# Graduate Mathematics (MATH)

*The minimum prerequisite for 50000-69999 level courses is graduate standing. The minimum prerequisite for courses 70000 and higher is doctoral standing. Additional prerequisites may apply and are listed in this catalog.*

**50011 Introduction to Probability Theory and Applications (3)**

Permutations and combinations, discrete and continuous distributions, random variables, conditional probabilities, Baye’s formula, mathematical expectation, law of large numbers, normal approximations, basic limit theorems. Prerequisite: MATH 12003.

**50012 Introduction to Statistical Concepts (3)**

Sample spaces, continuous distributions, sampling distributions, point and interval estimation, hypothesis testing, types of error, level and power of tests, sequential and nonparametric methods. Prerequisite: MATH 4/50011.

**50022 Linear Models and Statistical Analysis (3)**

Regression model. Multivariate normal distribution. Point and interval estimates. Gauss Markov theorem. Correlation and regression. Tests of hypotheses. Applications. Prerequisite: MATH 21001 and 4/50012.

**50031 Basic Non-parametric Statistics (3)**

Rank tests for different kinds of hypotheses, large sample theory, efficiency comparisons, tests of Kolmogorov Smirnov type. Prerequisite: MATH 4/50012

**50041 Statistical Methods for Experiments (3)**

Comparison of two groups, t and F-statistics, ANOVA, oneway and multiway layouts, randomization, blocking. Linear regression, correlation and analysis of covariance (ANCOVA). Repeated measures-analysis of variance. Prerequisite: MATH 30011.

**50042 Sampling Theory (3)
**This introductory course provides the methodology for the design and analysis of sampling and surveying studies. Simple random, stratified, cluster, PPS and two stage sampling techniques. Linear, ratio and regression estimators. Prerequisite: MATH 30011.

**50051 Topics in Probability Theory and Stochastic Processes (3)**

Topics from conditional expectations, Markov chains, Markov processes, Brownian Motion and Martingales and their applications to stochastic calculus. Prerequisite: MATH 4/50011.

**50055 Actuarial Mathematics I (4)**

Topics from survival models, stochastic analysis of annuities and life insurance and casualty models. Prerequisite: MATH 30055 and 50011.

**50056 Actuarial Mathematics II (4)**

Benefit premiums, benefit reserves and their analysis; decrement models, joint survivorship, risk models. Prerequisite: MATH 50055.

**50091 Seminar in Actuarial Mathematics (2)**

Seminar course designed to prepare students for the society of actuaries examination on actuarial mathematics. Prerequisite: MATH 50056.

**51001 Introduction to Modern Algebra I (3)**

Basic properties of groups, subgroups, factor groups. Basic properties of rings, integral domains and homomorphisms. Prerequisite: MATH 21001 and 22005.

**51002 Introduction to Modern Algebra II (3)**

A continuation of MATH 51001, emphasizing properties of rings, their ideals, polynomial ring extensions, fields, finite degree extensions, roots of polynomials, constructibility. Prerequisite: MATH 4/51001.

**51012 Finite Mathematics (3)**

(Cross-listed with CS 51012) Mathematics: emphasizing combinatorial techniques, graph applications in algorithms, finite algebra, number theory and probability. Covers useful mathematics for CS majors. Prerequisite: CS 31011 or MATH 31011.

**51021 Theory of Matirces (3)**

A rigorous study of the topics introduced in matrix algebra. Topics included are: canonical forms of matrices, diagonalizability criteria. Prerequisite: MATH 21001 and 22005.

**51045 Metalogic (3)**

(Cross-listed with CS 54045 and PHIL 51045).Consideration of various metatheorems including soundness and completeness of propositional and predicate calculus, undecidability of predicate calculus, and incompleteness of the theory of arithmetic.

**52001 Introduction to Analysis I (3)**

Topics include basic structure of the real numbers, Cauchy sequences, convergence, completeness of the real numbers,continuity, differentiation and Riemann integration. Prerequisite: MATH 21001 and 22005.

**52002 Introduction to Analysis II (3)**

Topics include further development of integration theory infinite series, uniform convergence, several variable calculus and metric spaces. Prerequisite: MATH 4/52001.

**52011 Mathematical Optimization (3)**

Analytical and numerical techniques for location of extreme points of functions and calculus of variations. Both constrained and unconstrained problems are considered. Prerequisite: MATH 21001 and 22005.

**52021 Graph Theory and Combinatorics (3)**

Fundamentals and applications of combinatorial mathematics. Topics include transversability, colorability, networks, inclusion and exclusion, matching and designs. Prerequisite: MATH 12003 and 21001.

