# 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 Credit Hours**

(Slashed with MATH 40011) 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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 50012 INTRODUCTION TO STATISTICAL CONCEPTS 3 Credit Hours**

(Slashed with MATH 40012) 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 40011 or MATH 50011; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 50051 TOPICS IN PROBABILITY THEORY AND STOCHASTIC PROCESSES 3 Credit Hours**

(Slashed with MATH 40051) Topics from conditional expectations, Markov chains, Markov processes, Brownian Motion and Martingales and their applications to stochastic calculus.

**Prerequisite: **MATH 50011 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 50055 ACTUARIAL MATHEMATICS I 4 Credit Hours**

(Slashed with MATH 40055) Topics from survival models, stochastic analysis of annuities and life insurance and casualty models.

**Prerequisite: **MATH 30055 and MATH 50011; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 50056 ACTUARIAL MATHEMATICS II 4 Credit Hours**

(Slashed with MATH 40056) Benefit premiums, benefit reserves and their analysis; decrement models, joint survivorship, risk models.

**Prerequisite: **MATH 50055 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 50059 STOCHASTIC ACTUARIAL MODELS 3 Credit Hours**

(Slashed with MATH 40059) Designed to prepare students for the Society of Actuaries examination on actuarial mathematics.

**Prerequisite: **Minimum C grade in MATH 50011 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 51001 INTRODUCTION TO MODERN ALGEBRA I 3 Credit Hours**

(Slashed with MATH 41001) Basic properties of groups, subgroups, factor groups. Basic properties of rings, integral domains and homomorphisms.

**Prerequisite: **MATH 21001 and MATH 22005 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 51002 INTRODUCTION TO MODERN ALGEBRA II 3 Credit Hours**

(Slashed with MATH 41002) A continuation of MATH 51001, emphasizing properties of rings, their ideals, polynomial ring extensions, fields, finite degree extensions, roots of polynomials, constructability.

**Prerequisite: **MATH 41001 or MATH 51001; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 51021 THEORY OF MATRICES 3 Credit Hours**

(Slashed with MATH 41021) A rigorous study of the topics introduced in matrix algebra. Topics included are: canonical forms of matrices, diagonalizability criteria.

**Prerequisite: **MATH 21001 and MATH 22005 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 51038 INTERMEDIATE LOGIC 3 Credit Hours**

(Cross-listed with CS 41038 and PHIL 41038 and PHIL 51038; slashed with MATH 41038; ) A detailed, systematic study of symbolic logic for philosophy majors, mathematics majors, computer science majors, and anyone else interested in advanced study in logic. The aim of the course is twofold: first, to develop a facility in understanding and using symbolic logic for various purposes, and second, to understand and appreciate symbolic logic as an area of study in itself. Topics include the distinction between syntactic, object-level proofs and semantic, meta-level proofs, the distinction between axiomatic systems and natural deduction systems of object-level proofs, various systems of modal logic and some non-classical logics.

**Prerequisite: **Graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 51045 METALOGIC 3 Credit Hours**

(Cross-listed with CS 41045 and CS 51045 and PHIL 41045 and PHIL 51045; slashed with MATH 41045) A detailed, systematic study of metalogic for philosophy majors, mathematics majors, computer science majors, and anyone else interested in advanced study in logic. Topics include the soundness and completeness of the propositional and predicate calculi, the decidablility of propositional calculus, the undecidability of predicate calculus, Gödel’s incompleteness proof for languages capable of expressing arithmetic, the co-extensionality of the set of general recursive functions, abacus computable functions, and Turing computable functions, and the philosophical motivations for the ChurchTuring Thesis that all computable functions are Turing computable

**Prerequisite: **Graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52001 INTRODUCTION TO ANALYSIS I 3 Credit Hours**

(Slashed with MATH 42001) 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 MATH 22005; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52002 INTRODUCTION TO ANALYSIS II 3 Credit Hours**

(Slashed with MATH 42002) Topics include further development of integration theory infinite series, uniform convergence, several variable calculus and metric spaces.

