Mathematics and Statistics Courses

Program | Faculty | Master's | Doctoral | Courses

Listed below are frequently given graduate-level courses in Mathematics and Statistics. Not all of these courses are offered regularly and many topics courses are offered under a "generic" designation. Please consult the department's Web site: http://www.math. umass.edu, which contains an updated list of regularly scheduled and topics courses.

Graduate students in other departments should consult the Undergraduate Catalog for mathematics courses which may carry graduate credit in their own departments or which in any event may be of interest, especially those in applied mathematics, probability, and statistics.

All courses carry 3 credits unless otherwise specified.

Mathematics

503 Topics in Computer-Connected Mathematics for Secondary Teachers

Computer-connected treatment of many topics appropriate for secondary school mathematics courses using computers, focusing on algorithms, round off error, and graphics. Topics chosen from number theory, linear algebra, geometry, analysis, probability, and statistics. Prerequisites: Math 233 and 235 or equivalents; working knowledge of Basic, PASCAL, or Fortran.

511 Abstract Algebra I

Introduction to various topics in abstract algebra, such as groups, rings, and fields. A deeper and more advanced treatment than Math 411. Prerequisites: Math 235 or 236; Math 300 or consent of instructor.

512 Abstract Algebra II

A continuation of Math 511.

513 Combinatorics and Graph Theory

Cross-listed with Cmpsci 575. A basic introduction to combinatorics and graph theory for advanced students in computer science, mathematics, and related fields. Topics include elements of graph theory, Euler and Hamiltonian circuits, graph coloring, matching, basic counting methods, generating functions, recurrences, inclusion-exclusion, Polya's theory of counting. Prerequisites: mathematical maturity, calculus, linear algebra, discrete mathematics course such as Cmpsci 250 or Math 455. Math 411 recommended but not required.

523 Introduction to Modern Analysis

Construction of the real number system, topology of R, Heine-Borel theorem; sequences, series, functions of one real variable, limits, continuity, differentiability, Riemann-Stieljes integral; elementary Lebesgue theory on R. Prerequisites: Math 233 and 235 or 236 and 300 or equivalent.

532 Topics in Ordinary Differential Equations

Topics chosen from: Sturm-Liouville the-ory, series solutions, stability theory and singular points, numerical methods, transform methods. Prerequisite: Math 431 and 235 or 236.

534 Introduction to Partial Differential Equations

Classification of second-order partial differential equations, wave equation, Laplace's equation, heat equation, sep-aration of variables. Prerequisites: Math 233 and 235 or 236 and 431.

545 Linear Algebra for Applied Mathematics

Basic concepts (over real or complex numbers): vector spaces, basis, dimension, linear transformations and matrices, change of basis, similarity. Study of a single linear operator: minimal and characteristic polynomial, eigenvalues, invariant subspaces, triangular form, Cayley-Hamilton theorem. Inner product spaces and special types of linear operators (over real or complex fields): orthogonal, unitary, self-adjoint, hermitian. Diagonalization of symmetric matrices, applications. Prerequisite: Math 235 or 236 or equivalent.

551 Numerical Analysis I

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis. Prerequisites: Math 233, and either Math 235 or 236; or consent of instructor; knowledge of a high level programming language.

552 Numerical Analysis II

Introduction to the application of computational methods to models arising in science and engineering, focusing mainly on the solution of partial differential equations. Topics include finite differences, finite elements, boundary value problems, fast Fourier transforms. Prerequisite: Math 551 or consent of instructor.

563 Differential Geometry

Differential geometry of curves and surfaces in Euclidean 3-space using vector methods. Prerequisites: Math 233 and 235 or 236.

611 Algebra I

Introduction to groups, rings, and fields. Direct sums and products of groups, cosets, Lagrange's theorem, normal subgroups, quotient groups. Polynomial rings, UFDs and PIDs, division rings. Fields of fractions, GCD and LCM, irreducibility criteria for polynomials. Prime field, characteristic, field extension, finite fields. Some topics from Math 612 included.

612 Algebra II

A continuation of Math 611. Topics in group theory (e.g., Sylow theorems, solvable and simple groups, Jordan-Holder and Schreier theorems, finitely generated Abelian groups). Topics in ring theory (matrix rings, prime and maximal ideals, Noetherian rings, Hilbert basis theorems). Modules, including cyclic, torsion, and free modules, direct sums, tensor products. Algebraic closure of fields, normal, algebraic, and transcendental field extensions, basic Galois theory. Prerequisite: Math 611 or equivalent.

621 Complex Analysis

Complex number field, elementary functions, holomorphic functions, integration, power and Laurent series, harmonic functions, conformal mappings, applications.

623 Real Analysis I

General theory of measure and integration and its specialization to Euclidean spaces and Lebesgue measure; modes of convergence, Lp spaces, product spaces, differentiation of measures and functions, signed measures, Radon-Nikodym theorem.