**52024 Numbers and Games (3)**

The study of partisan and impartial combinatorial games; games as numbers; Grundy-Sprague theory. Prerequisite: approval of instructor.

**52031 Mathematical Models and Dynamical Systems (3)**

Formulation and analysis of mathematical models for a variety of phenomena. Mathematical methods from optimization, dynamical systems and probability are developed and applied. Modern software tools are utilized. Prerequisite: MATH 32044. Special fee: $25.

**52041 Advanced Calculus (3)**

The calculus and applications of scalar and vector functions of several variables. Vector differential and integral calculus. Applications to field theories, electricity and magnetism, and fluid flow. Prerequisite: MATH 21001 and 22005.

**52045 Introduction to Partial Differential Equations (3)**

Introduction to Fourier series, Fourier transforms and partial differential equations. Wave, heat and potential equations of mathematical physics. Additional topics include Green’sfunctions and the Method of Characteristics for wave equations. Prerequisite: MATH 32044.

**52048 Introduction to Complex Variables (3)**

Algebra of complex numbers, analytic functions, mappings, Cauchy integral theory, residue theory and applications. Prerequisite: MATH 22005.

**52091 Seminar: Modeling Projects (3)**

Individual and small-group projects concerned with the formulation and analysis of mathematical models in a variety of areas. Written and oral reports required. IP permissible. S/U Grading permitted. Prerequisite: MATH 52031. Special fee: $25.

**52201 Introduction to Numerical Computing I (3)**

(Cross-listed with CS 52201) An introduction to numerical methods and software for solving many common scientific computing problems. Linear systems, least-square data fitting, nonlinear equations and systems and optimization problems. Prerequisite: MATH 12003, 21001 and CS 23021. Special fee: $25.

**52202 Introduction to Numerical Computing II (3)**

(Cross-listed with CS 52202) A continuation of MATH 52201. Topics include interpolation, numerical differentiation and intergration, and numerical solution of ordinary differential equations. Prerequisite: MATH 32044 and 52201. Special fee: $25.

**55011 Differential Geometry (3)
**Analytic and metric differential geometry of curves and surfaces. Prerequisite: MATH 22005.

**55021 Euclidean Geometry (3)**

Geometry of Euclid extended to advanced topics of the triangle, quadrilaterals and circles; crossratio, groups, constructions, geometric generalizations; inversion.

Prerequisite: MATH 21001.

**55022 Linear Geometry (3)**

Use of transformations as a tool to study geometry and to differentiate between different kinds of geometry. Linear algebra methods applied to geometry. Prerequisite: MATH 21001.

**56001 Elementary Topology (3)**

Metric spaces, introduction to topological spaces, separation axioms. Prerequisite: MATH 22005.

**57001 Mathematical Logic and Set Theory (3)**

Axiomatic set theory, relations, development of real numbers,cardinal number, axiom of choice. Prerequisite: special approval.

**57011 Theory of Numbers (3)**

Divisibility properties of the integers, prime numbers, congruences, quadratic reciprocity, Diophantine equations, number theoretic functions, simple continued fractions, rational approximations. Prerequisite: MATH 12003.

**57021 History of Mathematics (3)**

Survey from Babylonian and Egyptian mathematics to 20th century mathematics with emphasis on the development of algebra, geometry, calculus, number theory. Prerequisite: 3 hours of mathematics beyond 22005.

**57091 Selected Topics in Mathmeatics and Its Applications (1-3)**

Various special courses will be announced in the Schedule of Classes under this course number with different section numbers. Repeat registration permitted. Prerequisite: approval of instructor.

**59893 Variable Title Workshop in Mathematics (1-6)**

Studies in special topics in pure and applied mathematics. S/U grading. Prerequisite: special approval.

**60051/70051 Probability I (4)**

Distribution functions, measure theory, random variables, expectation, independence, convergence, concepts, law of large numbers. Prerequisite: MATH 4/50011 or 4/52002.

**60052/70052 Probability II (3)**

Characteristic functions, the central limit problem, conditional expectations, Martingale theory, Brownian motion. Prerequisite: MATH 6/70051.

**60061/70061 Mathematical Statistics I (4)
**Statistics, distributions of statistics. Sampling distributions. Decision spaces and loss functions. Sufficiency and completeness. Estimation theory. Rao Blackwell and the Cramer Rao theorems. Prerequisite: MATH 4/52002.

