# Graduate Course Descriptions

If you would like to know more about offered courses and seminars for the upcoming semester, check the schedule of classes on FlashLine, or contact the Graduate Secretary at math-gradinfo@math.kent.edu.

##### Course Permission

If you are trying to register for a course and it says you don't have the proper pre-requisites, see the Graduate Coordinator to request permission to take the class.

###### MATH 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; graduate standing.

###### MATH 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; graduate standing.

###### MATH 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 50011; graduate standing.

###### MATH 50055 - ACTUARIAL MATHEMATICS I (4)

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

###### MATH 50056 - ACTUARIAL MATHEMATICS II (4)

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

###### MATH 50091 - SEMINAR IN ACTUARIAL MATHEMATICS (2)

(Repeatable for credit) Seminar course designed to prepare students for the society of actuaries’ examination on actuarial mathematics. Prerequisite: MATH 50056; graduate standing.

###### MATH 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; graduate standing.

###### MATH 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, constructability. Prerequisite: MATH 4/51001; graduate standing.

###### MATH 51021 - THEORY OF MATRICES (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; graduate standing.

###### MATH 51038 - INTERMEDIATE LOGIC (3)

(Cross-listed with CS 41038 and MATH 41038 and PHIL 41038 and PHIL 51038) Study of first order predicate calculus with identity and function symbols. Prerequisite: graduate standing.

###### MATH 51045 – METALOGIC (3)

(Cross-listed with CS 41045 and CS 51045 and MATH 41045 and PHIL 41045 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. Prerequisite: graduate standing.

###### MATH 52001 - INTRODUCTION TO ANALYSIS I (3)

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

###### MATH 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; graduate standing.

###### MATH 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; graduate standing.

###### MATH 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; graduate standing.

###### MATH 52024 - NUMBERS AND GAMES (3)

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

###### MATH 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; graduate standing.

###### MATH 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 22205.

###### MATH 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's functions and the Method of Characteristics for wave equations. Prerequisite: MATH 32044; graduate standing.

###### MATH 52048 - INTRODUCTION TO COMPLEX VARIABLES (3)

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

###### MATH 52091 - SEMINAR: MODELING PROJECTS (3)

(Repeatable for credit)Individual and small-group projects concerned with the formulation and analysis of mathematical models in a variety of areas. Written and oral reports required. Prerequisite: MATH 52031; graduate standing.

###### MATH 52201 - INTRODUCTION TO NUMERICAL COMPUTING I (3)

(Cross-listed with CS 42201 and CS 52201 and MATH 42201)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; graduate standing.

###### MATH 52202 - INTRODUCTION TO NUMERICAL COMPUTING II (3)

(Cross-listed with CS 42202 and CS 52202 and MATH 42202) A continuation of MATH 52201. Topics include interpolation, numerical differentiation and integration, and numerical solution of ordinary differential equations. Prerequisite: MATH 32044 and 52201; graduate standing.

###### MATH 55011 - DIFFERENTIAL GEOMETRY (3)

Analytic and metric differential geometry of curves and surfaces. Prerequisite: MATH 22005; graduate standing.

###### MATH 55021 - EUCLIDEAN GEOMETRY (3)

Geometry of Euclid extended to advanced topics of the triangle, quadrilaterals and circles: cross-ratio, groups, constructions, geometric generalizations; inversion. Prerequisite: MATH 21001; graduate standing.

###### MATH 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; graduate standing.

###### MATH 56001 - ELEMENTARY TOPOLOGY (3)

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

###### MATH 57011 - THEORY OF NUMBERS (3)

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

###### MATH 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 credit hours of mathematics beyond MATH 22005; graduate standing.

###### MATH 57091 - SELECTED TOPICS IN MATHEMATICS AND ITS APPLICATIONS (1-3)

(Repeatable for credit) Various special courses will be announced in the schedule of classes under this course number with different section numbers. Prerequisite: Special approval of instructor; graduate standing.

###### MATH 59893 - VARIABLE TITLE WORKSHOP IN MATHEMATICS (1-6)

(Repeatable for credit) Studies in special topics in pure and applied mathematics. Prerequisite: Special approval; graduate standing.

