Fall and Spring

Typically offered in Fall and Spring

Study of special advanced topics in area of mathematical programming. Discussion of new techniques and current research in this area. The faculty responsible for this course select areas to be covered during semester according to their preference and interest. This course not necessarily taught by an individual faculty member but can, on occasion, be joint effort of several faculty members from this university as well as visiting faculty from other institutions. To date, a course of Theory of Networks and another on Integer Programming offered under the umbrella of this course. Anticipation that these two topics will be repeated in future together with other topics.

Prerequisite: IE(MA,OR) 505

Typically offered in Spring only

This course is offered alternate years

Advanced topics in some phase of system optimization. Identification of various specific topics and prerequisite for each section from term to term.

Typically offered in Fall and Spring

Independent study of an advanced mathematics topic under the direction of mathematics faculty member on tutorial basis. Requires a faculty sponsor and departmental approval.

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

Readings in advanced topics in mathematics

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

Teaching experience under the mentorship of faculty who assist the student in planning for the teaching assignment, observe and provide feedback to the student during the teaching assignment, and evaluate the student upon completion of the assignment.

Prerequisite: Doctoral student

Typically offered in Fall and Spring

For students who are preparing for and taking written and/or oral preliminary exams.

Prerequisite: Doctoral student

Typically offered in Fall and Spring

Instruction in research and research under the mentorship of a member of the Graduate Faculty.

Prerequisite: Doctoral student

Typically offered in Fall, Spring, and Summer

Dissertation Research

Prerequisite: Doctoral student

Typically offered in Fall and Spring

For graduate students whose programs of work specify no formal course work during a summer session and who will be devoting full time to thesis research.

Prerequisite: Doctoral student

Typically offered in Summer only

For students who have completed all credit hour requirements, full-time enrollment, preliminary examination, and residency requirements for the doctoral degree, and are writing and defending their dissertations.

Prerequisite: Doctoral student

Typically offered in Fall, Spring, and Summer