# Graduate Courses

Feel free to browse this list of graduate courses to help you plan your schedule for the next semester. If you would like to know more about seminars that will be available for the upcoming semester, check the schedule of classes on the university registrars site, or contact the graduate secretary at mathgrad@math.kent.edu for more information.

__REQUESTING A COURSE PERMIT__

Do you need a permit for a course? *Before* asking for a permit, make sure you check to see if you have the proper prerequisites for the course by looking up the course below. If you need a permit for a course, then ** USE THIS PERMIT FORM** to request a departmental permit.

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**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.

**MATH 50012**

INTRODUCTION TO STATISTICAL CONCEPTS (3)

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.

**MATH 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.

LINEAR MODELS AND STATISTICAL ANALYSIS (3)

Prerequisite: MATH 21001 and 4/50012.

**MATH 50031**

BASIC NON-PARAMETRIC STATISTICS (3)

Rank tests for different kinds of hypotheses, large sample theory, efficiency comparisons, tests of Kolmogorov Smirnov type.

BASIC NON-PARAMETRIC STATISTICS (3)

Prerequisite: MATH 4/50012.

**MATH 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.

STATISTICAL METHODS FOR EXPERIMENTS (3)

Prerequisite: MATH 30011 or permission.

**MATH 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.

SAMPLING THEORY (3)

Prerequisite: MATH 30011 or equivalent.

**MATH 50051**

TOPICS IN PROBABILITY THEORY AND STOCHASTIC PROCESSES (3)

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 or permission.

**MATH 51001**

INTRODUCTION TO MODERN ALGEBRA I (3)

Basic properties of groups, subgroups, factor groups. Basic properties of rings, integral domains and homomorphisms.

INTRODUCTION TO MODERN ALGEBRA I (3)

Prerequisite: MATH 21001 and 22005.

**MATH 51002**

INTRODUCTION TO MODERN ALGEBRA II (3)

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 or permission.

**MATH 51012**

FINITE MATHEMATICS (3)

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.

**MATH 51021**

THEORY OF MATRICES (3)

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 or permission.

**MATH 51045**

METALOGIC (3)

METALOGIC (3)

Cross-listed with CS 51045 and PHIL 51045 See PHIL 51045 for description.

**MATH 52001**

INTRODUCTION TO ANALYSIS I (3)

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 or permission.

**MATH 52002**

INTRODUCTION TO ANALYSIS II (3)

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.

**MATH 52011**

MATHEMATICAL OPTIMIZATION (3)

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 or permission.

**MATH 52021**

GRAPH THEORY AND COMBINATORICS (3)

Fundamentals and applications of combinatorial mathematics. Topics include transversability, colorability, networks, inclusion and exclusion, matching and designs.

GRAPH THEORY AND COMBINATORICS (3)

Prerequisite: MATH 12003 and 21001 or permission.

**MATH 52024**

NUMBERS AND GAMES (3)

NUMBERS AND GAMES (3)

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

Prerequisite: permission of instructor.

**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.

MATHEMATICAL MODELS AND DYNAMICAL SYSTEMS (3)

Prerequisite: MATH 32044.

**MATH 52041**

ADVANCED CALCULUS (3)

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.

**MATH 52045**

INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS (3)

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.

**MATH 52048**

INTRODUCTION TO COMPLEX VARIABLES (3)

Algebra of complex numbers, analytic functions, mappings, Cauchy integral theory, residue theory and applications.

INTRODUCTION TO COMPLEX VARIABLES (3)

Prerequisite: MATH 22005.

**MATH 52091**

SEMINAR: MODELING PROJECTS (3)

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 or permission.

**MATH 52201**

INTRODUCTION TO NUMERICAL COMPUTING I (3)

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 or permission. Special fee: $10 per semester (subject to change).

**MATH 52202**

INTRODUCTION TO NUMERICAL COMPUTING II (3)

INTRODUCTION TO NUMERICAL COMPUTING II (3)

Cross-listed with CS 52202. A continuation of MATH 52201. Topics include interpolation, numerical differentiation and integration, and numerical solution of ordinary differential equations.