###### MATH 60051 - PROBABILITY I (4)

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

###### MATH 60052 - PROBABILITY II (3)

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

###### MATH 60061 - 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; graduate standing.

###### MATH 60062 - MATHEMATICAL STATISTICS II (3)

Tests of statistical hypothesis. Neyman Pearson Lemma. Exponential families and invariance. Sequential tests. Non-parametric procedures. Prerequisite: MATH 6/70061; graduate standing.

###### MATH 60070 - 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; graduate standing.

###### MATH 60091 - SEMINAR IN STATISTICS & PROBABILITY (1-3)

(Repeatable for credit) Seminar on current research in statistics and probability. Prerequisite: Special approval; graduate standing.

###### MATH 60093 - VARIABLE TITLE WORKSHOP IN MATHEMATICS (1-3)

(Repeatable for credit) Studies of special topics in mathematics. Not acceptable for credit toward a graduate degree in mathematics without approval of the student's adviser. Prerequisite: Special approval; graduate standing.

###### MATH 60094 - COLLEGE TEACHING OF MATHEMATICS (1)

(Repeatable for credit)Techniques and problems in the teaching of college-level mathematics. Student presentations of mathematical papers and colloquia will be included. Prerequisite: graduate standing.

###### MATH 61051 - 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; graduate standing.

###### MATH 61052 - 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; graduate standing.

###### MATH 61091 - SEMINAR IN ALGEBRA (1-3)

(Repeatable for credit) Seminar on current research in algebra. Prerequisite: Special approval; graduate standing.

###### MATH 62041 - 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, MATH 52041, MATH 52045, and MATH 52048; graduate standing.

###### MATH 62042 - 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; graduate standing.

###### MATH 62051 - 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; graduate standing.

###### MATH 62052 - FUNCTIONS OF A REAL VARIABLE II (3)

A continuation of MATH 62051/72051. Included are basic topics in functional analysis and Hilbert space theory. Prerequisite: MATH 6/72051; graduate standing.

###### MATH 62091 - SEMINAR IN REAL ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in real analysis. Prerequisite: Special approval; graduate standing.

###### MATH 62151 - 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 52002; graduate standing.

###### MATH 62152 - 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; graduate standing.

###### MATH 62191 - SEMINAR IN COMPLEX ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in complex analysis. Prerequisite: Special approval; graduate standing.

###### MATH 62202 - NUMERICAL COMPUTING II (3)

(Cross-listed with CS 62202 and CS 72202 and MATH 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; graduate standing.

###### MATH 62203 - 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, MATH 21001, MATH 32044, and CS 10061; graduate standing.

###### MATH 62251 - NUMERICAL ANALYSIS I (4)

(Cross-listed with CS 62251 and CS 72251 and MATH 72251) Floating point computation, rounding error analysis, conditioning, interpolation (polynomial, trigonometric, spline). Numerical quadrature (Newton-Cotes, Gauss), extrapolation, Romberg integration. Prerequisite: MATH 4/52002; graduate standing.

###### MATH 62252 - NUMERICAL ANALYSIS II (3)

(Cross-listed with CS 62252 and CS 72252 and MATH 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; graduate standing.

###### MATH 62261 - NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (3)

(Cross-listed with CS 62261 and CS 72261 and MATH 72261) Discretization methods for ordinary differential equations and systems. Initial value and boundary value problems. Numerical implementation software and analysis. Prerequisite: MATH 32044 and MATH 4/52202; graduate standing.

###### MATH 62262 - NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS (3)

(Cross-listed with CS 62262 and CS 72262 and MATH 72262) Derivation and analysis of discrete methods (finite differences, finite elements) for the numerical solution of elliptic, hyperbolic and parabolic partial differential equations. Prerequisites: MATH 42045 or 52045; and MATH 42202 or 52202; graduate standing.

###### MATH 62263 - NUMERICAL SOLUTION OF LARGE SPARSE LINEAR SYSTEMS (3)

(Cross-listed with CS 62263 and CS 72263 and MATH 72263) Construction and analysis of iterative methods for large systems of linear algebraic equations. Jacobi, Gauss-Seidel, SOR. Polynomial acceleration methods, conjugate gradients. Multi-grid methods. Prerequisite: MATH 4/51021 and MATH 4/52202; graduate standing.

###### MATH 62264 - NUMERICAL SOLUTION OF NONLINEAR SYSTEMS (3)

(Cross-listed with CS 62264 and CS 72264 and MATH 72264) Construction and analysis of numerical methods for systems of nonlinear algebraic equations and optimization problems. Numerical implementation and software. Prerequisites: MATH 42041 or MATH 52041; and MATH 42202 or MATH 52202; graduate standing.