**60062/70062 Mathematical Statistics II (3)**

Tests of statistical hypothesis. Neyman Pearson Lemma. Exponential families and invariance. Sequential tests. Nonparametric procedures. Prerequisite: MATH 6/70061.

**60070/70070 Financial Mathematics (3)**

Topics from replication of trading strategies, arbitrage, completeness, martingale representation theorem, fundamental theorem of finance, stochastic differential equations, Black and Scholes formula of option pricing. Prerequisite: MATH 50051.

**60091/70091 Seminar in Statistics and Probability (1-3)**

Seminar on current research in statistics and probability.Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**60093 Variable Title Workshop in Mathematics (1-3)**

Studies of special topics in mathematics. Not acceptable for credit toward a graduate degree in mathematics without approval of the student’s advisor. S/U grading. Prerequisite: special approval.

**60094/70094 College Teaching of Mathematics (1)**

Techniques and problems in the teaching of college-level mathematics. Student presentations of mathematical papers and colloquia will be included.

**61051/71051 Abstract Algebra I (4)**

Advanced topics in group theory including Sylow Theorems,finite Abelian groups, divisible groups and related concepts. Includes one-hour problem session each week. Prerequisite: special approval.

**61052/71052 Abstract Algebra II (3)**

Advanced topics, commutative ring theory, and field theory, including polynomial rings, unique factorization domains, matrix rings, Galois theory. Prerequisite: MATH 6/71051.

**61091/71091 Seminar in Algebra (1-3)**

Seminar on current research in algebra. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**62041/72041 Methods of Applied Mathematics I (3)**

Analysis and applications of ordinary and partial differential equations and related topics. Dimensional analysis (Buckingham Pi Theorem). Perturbation methods (singular perturbations, matched asymptotic expansions, WKB approximation). Variational methods (Euler-Lagrange equations). Prerequisite: MATH 51021 and 52041 and 52045 and 52048.

**62042/72042 Methods of Applied Mathematics II (3)**

Continuation of MATH 6/72041. Integral equations and Green’s functions (Fredholm alternative, compact operators, distributions, weak solutions). Wave phenomena (dispersion, KdV equation). Stability and bifurcation (linearized stability analysis, turning points, Hopf bifurcation). Prerequisite: MATH 6/72041.

**62051/72051 Functions of a Real Variable I (4)**

Introduction to modern concepts of real analysis including metric spaces, measure and integration theory. Prerequisite: MATH 4/52002.

**62052/72052 Functions of a Real Variable II (3)
**A continuation of MATH 6/72051. Included are basic topics in functional analysis and Hilbert space theory. Prerequisite: MATH 6/72051.

**62091/72091 Seminar in Real Analysis (1-3)**

Seminar on current research in real analysis. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**62151/72151 Functions of a Complex Variable I (4)**

Topological properties of the complex plane; analytic, entire, meromorphic functions; analytic continuation; conformal mappings; Picard’s Theorem; Riemann surfaces. Prerequisite: MATH 4/52002.

**62152/72152 Functions of a Complex Variable II (3)**

Topological properties of the complex plane; analytic, entire, meromorphic functions; analytic continuation; conformal mappings; Picard’s Theorem; Riemann surfaces. Prerequisite: MATH 62151.

**62191/72191 Seminar in Complex Analysis (1-3)**

Seminar on current research in complex analysis. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**62201 Numerical Computing I (3)**

(Cross-listed with CS 6/72201) Numerical methods for scientific computing emphasizing modern mathematical software and environments. Computer arithmetic and errors, linear systems of equations, least squares data fitting, eigenvalues and singular values, nonlinear systems, optimization. Computer programming experience (C or C++ or Fortran) required. Prerequisite: MATH 21001 and 22005.

**62202/72202 Numerical Computing II (3)**

(Cross-listed with CS 6/72202) A continuation of MATH 6/72201. Interpolation, numerical integration and differentiation, ordinary differential equations, initial and boundary value problems, partial differential equations, fast Fourier transforms and applications. Prerequisite: MATH 52045 and 6/72201.

**62203/72203 Computational Finance (3)**

Basic numerical methods (numerical linear algebra, nonlinear equations, curve fitting, ODEs, integration, Monte-Carlo methods), numerical solution of PDEs (stability, convergence, Black-Scholes, American options, SDEs), probabilistic methods.

Prerequisite: MATH 22005 and 21001 and 32044; and CS 10061. Special fee: $25.

