Graduate Courses

This course is an introduction to the basic ideas of pre-calculus and calculus for the people who need preparation or review before taking more advanced courses. The exact content depends upon the particular needs of those enrolled and the requirements of degree programs they are pursuing. Topics covered will be selected from the following: algebra, functions, and graphs; slopes and secant lines; derivatives; chain rule; optimization; curve sketching; integration; the exponential and natural logarithm; and probability density functions and integration by parts.

This course provides the necessary mathematical prerequisites for the computer science master’s program and also serves as a foundation for further study in mathematics. The topics covered include prepositional calculus: predicates and quantifiers; elementary number theory and methods of proof; mathematical induction; elementary set theory; combinatorics; functions and relations; countability; recursion and O-notation. Applications to computer science are stressed.

Topics include basic discrete probability, including urn models and random mappings; a brief introduction to information theory; elements of number theory, including the prime number theorem, the Euler phi function, the Euclidean algorithm, and the Chinese remainder theorem; and elements of abstract algebra and finite fields including basic fundamentals of groups, rings, polynomial rings, vector spaces, and finite fields. Carries credit toward the Applied Mathematics degree only when followed by CS 668. Recommended for high-level undergraduate students.

Elementary mathematical techniques important to applied mathematics. Topics covered include review of functions and continuity; ordinary and partial derivatives; integration; ordinary and partial differential equations; infinite series and numerical techniques for solving differential equations; and multiple integration and surface integrals. Applications to problems of applied mathematics are given where feasible.

This course is primarily for students interested in using numerical methods to solve problems in mathematics, science, engineering, and management. Computational projects will be a significant part of this course and it is expected that students already have experience programming in at least one high level language. Standard topics include numerical solutions of ordinary and partial differential equations, techniques in numerical linear algebra, the Fast Fourier Transform, optimization methods, and an introduction to parallel programming. Additional topics will depend on the interests of the instructor and students.

Review of limits, continuity, partial differentiation, Leibnitz’s rule; implicit functions and Jacobians; gradients, divergence, curl, line and surface integrals; theorems of Stokes, Gauss and Green; complex numbers, elementary functions, analytic functions, complex integration, power series, residue theorem, evaluation of real definite integrals; systems of linear equations, rank, eigenvalues and eigenvectors.

Review of first order and second order constant coefficient differential equations, nonhomogeneous equations; series solutions, Bessel and Legendre functions; boundary value problems, Fourier-Bessel series and separation of variables for partial differential equations; classification of partial differential equations; Laplace transform methods; calculus of variations; introduction to finite-difference methods.

Difference equations; calculus of variations; integral equations; and applications to engineering and science.

Sample space, events, and probability; basic counting techniques and combinatorial probability; random variables, discrete and continuous; probability mass, probability density, and cumulative distribution functions; expectation and moments; some common distributions; jointly distributed random variables, conditional distributions and independence, bivariate normal, and transformations of variables; and Central Limit Theorem. Some additional topics may include an introduction to confidence intervals and hypothesis testing.

This course offers an introduction to exploratory data analysis and the use of basic statistical tools. Topics will include: data collection; descriptive statistics, and graphical and tabular treatment of quantitative, qualitative, and count data; detecting relations between variables; confidence intervals and hypothesis testing for one and two samples; simple and multiple linear regression; analysis of variance; design of experiments; and nonparametric methods. Selected topics, such as quality control and time series analysis, may also be included. Statistical software will be used throughout the course and statistical inference will be based on examples using real data. Students will participate in group projects of data analysis. They will be trained in the different phases of the professional statistician’s work, namely: data collection, description, analysis, testing, and presentation of the conclusions.

Elementary topology of Euclidean spaces; differential calculus of functions of several variables; inverse and implicit function theorems; integration; differential forms; and theorems of Gauss, Green, and Stokes.

A continuation of MA 547, but with greater emphasis on mathematical rigor. Topics covered may include convergence of series, Riemann-Stieltjes integration, functions of bounded variation, metric spaces, introduction to measure theory, and functional analysis.

Fields and vector spaces; subspaces and quotient spaces; basis and dimension; linear transformations and matrices; determinants; and the theory of a single linear transformation.