###### MATH 62291 - SEMINAR IN NUMERICAL ANALYSIS (1-3)

(Repeatable for credit)Seminar on current research in numerical analysis. Prerequisite: Special approval; graduate standing.

###### MATH 62391 - SEMINAR IN MEASURE THEORY (1-3)

(Repeatable for credit) Seminar on current research in measure theory. Prerequisite: Special approval; graduate standing.

###### MATH 62491 - SEMINAR IN NONLINEAR ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in nonlinear analysis. Prerequisite: Special approval; graduate standing.

###### MATH 62591 - SEMINAR IN OPERATOR THEORY (1-3)

(Repeatable for credit) Seminar on current research in operator theory. Prerequisite: Special approval; graduate standing.

###### MATH 62691 - SEMINAR IN HARMONIC ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in harmonic analysis. Prerequisite: Special approval; graduate standing.

###### MATH 63291 - SEMINAR IN APPLIED MATHEMATICS (1-3)

(Repeatable for credit) Seminar on current research in applied mathematics. Prerequisite: Special approval; graduate standing.

###### MATH 64091 - SEMINAR IN MATHEMATICS EDUCATION (3)

(Repeatable for credit) Studies in geometry, algebra, mathematics for junior and senior high school. Mathematics content professionalized for teachers. Prerequisite: MATH 34001; graduate standing.

###### MATH 66051 – INTRODUCTION TO TOPOLOGY I (3)

Set theory, topological spaces, continuity, product spaces, quotient spaces, separation axioms, compactness and metrizability. Prerequisite: Special approval; graduate standing.

###### MATH 66052 - INTRODUCTION TO TOPOLOGY II (3)

Geometric topology, including connectedness, continua, homotopy, the plane and 2 manifolds. Prerequisite: MATH 6/76051; graduate standing.

###### MATH 66091 - SEMINAR IN TOPOLOGY (1-3)

(Repeatable for credit) Seminar on current research in topology. Prerequisite: Special approval; graduate standing.

###### MATH 67091 - SEMINAR IN NUMBER THEORY (1-3)

(Repeatable for credit) Seminar on current research in number theory. Prerequisite: Special approval; graduate standing.

###### MATH 67098 - RESEARCH (1-15)

(Repeatable for credit) Research or individual investigation. Credits are applied toward degree requirements with approval if letter grade of "S" is given. Prerequisite: graduate standing.

###### MATH 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. Prerequisite: graduate standing.

###### MATH 67299 - THESIS II (2)

Thesis students must continue registration each semester until all degree requirements are met. Prerequisite: MATH 67199; graduate standing.

###### MATH 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; doctoral standing.

###### MATH 70052 - PROBABILITY II (3)

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

###### MATH 70061 - MATHEMATICAL STATISTICS I (4)

Statistics, distributions of statistics. Sampling distributions. Decision spaces and loss functions. Sufficiency and completeness. Estimation theory. Rao Bblackwell and the Cramer Rao theorems. Prerequisite: MATH 4/52002.

###### MATH 70062 - MATHEMATICAL STATISTICS II (3)

Tests of statistical hypothesis. Neyman Pearson Lemma. Exponential families and invariance. Sequential tests. Non parametric procedures. Prerequisite: MATH 6/70061; doctoral standing.

###### MATH 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; doctoral standing.

###### MATH 70091 - SEMINAR IN STATISTICS & PROBABILITY (1-3)

(Repeatable for credit) Seminar on current research in statistics and probability. Prerequisite: Special approval; doctoral standing.

###### MATH 70094 - COLLEGE TEACHING OF MATHEMATICS (1)

(Repeatable for credit) Techniques and problems in the teaching of college-level mathematics. Student presentations of mathematical papers and colloquia will be included. Prerequisite: doctoral standing.

###### MATH 70095 - SELECTED TOPICS IN STATISTICS AND PROBABILITY (1-3)

(Repeatable for credit) Contents will vary with each offering and will complement topics covered in MATH 6/70051, 6/70052 and MATH 6/70061, 6/70062. Prerequisite: Special approval; doctoral standing.

###### MATH 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; doctoral standing.

###### MATH 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; doctoral standing.

###### MATH 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. Prerequisite: MATH 6/71051 and MATH 6/71052; doctoral standing.