**62251/72251 Numerical Analysis I (4)**

(Cross-listed with CS 6/72251) Floating point computation, rounding error analysis, conditioning, interpolation (polynomial, trigonometric, spline). Numerical quadrature (Newton-Cotes, Gauss), extrapolation, Romberg integration. Prerequisite: MATH 4/52002.

**62252/72252 Numerical Analysis II (3)**

(Cross-listed with CS 6/72252) Numerical solution of linear systems of equations (LU factorization, error analysis). Least squares, orthogonalization methods. Algebraic eigenvalue problems, QR algorithm, singular value decomposition. Prerequisite: MATH 4/51021 and 6/72251.

**62261/72261 Numerical Solution of Ordinary Differential Equations (3)**

(Cross-listed with CS 6/72261) Discretization methods for ordinary differential equations and systems. Initial-value and boundary value problems. Numerical implementation, software and analysis. Prerequisite: MATH 32044 and 4/52202.

**62262/72262 Numerical Solution of Partial Differential Equations (3)**

(Cross-listed with CS 6/72262) Derivation and analysis of discrete methods (finite differences, finite elements) for the numerical solution of elliptic, hyperbolic and parabolic partial differential equations. Prerequisite: MATH 4/52045 and 4/52202.

**62263/72263 Numerical Solution of Large Sparse Linear Systems (3)**

(Cross-listed with CS 6/72263) Construction and analysis of iterative methods for large systems of linear algebraic equations. Jacobi, Gauss-Seidel, SOR.Polynomial acceleration methods, conjugate gradients. Multigrid methods. Prerequisite: MATH 4/51021 and 4/52202.

**62264/72264 Numerical Solution of Nonlinear Systems (3)**

(Cross-listed with CS 6/72264) Construction and analysis of numerical methods for systems of nonlinear algebraic equations and optimization problems. Numerical implementation and software. Prerequisite: MATH 4/52041 and 4/52202.

**62291/72291 Seminar in Numerical Analysis (1-3)**

Seminar on current research in numerical analysis. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**62391/72391 Seminar in Measure Theory (1-3)**

Seminar on current research in measure theory. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**62491/72491 Seminar in Nonlinear Analysis (1-3)**

Seminar on current research in nonlinear analysis. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**62591/72591 Seminar in Operator Theory (1-3)**

Seminar on current research in operator theory. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**62691/72691 Seminar in Harmonic Analysis (1-3)**

Seminar on current research in harmonic analysis. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.63291/73291 Seminar in Applied Mathematics (1-3)

Seminar on current research in applied mathematics. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**64091 Seminar in Mathematics Education (3)
**Studies in geometry, algebra, mathematics for junior and senior high school. Mathematics content professionalized for teachers. Repeat registration permitted. Prerequisite: MATH 34001.

**66051/76051 Introduction to Topology I (4)**

Set theory, topological spaces, continuity, product spaces, quotient spaces, separation axioms, compactness and metrizability. Prerequisite: special approval.

**66052/76052 Introduction to Topology II (3)
**Geometric topology, including connectedness, continua, homotopy, the plane and 2 manifolds. Prerequisite: MATH 6/76051.

**66091/76091 Seminar in Topology (1-3)**

Seminar on current research in topology. Repeated registration permitted. S/U grading permissible. Prerequisite: special approval.

**67098/77098 Research (1-15)**

Research or individual investigation. Credits are applied toward degree requirements, with approval, if letter grade of “S” is given. Repeat registration permitted.

**67199 Thesis I (2-6)**

Thesis student must register for a total of 6 hours, 2 to 6 hours in a single semester, distributed over several semesters if desired. S/U grading; IP permissible.

**67299 Thesis II (2)**

Thesis students must continue registration each semester until all degree requirements are met. S/U grading; IP permissible. Prerequisite: MATH 67199.

**70095 Selected Topics in Statistics and Probability (1-3)**

Contents will vary with each offering and will complement topics covered in MATH 6/70051, 6/70052, 6/70061 and 6/70062. Repeat registration permitted. Prerequisite: special approval.

**71001 Structure of Rings and Algebras I (3)**

Advanced topics in ring theory, including Artinian rings, Noetherian rings, advanced commutative ring theory. Prerequisite: special approval.

**71002 Structure of Rings and Algebras II (3)**

Advanced topics in ring theory including an introduction to homological algebra. Includes Dedekind domains, regular rings, torsion theory. Prerequisite: MATH 71001.

**71011 Advanced Group Theory (3)**

Advanced topics in group theory. Topics include permutation arguments, coprime actions, transfer theorems, nonsimplicity criteria. Properties of families of groups: solvable, p solvable, nilpotent, p groups. Repeat registration permitted. Prerequisite: MATH 6/71051 and 6/71052.