624 Real Analysis II

Continuation of Math 623. Introduction to functional analysis; elementary theory of Hilbert and Banach spaces; functional analytic properties of Lp-spaces, appli-cations to Fourier series and integrals; interplay between topology, and measure, Stone-Weierstrass theorem, Riesz representation theorem. Further topics depending on instructor.

645 Differential Equations and Dynamical Systems I

Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Linear systems and exponential matrix solutions. Well-posedness for nonlinear systems. Qualitative theory: limit sets, invariant set and manifolds. Stability theory: linearization about an equilibrium, Lyapunov functions. Autonomous two-dimensional systems and other special systems. Prerequisites: advanced calculus, linear algebra and basic ODE.

646 Differential Equations and Dynamical Systems II

A continuation of Math 645. Topics in the theory of dynamical systems, possibly including: Floquet theory of periodic solutions; bifurcation theory; complex behavior of higher dimensional systems, attractors and chaos. Applications from physical and biological sciences illustrate and motivate the theoretical material. Prerequisite: Math 645.

651 Numerical Analysis I

The analysis and application of the fundamental numerical methods used to solve a common body of problems in science. Linear system solving: direct and iterative methods. Interpolation of data by function. Solution of nonlinear equations and systems of equations. Numerical integration techniques. Solution methods for ordinary differential equations. Emphasis on computer representation of numbers and its consequent effect on error propagation. Prerequisites: advanced calculus, knowledge of a scientific programming language.

652 Numerical Analysis II

Presentation of the classical finite difference methods for the solution of the prototype linear partial differential equations of elliptic, hyperbolic, and parabolic type in one and two dimensions. Finite element methods developed for two dimensional elliptic equations. Major topics include consistency, convergence and stability, error bounds, and efficiency of algorithm. Prerequisites: Math 651, familiarity with partial differential equations.

671 Topology

Topological spaces. Metric spaces. Compactness, local compactness. Product and quotient topology. Separation axioms. Connectedness. Function spaces. Fundamental group and covering spaces.

696 Independent Study

Credit, 1-6.

699 Master's Thesis

701, 702 Topics in Algebra

Consent of instructor required. Credit, 1-3 each semester.

703, 704 Theory of Manifolds I, II

Inverse and implicit functions theorems, rank of a map. Regular and critical values. Sard's theorem. Differentiable manifolds, submanifolds, embeddings, immersions and diffeomorphisms. Tangent space and bundle, differential of a map. Partitions of unity, orientation, transversality embed-dings in Rⁿ. Vector fields, local flows. Lie bracket, Frobenius theorem. Lie groups, matrix Lie groups, left invariant vector fields, tensor fields, differential form, integration, closed and exact forms. DeRham's cohomology, vector bundles, connections, curvature.

705, 706 Topics in Analysis

Consent of instructor required. Credit, 1-3 each semester.

711, 712 Advanced Algebra

Consent of instructor required. Credit, 3 each semester.

713 Introduction to Algebraic Number Theory

Valuations, rings of integral elements, ideal theory in algebraic number fields of algebraic functions of one variable, Dirichlet-Hasse unit theorem and Riemann-Roch theorem for curves. Prerequisites: Math 611, 612 or equivalent.

725 Introduction to Functional Analysis

Banach and Hilbert spaces, continuous linear operators, spectral theory, Banach algebras. Prerequisite: Math 623 or 705.

731 Introduction to Partial Differential Equations I

Introduction to the modern methods in partial differential equations. Calculus of distributions: weak derivatives, mollifiers, convolutions and Fourier transform. Prototype linear equations of hyperbolic, parabolic and elliptic type, and their fundamental solutions. Initial value problems: Cauchy problem for wave and diffusion equations; well-posedness in the Hilbert-Sobolev setting. Boundary value problems: Dirichlet and Neumann problems for Laplace and Poisson equations; variational formulation and weak solutions; basic regularity theory; Green functions and operators; eigenvalue problems and spectral theorem. Prerequisites: advanced calculus and Math 623-624.

732 Introduction to Partial Differential Equations II

A continuation of Math 731. General class of equations and systems, modeled on the prototypes studied in 731. Linear hyperbolic systems. Parabolic evolution equations, and semigroups of operators. Linear elliptic equations of second order. Topics in nonlinear equations. Possible topics include: hyperbolic conservation laws; nonlinear parabolic systemsreaction-diffusion equations, Navier-Stokes equations; mean curvature equations; free-boundary problems. Prerequisite: Math 731.

781 Algebraic Topology I

Homotopy theory, simplicial and Cech homology theories. Prerequisites: Math 611, 671.

782 Algebraic Topology II

General homology theory, universal co-efficient theorems, singular homology theories, ring structure in cohomology theories, spectral sequences. Steenrod operations. Prerequisite: Math 781.

796, 896 Independent Study

Maximum credit, 6.

861, 862 Advanced Geometry

Credit, 3 each semester.

892, 893, 894, 895 Research Seminar

Presentation by advanced graduate students of research articles, perhaps own research. Credit, 1 each semester.

899 Doctoral Dissertation

Credit, 18.