###### MATH 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 MATH 6/71052; doctoral standing.

###### MATH 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; doctoral standing.

###### MATH 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; doctoral standing.

###### MATH 71091 - SEMINAR IN ALGEBRA (1-3)

(Repeatable for credit) Seminar on current research in algebra. Prerequisite: Permission; doctoral standing.

###### MATH 71095 - SELECTED TOPICS IN ALGEBRA (1-3)

(Repeatable for credit) Contents will vary with each offering and will complement topics covered in MATH 6/71051, 6/71052, 71001 and 71002. Prerequisite: Permission; doctoral standing.

###### MATH 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 72052; doctoral standing.

###### MATH 72002 - FUNCTIONAL ANALYSIS II (3)

A study of principles of linear analysis in the setting of normed linear spaces and topological vector spaces. Prerequisite: MATH 72001; doctoral standing.

###### MATH 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, MATH 52041, MATH 52045, and MATH 52048; doctoral standing.

###### MATH 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; doctoral standing.

###### MATH 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; doctoral standing.

###### MATH 72052 - FUNCTIONS OF A REAL VARIABLE II (3)

A continuation of MATH 62051/72051. Included are basic topics in functional analysis and Hilbert space theory. Prerequisite: MATH 6/72051; doctoral standing.

###### MATH 72091 - SEMINAR IN REAL ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in real analysis. Prerequisite: Special approval; doctoral standing.

###### MATH 72095 - SELECTED TOPICS IN REAL ANALYSIS (1-3)

(Repeatable for credit) Contents will vary with each offering and will implement topics covered in MATH 6/72051, 6/72052 and 6/72002. Prerequisite: Special approval; doctoral standing.

###### MATH 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 52002; doctoral standing.

###### MATH 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: 72151; doctoral standing.

###### MATH 72191 - SEMINAR IN COMPLEX ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in complex analysis. Prerequisite: Special approval; doctoral standing.

###### MATH 72195 - SELECTED TOPICS IN COMPLEX ANALYSIS (1-3)

(Repeatable for credit) Contents will vary with each offering and will complement topics covered in MATH 6/72151, 6/72152 and 72021. Prerequisite: Special approval; doctoral standing.

###### MATH 72202 - NUMERICAL COMPUTING II (3)

(Cross-listed with CS 62202 and CS 72202 and MATH 62202) 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 MATH 6/72201; doctoral standing.

###### MATH 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, MATH 21001, MATH 32044 and CS 10061; doctoral standing.

###### MATH 72251 - NUMERICAL ANALYSIS I (4)

(Cross-listed with CS 62251 and CS 72251 and MATH 62251) Floating point computation, rounding error analysis, conditioning, interpolation (polynomial, trigonometric spline). Numerical quadrature (Newton-Cotes, Gauss), extrapolation, Romberg integration. Prerequisite: MATH 4/52002; doctoral standing.

###### MATH 72252 - NUMERICAL ANALYSIS II (3)

(Cross-listed with CS 62252 and CS 72252 and MATH 62252) 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 MATH 6/72251; doctoral standing.

###### MATH 72261 - NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (3)

(Cross-listed with CS 62261 and CS 72261 and MATH 62261) Discretization methods for ordinary differential equations and systems. Initial value and boundary value problems. Numerical implementation software and analysis. Prerequisite: MATH 32044 and MATH 4/52202; doctoral standing.

###### MATH 72262 - NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS (3)

(Cross-listed with CS 62262 and CS 72262 and MATH 62262) Derivation and analysis of discrete methods (finite differences, finite elements) for the numerical solution of elliptic, hyperbolic and parabolic partial differential equations. Prerequisites: MATH 42045 or 52045; and MATH 42202 or 52202; doctoral standing.

###### MATH 72263 - NUMERICAL SOLUTION OF LARGE SPARSE LINEAR SYSTEMS (3)

(Cross-listed with CS 62263 and CS 72263 and MATH 62263) Construction and analysis of iterative methods for large systems of linear algebraic equations. Jacobi, Gauss-Seidel, SOR. Polynomial acceleration methods, conjugate gradients. Multi-grid methods. Prerequisite: MATH 4/51021 and MATH 4/52202; doctoral standing.