Special topics in mathematics not covered in regularly scheduled courses and suitable for both graduates and advanced undergraduates. May be taken more than once.

A unified development of mathematical tools for treating a variety of problems in physics and engineering. Linear algebra, normed and inner product spaces, and spectral theory of operators; integral equations; boundary value problems for ordinary and partial differential equations; Green’s functions; calculus of variations; and other related topics as time permits. Problem solving is stressed.

A unified development of mathematical tools for treating a   variety of problems  in physics and engineering; linear algebra, normed and   inner product spaces,  spectral theory of operators; integral equations; boundary   value problems for  ordinary and partial differential equations; Green's   functions; calculus of  variations; other related topics as time permits; problem   solving is stressed.

Topics covered in the sequence MA 605-606 include: elementary number theory, basic group theory, Lagrange’s theorem, isomorphism theorems, solvability, direct products, Jordan-Holder theorem, Sylow theorems, basic properties of rings, quotient rings, field of quotients of an integral domain, polynomial rings, factorization, elementary properties of fields, field extensions, and Galois theory. 

Topics covered in the sequence MA 605-606 include: elementary number theory, basic group theory, Lagrange’s theorem, isomorphism theorems, solvability, direct products, Jordan-Holder theorem, Sylow theorems, basic properties of rings, quotient rings, field of quotients of an integral domain, polynomial rings, factorization, elementary properties of fields, field extensions, and Galois theory.

Foundations of probability, random variables and their distributions, discrete and continuous random variables, independence, expectation and conditioning, generating functions, multivariate distributions, convergence of random variables, and classical limit theorems.

Point estimation, method of moments, maximum likelihood, and properties of point estimators; confidence intervals and hypothesis testing; sufficiency; Neyman-Pearson theorem, uniformly most powerful tests, and likelihood ratio tests; and Fisher information and the Cramer-Rao inequality. Additional topics may include nonparametric statistics, decision theory, and linear models.

The MA 615-616 sequence covers topics in numerical analysis and numerical methods including: errors and accuracy; polynomial approximation; interpolation; numerical differentiation and integration; numerical solution of differential equations; least square and minimum-maximum error approximations; nonlinear equations; simultaneous linear equations; summing series, Fourier series, filter design, the frequency approach, design of numerical tools, and statistics of error analysis; eigenvalues and eigenvectors of matrices; and the orientation throughout is toward computers.

The MA 615-616 sequence covers topics in numerical analysis and numerical methods including: errors and accuracy; polynomial approximation; interpolation; numerical differentiation and integration; numerical solution of differential equations; least square and minimum-maximum error approximations; nonlinear equations; simultaneous linear equations; summing series, Fourier series, filter design, the frequency approach, design of numerical tools, and statistics of error analysis; eigenvalues and eigenvectors of matrices; and the orientation throughout is toward computers.

This course covers basic ideas in sampling theory and uses only elementary mathematics. Topics include multistage sampling, stratified sampling, systematic sampling, self-weighting samples, and optimum allocation.

Random walks and Markov chains; Brownian motions and Markov processes; and applications, stationary (wide sense) processes, infinite divisibility, and spectral decomposition.

Absolute geometry as founded on axioms of incidence, order, congruence, and continuity; models of absolute geometry and problems of consistency; independence and categoricity of an axiom system; Euclidean and non-Euclidean geometry; brief description of the Erlangen program; and classical differential geometry of surfaces.

Fundamental laws of counting, permutations, combinations, recurrence relations, Mšbius inversion, probleme des menages, probleme des recontres, partitions, trees, generating functions, Ramsey theory, transversal theory, and matroid theory.

This course introduces the students to the foundation of optimization. The first part of the class focuses on basic results of convex analysis and their application to the development of necessary and sufficient conditions of optimality and Lagrangian duality theory. The main numerical methods of optimization and their convergence constitute the second portion of the class. Along with the theoretical results and methods, examples of optimization models in probability, statistics, and approximation theory will be discussed as well as some basic models from management, finance, and other practical situations will be introduced in order to illustrate the discussed notions and phenomena, and to demonstrate the scope of applications. Linear optimization techniques will be treated as a special case. Some attention will be paid to using optimization software such as AMPL and CPLEX in the numerical assignments.