Prerequisite: MATH 32044 and 52201. Special fee: $10 per semester (subject to change).

**MATH 55011**

DIFFERENTIAL GEOMETRY (3)

DIFFERENTIAL GEOMETRY (3)

Analytic and metric differential geometry of curves and surfaces.

Prerequisite: MATH 22005.

**MATH 55021**

EUCLIDEAN GEOMETRY (3)

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 or permission of instructor.

**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.

LINEAR GEOMETRY (3)

Prerequisite: MATH 21001 or permission.

**MATH 56001**

ELEMENTARY TOPOLOGY (3)

ELEMENTARY TOPOLOGY (3)

Metric spaces, introduction to topological spaces, separation axioms.

Prerequisite: MATH 22005.

**MATH 57001**

MATHEMATICAL LOGIC AND SET THEORY (3)

MATHEMATICAL LOGIC AND SET THEORY (3)

Axiomatic set theory, relations, development of real numbers, cardinal number, axiom of choice.

Prerequisite: Permission.

**MATH 57011**

THEORY OF NUMBERS (3)

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 or permission.

**MATH 57021**

HISTORY OF MATHEMATICS (3)

HISTORY OF MATHEMATICS (3)

Survey from Babylonian and Egyptian mathematics to 20thcentury mathematics with emphasis on the development of algebra, geometry, calculus, number theory.

Prerequisite: 3 hours of mathematics beyond 22005 or permission.

**MATH 57091**

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

SELECTED TOPICS IN MATHEMATICS 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: Permission of instructor.

**MATH 59893**

VARIABLE TITLE WORKSHOP IN MATHEMATICS (1-6)

VARIABLE TITLE WORKSHOP IN MATHEMATICS (1-6)

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

Prerequisite: Permission.

**MATH 60051/70051**

PROBABILITY I (4)

Distribution functions, measure theory, random variables, expectation, independence, convergence, concepts, law of large numbers.

PROBABILITY I (4)

Prerequisite: MATH 4/50011 or 4/52002 or permission.

**MATH 60052/70052**

PROBABILITY II (3)

PROBABILITY II (3)

Characteristic functions, the central limit problem, conditional expectations, Martingale theory, Brownian motion.

Prerequisite: MATH 6/70051.

**MATH 60061/70061**

MATHEMATICAL STATISTICS I (4)

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 or permission.

**MATH 60062/70062**

MATHEMATICAL STATISTICS II (3)

MATHEMATICAL STATISTICS II (3)

Tests of statistical hypothesis. Neyman Pearson Lemma. Exponential families and invariance. Sequential tests. Nonparametric procedures.

Prerequisite: MATH 6/70061 or permission.

**MATH 60070/70070**

FINANCIAL MATHEMATICS (3)

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.

**MATH 60091/70091**

SEMINAR IN STATISTICS & PROBABILITY (1-3)

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

SEMINAR IN STATISTICS & PROBABILITY (1-3)

Prerequisite: permission.

**MATH 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.

VARIABLE TITLE WORKSHOP IN MATHEMATICS (1-3)

Prerequisite:Permission.

**MATH 60094/70094**

COLLEGE TEACHING OF MATHEMATICS (1)

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.

**MATH 61051/71051**

ABSTRACT ALGEBRA I (4)

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: Permission.

**MATH 61052/71052**

ABSTRACT ALGEBRA II (3)

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.

**MATH 61091/71091**

SEMINAR IN ALGEBRA (1-3)

Seminar on current research in algebra. Repeated registration permitted. S/U grading permissible.

SEMINAR IN ALGEBRA (1-3)

Prerequisite: permission.

**MATH 62041/72041**

METHODS OF APPLIED MATHEMATICS I (3)

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, 52041, 52045 and 52048.

**MATH 62042/72042**

METHODS OF APPLIED MATHEMATICS II (3)

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.

**MATH 62051/72051**

FUNCTIONS OF A REAL VARIABLE I (4)

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 or permission.

**MATH 62052/72052**

FUNCTIONS OF A REAL VARIABLE II (3)

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.