###### MATH 72264 - NUMERICAL SOLUTION OF NONLINEAR SYSTEMS (3)

(Cross-listed with CS 62264 and CS 72264 and MATH 72264) Construction and analysis of numerical methods for systems of nonlinear algebraic equations and optimization problems. Numerical implementation and software. Prerequisites: MATH 4/52041 and MATH 4/52202; doctoral standing.

###### MATH 72291 - SEMINAR IN NUMERICAL ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in numerical analysis. Prerequisite: Special approval; doctoral standing.

###### MATH 72295 - SELECTED TOPICS IN NUMERICAL ANALYSIS (1-3)

(Repeatable for credit) Contents will vary with each offering and will complement topics covered in MATH 6/72251-6/72252 and MATH 72201. Prerequisite: Special approval; doctoral standing.

###### MATH 72391 - SEMINAR IN MEASURE THEORY (1-3)

(Repeatable for credit) Seminar on current research in measure theory. Prerequisite: Special approval; doctoral standing.

###### MATH 72491 - SEMINAR IN NONLINEAR ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in nonlinear analysis. Prerequisite: Special approval; doctoral standing.

###### MATH 72591 - SEMINAR IN OPERATOR THEORY (1-3)

(Repeatable for credit) Seminar on current research in operator theory. Prerequisite: Special approval; doctoral standing.

###### MATH 72691 - SEMINAR IN HARMONIC ANALYSIS (1-3)

(Repeatable for credit) Seminar on current research in harmonic analysis. Prerequisite: Special approval; doctoral standing.

###### MATH 73291 - SEMINAR IN APPLIED MATHEMATICS (1-3)

(Repeatable for credit) Seminar on current research in applied mathematics. Prerequisite: Special approval; doctoral standing.

###### MATH 76051 - INTRODUCTION TO TOPOLOGY I (4)

Set theory, topological spaces, continuity, product spaces, quotient spaces separation axioms, compactness and metrizability. Prerequisite: Special approval; doctoral standing.

###### MATH 76052 - INTRODUCTION TO TOPOLOGY II (3)

Geometric topology, including connectedness, continua, homotopy, the plane and 2 manifolds. Prerequisite: MATH 6/76051; doctoral standing.

###### MATH 76091 - SEMINAR IN TOPOLOGY (1-3)

(Repeatable for credit) Seminar on current research in topology. Prerequisite: Permission; doctoral standing.

###### MATH 76095 - SELECTED TOPICS IN TOPOLOGY (1-3)

(Repeatable for credit) Contents will vary with each offering and will complement topics covered in MATH 6/76051, 6/76052, 76001 and 76002. Prerequisite: Permission; doctoral standing.

###### MATH 77011 - ALGEBRAIC NUMBER THEORY (3)

Number fields and Dedekind domains; conjugates, norm and trace, discriminant, integral bases; arithmetic or quadraic and cyclotomic number fields; theory of ideals and class group; Dirichlet's theorem on units. Prerequisite: MATH 57011 and MATH 6/71052; doctoral standing.

###### MATH 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 primes in arithmetic progressions, asymptotic formula for partitions. Prerequisite: MATH 57011 and MATH 6/72151.

###### MATH 77091 - SEMINAR IN NUMBER THEORY (1-3)

(Repeatable for credit) Seminar on current research in number theory. Prerequisite: Special approval; doctoral standing.

###### MATH 77095 - SELECTED TOPICS IN MATHEMATICS (1-3)

(Repeatable for credit) Contents will vary with each offering. Prerequisite: Permission.

###### MATH 77098 - RESEARCH (1-15)

(Repeatable for credit) Research or individual investigation. Credits are applied toward degree requirements with approval if letter grade of "S" is given. Prerequisite: doctoral standing.

###### MATH 77195 - SELECTED TOPICS IN NUMBER THEORY (1-3)

(Repeatable for credit) Content varies with each offering and complements topics covered in MATH 77011 and MATH 77012. Prerequisite: Special approval; doctoral standing.

###### MATH 87098 - RESEARCH (1-15)

(Repeatable for credit) 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. Prerequisite: doctoral standing.

###### MATH 87199 - DISSERTATION I (15)

(Repeatable for credit) 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. Prerequisite: Admission to doctoral candidacy; doctoral standing.

###### MATH 87299 - DISSERTATION II (15)

(Repeatable for credit) Continuing registration required of doctoral students who have completed the initial 30 hours of dissertation and continuing until all degree requirements are met. Prerequisite: MATH 87199 and doctoral standing.