This course introduces the students to the several advanced topics in the theory and methods of optimization. The first portion of the class focuses on subgradient calculus for non-smooth convex functions, optimality conditions for non-smooth optimization problems, conjugate and Lagrangian convex duality. The second part of the class discusses numerical methods for non-smooth optimization as well as approaches to large-scale optimization problems. The latter include decomposition methods, design of distributed and parallel methods of optimization, as well as stochastic approximation methods. Along with the theoretical results and methods, examples of optimization models in statistical learning and data mining, compressed sensing and image reconstruction will be discussed in order to illustrate the challenges and the phenomena, and to demonstrate the scope of applications. Some attention will be paid to using optimization software such as AMPL, CPLEX and SNOPT in the numerical assignments. 

Strategic games and Nash equilibrium, strictly competitive (zero-sum) games and max-minimization, sStrategic games with imperfect information (Bayesian games), extensive games with perfect information (bargaining and repeated games), extensive games with imperfect information and signaling games, coalitional games (the core, stable sets, and bargaining sets), and auctions.

Modern theory of the delta function and other generalized functions: Fourier and Laplace transforms and applications to ordinary and partial differential equations.

Queuing theory, transportation problem, traffic theory, inventory control, search theory, and methods of optimization.

The real number system. Introduction to metric spaces and their applications. Lebesque measure and integral from a classical and/or modern approach.

L-p spaces and applications to Fourier series and Lebesque-Stieltjes integral.

Prepositional calculus; syntax and semantics of first order theories; completeness theorem; elementary model theory: axiomatic development of Zermelo-Fraenkel or Bernays-Gödel set theory; and ordinals, cardinals, the axiom of choice, and several equivalent axioms.

First order number theory; primitive and general recursive functions; arithmetization; Gödel’s incompleteness theorems; Tarski’s theorems; and syntax and semantics of second order theories.

Scope and applications of time series analysis: process control, financial data analysis and forecasting, and signal processing. Exploratory data analysis: graphical analysis, trend and seasonality detection and removal, and moving-average filtering. Review of basic statistical concepts related to the characterization of stationary processes. ARMA models and prediction of stationary processes. Estimation of ARMA models and model building and forecasting with ARMA models. Spectral analysis: periodogram testing for seasonality and periodicities and the maximum entropy and maximum-likelihood estimators. Asymptotic convergence. Selected topics, such as multivariate time series, nonlinear models, Kalman filtering, econometric forecasting, and long-memory processes. Selected applications, such as the unit-root problem in economics, forecasting and testing for market efficiency in financial time series, process control, and quality control.

 Scope and applications of time series analysis: process   control, financial  data analysis and forecasting, signal processing.   Exploratory data analysis:  graphical analysis, trend and seasonality detection and   removal,  moving-average filtering.  Review of basic statistical   concepts related to the  characterization of stationary processes.  ARMA models,   prediction of  stationary processes.  Estimation of ARMA models, model   building and  forecasting with ARMA models.  Spectral analysis:   periodogram testing for  seasonality and periodicities, the maximum entropy and   maximum-likelihood  estimators.  A symptotic convergence.  Selected topics   such as multivariate  time series, nonlinear models, Kalman filtering,   econometric forecasting, and  long-memory processes.  Selected applications such as the   unit-root problem in  economics, forecasting and testing for market efficiency   in financial time  series, process control and quality control.

Theory and application of ordinary differential equations (ODEs) with an emphasis on ODEs as continuous dynamical systems on a finite-dimensional phase space. Standard topics include existence and uniqueness theorems, general theory for linear equations, the exponential of linear map, stability of equilibrium points, hyperbolicity and structural stability, Lyapunov’s method, invariant manifolds, Floquet theory for periodic orbits, and Poincare-Bendixon theorem.

This course discusses the classical theory and applications of partial differential equations and introduces the student to the modern theory. Classification of second order equations; well-posedness; existence and uniqueness for the Cauchy problem; Riemann function; Dirichlet and Neumann problems; Green’s functions; perturbation theory; elliptic operators; variational formulation for the Laplace equation; weak solutions; and Sobolev spaces.