**MATH 62091/72091**

SEMINAR IN REAL ANALYSIS (1-3)

SEMINAR IN REAL ANALYSIS (1-3)

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

Prerequisite: permission.

**MATH 62151/72151**

FUNCTIONS OF A COMPLEX VARIABLE I (4)

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.

**MATH 62152/72152
FUNCTIONS OF A COMPLEX VARIABLE II (3)**

A continuation of MATH 62151/72151. Topological properties of the complex plane; analytic, entire, meromorphic functions; analytic continuation; conformal mappings; Picard's Theorem; Riemann surfaces.

Prerequisite: MATH 4/52002

**MATH 62191/72191**

SEMINAR IN COMPLEX ANALYSIS (1-3)

SEMINAR IN COMPLEX ANALYSIS (1-3)

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

Prerequisite: permission.

**MATH 62201**

NUMERICAL COMPUTING I (3)

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.

Prerequisite: MATH 21001, 22005 and computer programming (C or C++ or Fortran).

**MATH 62202**

NUMERICAL COMPUTING II (3)

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, 6/72201.

**MATH 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.

COMPUTATIONAL FINANCE (3)

Prerequisite: MATH 22005, 21001, 32044 and CS 10061.

**MATH 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.

NUMERICAL ANALYSIS I (4)

Prerequisite: MATH 4/52002.

**MATH 62252/72252**

NUMERICAL ANALYSIS II (3)

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.

**MATH 62261/72261**

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (3)

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 or equivalent.

**MATH 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.

NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS (3)

Prerequisite: MATH 4/52045 and 4/52202.

**MATH 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.

NUMERICAL SOLUTION OF LARGE SPARSE LINEAR SYSTEMS (3)

Prerequisite: MATH 4/51021 and 4/52202.

**MATH 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.

NUMERICAL SOLUTION OF NONLINEAR SYSTEMS (3)

Prerequisite: MATH 4/52041 and 4/52202 or equivalent.

**MATH 62291/72291**

SEMINAR IN NUMERICAL ANALYSIS (1-3)

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

SEMINAR IN NUMERICAL ANALYSIS (1-3)

Prerequisite: permission.

**MATH 62391/72391**

SEMINAR IN MEASURE THEORY (1-3)

SEMINAR IN MEASURE THEORY (1-3)

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

Prerequisite: permission.

**MATH 62491/72491**

SEMINAR IN NONLINEAR ANALYSIS (1-3)

SEMINAR IN NONLINEAR ANALYSIS (1-3)

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

Prerequisite: permission.

**MATH 62591/72591**

SEMINAR IN OPERATOR THEORY (1-3)

SEMINAR IN OPERATOR THEORY (1-3)

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

Prerequisite: permission.

**MATH 62691/72691**

SEMINAR IN HARMONIC ANALYSIS (1-3)

Seminar on current research in harmonic analysis. Repeated registration permitted. S/U grading permissible.

SEMINAR IN HARMONIC ANALYSIS (1-3)

Prerequisite: permission.

**MATH 63291/73291**

SEMINAR IN APPLIED MATHEMATICS (1-3)

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

SEMINAR IN APPLIED MATHEMATICS (1-3)

Prerequisite: permission.

**MATH 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.

SEMINAR IN MATHEMATICS EDUCATION (3)

Prerequisite: MATH 34001 or equivalent and one year of teaching experience in high school mathematics.

**MATH 66051/76051**

INTRODUCTION TO TOPOLOGY I (4)

INTRODUCTION TO TOPOLOGY I (4)

Set theory, topological spaces, continuity, product spaces, quotient spaces, separation axioms, compactness and metrizability.

Prerequisite: Permission.

**MATH 66052/76052**

INTRODUCTION TO TOPOLOGY II (3)

INTRODUCTION TO TOPOLOGY II (3)

Geometric topology, including connectedness, continua, homotopy, the plane and 2 manifolds.

Prerequisite: MATH 6/76051.