Statistics

501 Methods of Applied Statistics

For graduate and upper-level undergraduate students, with focus on practical aspects of statistical methods. Topics include: data description and display, probability, random variables, random sampling, estimation and hypothesis testing, one and two sample problems, analysis of variance, simple and multiple linear regression, contingency tables. Includes data analysis using a computer package. Prerequisites: high school algebra; junior standing or higher.

505 Regression Analysis

Inferences in simple and multiple regression models, model fitting, checking and selection, diagnostics, presentation of the multiple linear regression model in matrix form. Has a strong applied component involving the use of a statistical package for data analysis. Prerequisite: previous coursework in statistics.

506 Design of Experiments

Planning, statistical analysis and interpretation of experiments. Designs considered include factorial designs, randomized blocks, latin squares, incomplete balanced blocks, nested and crossover designs, mixed models. Has a strong applied component involving the use of a statistical package for data analysis. Prerequisite: previous coursework in statistics.

511 Multivariate Statistical Methods

Introduction to the analysis of multivariate data. Topics include description of multivariate data; random vectors; multivariate analysis of variance, repeated measures/profile analysis; and topics from mul-tivariate regression, discriminant analysis, clustering, (principal components, factor analysis, and canonical correlation). Has a strong applied component involving the use of a statistical package for data analysis. Prerequisite: previous background in statistics, or consent of instructor.

515 Introduction to Statistics I

First semester of a two-semester sequence. Emphasis on those parts of probability theory necessary for statistical inference. Probability models, sample spaces, conditional probability, independence, random variables, expectation, variance, discrete and continuous probability distributions, joint distributions, sampling distributions, the central limit theorem. Prerequisites: Math 131, 132.

516 Introduction to Statistics II

Basic ideas of point and interval estimation and hypothesis testing; one and two sample problems, simple linear regression, topics from among one-way analysis of variance, discrete data analysis and nonparametric methods. Prerequisite: Statis 515 or equivalent.

605 Probability Theory

A modern treatment of probability theory based on abstract measure and integration. Random variables, expectations, independence, laws of large numbers, central limit theorem, and general conditioning using the Radon-Nikodym theorem. Introduction to stochastic processes: martingales, Brownian motion. Prerequisite: Math 623.

607 Mathematical Statistics I

Probability theory, including random variables, independence, laws of large numbers, central limit theorem; statistical models; introduction to point estimation, confidence intervals, and hypothesis testing. Prerequisite: advanced calculus and linear algebra, or consent of instructor.

608 Mathematical Statistics II

Point and interval estimation, hypothesis testing, large sample results in estimation and testing; decision theory; Bayesian methods; analysis of discrete data. Also, topics from nonparametric methods, sequential methods, regression, analysis of variance. Prerequisite: Statis 607 or equivalent.

640 Sampling Theory

Introduction to the theory and practice of sampling from finite populations. Designs covered include simple random sampling, systematic sampling, cluster sampling, stratified sampling, sampling with unequal probabilities, multistage and double sampling. Also, ratio, regression and jackknife estimators and the determination of samples sizes. Prerequisite: calculus-based background in probability and statistics.

705 Linear Models I

First semester of two-semester sequence in the theory of linear models. Basic results on the multivariate normal distribution; linear and quadratic forms; noncentral Chi-square and F distributions; inference in linear models, including point and interval estimation, hypothesis testing, etc. Prerequisites: Statis 607-608 or equivalent; linear algebra.

706 Linear Models II

Second semester of sequence in theory of linear models with focus on "analysis of variance/design of experiments" models. Includes factorial experiments (balanced and unbalanced designs, notions of interaction, etc.), randomized block designs, incomplete designs (incomplete block designs and latin squares), random effects, nested models, and mixed models.

708 Applied Stochastic Models and Methods

Stochastic processes and their applications to engineering, the physical and social sciences, and finance. General topics: discrete-time processes (e.g., Markov chains, stationary sequences); continuous-time processes (e.g., Markov, Gaussian); spatial processes (e.g., Markov random fields). Specific applications, depending on the composition of the class may include: statistical physics, queuing theory, neural networks, time series prediction, stochastic algorithms in optimization, image processing, stochastic differential equations in finance, control theory. Prerequisites: Statis 607-608.

712 Multivariate Analysis

The fundamentals of distribution theory, estimation and hypothesis testing in the multivariate framework. Includes multivariate regression, multivariate anova, discriminant analysis/classification, principal components, factor analysis, cluster analysis and canonical correlation. Prerequisites: Statis 607-608.

725-726 Estimation Theory and Hypothesis Testing

The advanced theory of statistics, including methods of estimation (unbiasedness, equivariance, maximum likelihood, Bayesian, minimax), optimality properties of estimators, hypothesis testing, uniformly most powerful tests, unbiased tests, invariant tests, relationship between confidence regions and tests, large sample properties of tests and estimators. Prerequisites: Statis 605 and 608.

741, 742 Recent Developments in Statistics

Content varies with instructor. Possible topics include nonparametric statistics, reliability/survival analysis, time series, generalized linear models.