**71012 Characters of Finite Groups (3)
**Development of characters of finite groups, their properties, orthogonality relations, integrality conditions. Applications include Burnside’s paqb theorem and existence of Frobenius kernels in Frobenius groups. Prerequisite: MATH 6/71051 and 6/71052.

**71095 Selected Topics in Algebra (1-3)**

Contents will vary with each offering and will complement topics covered in MATH 6/71051, 6/71052, 71001 and 71002. Repeat registration permitted. Prerequisite: special approval.

**72001 Functional Analysis I (3)
**A study of principles of linear analysis in the setting of normed linear spaces and topological vector spaces. Prerequisite: MATH 6/72052.

**72002 Functional Analysis II (3)**

A continuation of MATH 72001. A study of principles of linear analysis in the setting of normed linear spaces and topological vector spaces. Prerequisite: MATH 6/72052.

**72011 Advanced Differential Equations (3)**

Advanced topics in ordinary differential equations in real and complex plane. Prerequisite: MATH 6/72051 and 6/72151.

**72021 Summability Theory (3)**

Generalizations of concept of convergent sequence; properites of classical summability methods and of related sequence spaces. Prerequisite: approval of instructor.

**72095 Selected Topics in Real Analysis (1-3)**

Contents will vary with each offering and will implement topics covered in MATH 6/72051, 6/72052 and 6/72002. Repeat registration permitted. Prerequisite: special approval.

**72195 Selected Topics in Complex Analysis (1-3)**

Contents will vary with each offering and will complement topics covered in MATH 6/72151, 6/72152 and 72021. Repeat registration permitted. Prerequisite: special approval.

**72201 Approximation Theory (3)**

Introduction to polynomial, trigonometric and spline approximations; direct and inverse theorems of constructive function theory; other topics chosen according to interest of students and instructors. Prerequisite: special approval.

**72295 Selected Topics in Numerical Analysis (1-3)**

Contents will vary with each offering and will complement topics covered in MATH 6/72251, 6/72252 and MATH 72201. Repeat registration permitted. Prerequisite: special approval.

**76001 Algebraic Topology I (3)**

Singular and simplicial homology and cohomology theory. Homotopy theory. Prerequisite: special approval.

**76002 Algebraic Topology II (3)**

A continuation of MATH 76001. Singular and simplicial homology and cohomology theory. Homotopy theory. Prerequisite: special approval.

**76095 Selected Topics in Toplogy (1-3)**

Contents will vary with each offering and will complement topics covered in MATH 6/76051, 6/76052, 76001 and 76002. Repeat registration permitted. Prerequisite: special approval.

**77011 Algebraic Number Theory (3)**

Number fields and Dedekind domains; conjugates, norm and trace, discriminant, integral bases; arithmetic of quadratic and cyclotomic number fields; theory of ideals and class group; Dirichlet’s theorem on units. Prerequisite: MATH 57011 and

MATH 6/71052.

**77012 Analytic Number Theory (3)**

Multiplicative functions and summatory functions, Riemann’s Zeta Function and the Prime Number Theorem, L-functions and Dirichlet’s theorem on prmes in arithmetic progressions, asymptotic formula for partitions. Prerequisite: MATH 57011 and MATH 6/72151.

**77095 Selected Topics in Mathematics (1-3)
**Contents will vary with each offering. Repeat registration permitted. Prerequisite: special approval.

**77098 Research (1-15)**

Research or individual investigation. Credits are applied toward degree requirements with approval if letter grade of “S” is given. Repeated registration permitted.

**77195 Selected Topics in Number Theory (1-3)**

Content varies with each offering and complements topics covered in MATH 77011 and MATH 77012. Repeat registration permitted. Prerequisite: special approval.

**87098 Research (1-15)**

Research or individual investigation for doctoral students who have not yet passed their candidacy examinations. Credits earned may be applied toward degree if department approves. Repeat registration permitted.

**87199 Dissertation (15)**

Doctoral dissertation, for which registration in at least two semesters is required, first of which will be semester in which dissertation work is begun and continuing until the completion of 30 hours. S/U grading; IP permissible. Prerequisite: Admission to candidacy for doctoral degree.

**87299 Dissertation II (1 or 15)**

Continuing registration required of doctoral students who have completed the initial 30 hours of dissertation and continuing until all degree requirements are met. S/U grading; IP permissible. Prerequisite: MATH 87199.

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