**MATH 66091/76091**

SEMINAR IN TOPOLOGY (1-3)

SEMINAR IN TOPOLOGY (1-3)

Seminar on current research in topology. Repeated registration permitted. S/U grading permissible.

Prerequisite: permission.

**MATH 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.

RESEARCH (1-15)

**MATH 67199**

THESIS I (2-6)

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.

**MATH 67299**

THESIS II (2)

THESIS II (2)

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

Prerequisite: MATH 67199.

**MATH 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.

SELECTED TOPICS IN STATISTICS AND PROBABILITY (1-3)

Prerequisite: Permission.

**MATH 71001**

STRUCTURE OF RINGS AND ALGEBRAS I (3)

STRUCTURE OF RINGS AND ALGEBRAS I (3)

Advanced topics in ring theory, including Artinian rings, Noetherian rings, advanced commutative ring theory.

Prerequisite: Permission.

**MATH 71002**

STRUCTURE OF RINGS AND ALGEBRAS II (3)

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.

**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. Repeat registration permitted.

ADVANCED GROUP THEORY (3)

Prerequisite: MATH 6/71051, 6/71052 or permission.

**MATH 71012**

CHARACTERS OF FINITE GROUPS (3)

CHARACTERS OF FINITE GROUPS (3)

Development of characters of finite groups, their properties, orthogonality relations, integrality conditions. Applications include Burnside’s p

^{a}q

^{b}theorem and existence of Frobenius kernels in Frobenius groups.

Prerequisite: MATH 6/71051 and 6/71052 or permission.

**MATH 71095**

SELECTED TOPICS IN ALGEBRA (1-3)

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: Permission.

**MATH 72001**

FUNCTIONAL ANALYSIS I (3)

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 or permission of instructor.

**MATH 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 or special permission.

**MATH 72011**

ADVANCED DIFFERENTIAL EQUATIONS (3)

Advanced topics in ordinary differential equations in real and complex plane.

ADVANCED DIFFERENTIAL EQUATIONS (3)

Prerequisite: MATH 6/72051, 6/72151.

**MATH 72021**

SUMMABILITY THEORY (3)

Generalizations of concept of convergent sequence; properties of classical summability methods and of related sequence spaces.

SUMMABILITY THEORY (3)

Prerequisite: Permission of instructor.

**MATH 72095**

SELECTED TOPICS IN REAL ANALYSIS (1-3)

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: Permission.

**MATH 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.

SELECTED TOPICS IN COMPLEX ANALYSIS (1-3)

Prerequisite: Permission.

**MATH 72201**

APPROXIMATION THEORY (3)

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: Permission.

**MATH 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.

SELECTED TOPICS IN NUMERICAL ANALYSIS (1-3)

Prerequisite: Permission.

**MATH 76001**

ALGEBRAIC TOPOLOGY I (3)

Singular and simplicial homology and cohomology theory. Homotopy theory.

ALGEBRAIC TOPOLOGY I (3)

Prerequisite: Permission.

**MATH 76002
ALGEBRAIC TOPOLOGY II (3)
**A continuation of MATH 76001. Singular and simplicial homology and cohomology theory. Homotopy theory.

Prerequisite: Permission.

**MATH 76095**

SELECTED TOPICS IN TOPOLOGY (1-3)

SELECTED TOPICS IN TOPOLOGY (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: Permission.

**MATH 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, or permission.

**MATH 77012
ANALYTIC NUMBER THEORY (3)**

Multiplicative functions and summatory functions, Riemann's Zeta Function and the Prime Theorem, L-functions and Dirichlet's theorem on prmes in arithmetic progressions, asymptotic formula for partitions.

Prerequisite: MATH 57011 and MATH 6/71052, or permission.

**MATH 77095**

SELECTED TOPICS IN MATHEMATICS (1-3)

Contents will vary with each offering. Repeat registration permitted.

SELECTED TOPICS IN MATHEMATICS (1-3)

Prerequisite: Permission.

**MATH 87098**

RESEARCH (1-15)

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.

**MATH 87199**

DISSERTATION I (15)

DISSERTATION I (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.

**MATH 87299**

DISSERTATION II (1 or 15)

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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