Metric spaces and topological spaces, bases and sub-bases, connectivity, local (path) connectivity, separation axioms, compactness and local compactness, concepts of convergence, Tychonoff’s theorem, Urysohn’s lemma, Tietze extension theorem, and selected topics as time permits.

Metric spaces and topological spaces, bases and sub-bases,   connectivity, local  (path) connectivity, separation axioms, compactness and   local compactness,  concepts of convergence, Tychonoff's theorem, Ury-sohn's   lemma, Tietze  extension theorem; homotopy type, fundamental group,   covering spaces; topology  of Euclidean space and manifold; selected topics as time   permits.            Spring semester.

This course is an introduction to methods and theory in numerical solutions of partial differential equations. The finite difference and pseudo-spectral methods will be used as examples to solve partial differential equations, including parabolic, hyperbolic, and elliptic equations in one or higher dimensional space. The theory on consistency, convergence, and Von Neumann stability analysis of numerical schemes will be emphasized for a basic understanding about how to control numerical errors and to achieve higher order accuracy for numerical solutions. Students will also be assigned projects to obtain the first-hand experience in numerical computations.

The main purpose of this course is to present the foundations of the optimal control theory, some applications, and their solutions. The students will be introduced to the core concepts and results of control and system theory. The foundational and basic results will be derived for discrete and continuous time scales, and state variables. Topics to be covered: proportional-derivative control; state-space and spectrum assignment; outputs and dynamic feedback; reachability; controllability; feedback and stability; Lyapunov theory; linearization principle of observability; dynamic programming algorithm; multipliers for unconstrained and constrained controls; and Pontryagin maximum principle.

The main purpose of this course is to present the foundations of the stochastic control theory, the corresponding numerical methods, and some applications. The focus will be on the idea of dynamic programming which will be developed starting from deterministic models, through finite-horizon stochastic problems, to infinite-horizon stochastic problems of various types. Applications to queuing systems, network design, and routing; supply-chain management and others will be discussed in detail. Topics to be covered: basic concepts of control theory for stochastic dynamic systems; controlled Markov chains; dynamic programming for finite horizon problems; infinite horizon discounted problems; numerical methods for infinite horizon problems; linear stochastic dynamic systems in discrete time; tracking and Kalman filtering; linear quadratic models; controlled Markov processes in continuous time; and elements of stochastic control theory in continuous time and state space.

This course introduces students to modeling and numerical techniques for optimization under uncertainty and risk. Topics include: generalized concavity of measures, optimization problems with probabilistic constraints (convexity, differentiability, optimality, and duality), numerical methods for solving problems with probabilistic constraints, two-stage and multi-stage models (structure, optimality, duality), decomposition methods for two-stage and multi-stage models, risk averse optimization models,

Complex numbers; elementary functions; Möbius transformations; analytic functions; power series; integration; Cauchy-Goursat theorems; Cauchy integral formula; Taylor and Laurent series; singularities; residue theory; and meromorphic and entire functions.

Analytic continuation; Riemann surfaces; elliptic   functions; gamma function; conformal mapping.

Theory and methods in continuous and discrete dynamical systems. Topics may vary, but will typically include local bifurcation theory for vector fields and maps, center manifold reductions, normal forms, periodic orbits and Poincaré maps, averaging methods, Melnikov methods, chaotic dynamics, the Smale horseshoe map, and symbolic dynamics.

Study of the classical transforms, the Laplace, Fourier, Hilbert, and other transforms; inversion and application to solution of differential, difference, and integral equations; and Abelian and Tauberian theorems, including Wiener’s theory.

Geometry of Hilbert space; spectral theory of self-adjoint and normal operators; applications to differential operators; multiplicity theory; and families of operators, Stone’s theorem, and introduction to rings of operators.

This course introduces basic concepts and techniques to solve inverse problems for both integral and differential equations.  Topics include: Ill-posed problems, Tikhonov regularization, collocation methods, Galerkin methods, inverse eigenvalue problems, inverse boundary value problems, conditions on dense solvability.  Computational projects may be assigned.

The course will introduce the students to the fundamental mathematical models of risk and approaches to decision-making under uncertainty and risk-aversion. The mathematical models will range from classical models as Expected Utility Theory, Prospect Theory, Dual Utility Theory, to state-of-the-art work on stochastic dominance, the theory of coherent risk measures, and general deviation measures. The course also surveys recent developments in particular applied areas as portfolio optimization, asset pricing, nuclear safety, reliability, etc.

Linear topological spaces, local convexity, and spaces of distribution; Banach spaces; three fundamental theorems and applications to classical analysis; operators, operational calculus, compact operators, and applications to integral equations; Klein-Milman theorems; and fixed point theorems with applications to nonlinear problems.

Notion of simplicial complex, absolute, and relative homology groups of a space; exact sequences; cohomology; axioms for homology theory; introduction to homological algebra; and homotopy and the fundamental group.

Martingales; generalized weak and strong laws; infinitely   divisible  distribution; stable distributions, limiting distributions   for triangular  arrays; semigroup theory applications; bilateral Laplace   transforms; renewal  equation; random walks;. Markov processes.

Selected topics may include: distribution theory; theory   of inference;  foundations of probability; spectral analysis;   multivariant analysis.

Existence and uniqueness of solutions; dependence on parameters; periodic solutions; nonlinear autonomous systems; Poincare-Bendixon theory; continuous transformation groups; linear systems; Floquet theory; linear systems in complex domain; regular and irregular singularities; asymptotic expansions; Stokes' phenomenon; boundary value problems.

Characteristics and classification of equations; Cauchy-Kowalewski theorem; linear and quasilinear systems; elliptic equations and potential theory; Green's function; mean value theorems; a priori estimates; functions space methods; hyperbolic equations; Riemann's solution of the Cauchy problem; discontinuities and shocks; Huyghen's principle; method of spherical means; parabolic equations.

Selected topics in numerical analysis not treated in   MA615-616; topics may include: numerical solution of partial differential   equations, boundary value problems, approximation theory; Monte Carlo methods, power   spectral methods as they apply to numerical analysis, optimal search problems.

Algebraic number fields; rings of algebraic integers and integral basis of field discriminant; unique factorication for ideals; splitting and ramifications of primes; Kummer’s theorem with applications to quadratic and roots of unity fields; padic numbers; Hensel’s lemMA ; geometry of numbers; units in an algebraic extension; finiteness of class numbers of a field; and computation of class numbers in special cases.

Selected topics in advanced analysis not treated in other   courses; topics may include: integral transforms, general convolution   transform, approximation theory, theorems of Jackson and Bernstein, functions of   exponential type, Nevalinna's theory of meromoporhic functions, asymptotic   development, perturbation theory.

 Selected topics in algebra not treated in other courses;   topics may include: group representations, Lie algebra, structure of rings,   valuation theory, algebraic curves, Galois theory of non-commutative fields,   polynomial ideals, elimination theory.

 Selected topics in mathematical logic; topics may include:   a study of the
 connection between the semantical and syntactical   treatments of propositional calculus and quantification theory, including references   to the works of Harbrand, Dreben and Hintikka, Gödel's completeness for   theorem for the first order and predicate calculus, recursive function theory,   decidable theories, and Gödel's incompleteness theorem for arithmetic,   axiomatic set theory, model theory.

 Selected topics in topology; topics may include: K theory,   infinite
 dimensional analysis, knot theory, applications of   algebraic topology to
 algebraic geometry.

This course will focus on one or more topics of current   interest in graph
 theory and its applications.  Possible topics include:   linear algebra and
 graph theory; graphs and groups, graphical enumeration;   extremal graph theory; graph equations; covering and packing problems; graph algorithms; graph

 Existence and uniqueness of solutions to nonlinear partial   differential
 equations with applications to equations from physics and   engineering.  Topics covered will include Degree Theory, The Mountain Pass   Lemma, Variational Methods, Index Theory, Nash-Moser Iteration Schemes.  The   course will also include a review of Hilbert space methods.

One to six credits. Limit of six credits for the degree of Master of Science.

One to six credits. Limit of six credits for the degree of Doctor of Philosophy.

Special topics in mathematics not covered in regularly scheduled courses and suitable for both graduates and advanced undergraduates. May be taken more than once.

For the degree of Master of Science. Five to ten credits with departmental approval.

Original research carried out under the guidance of a member of the faculty which may serve as the basis for the dissertation required for the degree of Doctor of Philosophy. Hours and credits to be arranged.