**Prerequisite: **MATH 42001 or MATH 52001; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52011 MATHEMATICAL OPTIMIZATION 3 Credit Hours**

(Slashed with MATH 42011) 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 MATH 22005; graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52021 GRAPH THEORY AND COMBINATORICS 3 Credit Hours**

(MATH 42021) Fundamentals and applications of combinatorial mathematics. Topics include transversability, colorability, networks, inclusion and exclusion, matching and designs.

**Prerequisite: **MATH 12003 and MATH 21001 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52024 NUMBERS AND GAMES 3 Credit Hours**

(Slashed with AMTH 42024) The study of partisan and impartial combinatorial games; games as numbers; Grundy-Sprague theory.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52031 MATHEMATICAL MODELS AND DYNAMICAL SYSTEMS 3 Credit Hours**

(Slashed with MATH 42031) 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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52039 MODELING PROJECTS 3 Credit Hours**

(Slashed with MATH 42039) 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: **Minimum C grade in MATH 52031; and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **3 other

**Grade Mode: **Standard Letter-S/U-IP

** **

**MATH 52041 ADVANCED CALCULUS 3 Credit Hours**

(Slashed with MATH 42041) 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 MATH 22205.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52045 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 3 Credit Hours**

(Slashed with MATH 42045) 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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52048 INTRODUCTION TO COMPLEX VARIABLES 3 Credit Hours**

(Slashed with MATH 42048) Algebra of complex numbers, analytic functions, mappings, Cauchy integral theory, residue theory and applications.

**Prerequisite: **MATH 22005 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52201 INTRODUCTION TO NUMERICAL COMPUTING I 3 Credit Hours**

(Cross-listed with CS 42201 and CS 52201; slashed with 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 and MATH 21001 and CS 23021 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 52202 INTRODUCTION TO NUMERICAL COMPUTING II 3 Credit Hours**

(Cross-listed with CS 42202 and CS 52202; slashed with 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 MATH 52201 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 55011 DIFFERENTIAL GEOMETRY 3 Credit Hours**

(Slashed with MATH 45011) Analytic and metric differential geometry of curves and surfaces.

**Prerequisite: **MATH 22005 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 55021 EUCLIDEAN GEOMETRY 3 Credit Hours**

(Slashed with MATH 45021) Geometry of Euclid extended to advanced topics of the triangle, quadrilaterals and circles: crossratio, groups, constructions, geometric generalizations; inversion.

**Prerequisite: **MATH 21001 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 55022 LINEAR GEOMETRY 3 Credit Hours**

(Slashed with MATH 45022) 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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 56001 ELEMENTARY TOPOLOGY 3 Credit Hours**

(Slashed with MATH 45001) Metric spaces, introduction to topological spaces, separation axioms.

**Prerequisite: **MATH 22005 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 57011 THEORY OF NUMBERS 3 Credit Hours**

(Slashed with MATH 47011) Divisibility properties of the integers, prime numbers, congruences, quadratic reciprocity, Diophantine equations, number theoretic functions, simple continued fractions, rational approximations.

**Prerequisite: **MATH 12003 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 57021 HISTORY OF MATHEMATICS 3 Credit Hours**

(Slashed with MATH 47021) Survey from Babylonian and Egyptian mathematics to 20th-century mathematics with emphasis on the development of algebra, geometry, calculus, number theory.

**Prerequisite: **one course MATH 22006 to 59999 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 57091 SELECTED TOPICS IN MATHEMATICS AND ITS APPLICATIONS 1-3 Credit Hours**

(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 and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter

** **

**MATH 59893 VARIABLE TITLE WORKSHOP IN MATHEMATICS 1-6 Credit Hours**

(Repeatable for credit) Studies in special topics in pure and applied mathematics.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Workshop

**Contact Hours: **1-6 other

**Grade Mode: **Satisfactory/Unsatisfactory

** **

**MATH 60051 PROBABILITY I 4 Credit Hours**

(Slashed with MATH 70051) Distribution functions, measure theory, random variables, expectation, independence, convergence, concepts, law of large numbers.

**Prerequisite: **MATH 40011 or MATH 42002 or MATH 50011 or MATH 52002; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 60052 PROBABILITY II 3 Credit Hours**

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

**Prerequisite: **MATH 6/70051 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 60061 MATHEMATICAL STATISTICS I 4 Credit Hours**

Statistics, distributions of statistics. Sampling distributions. Decision spaces and loss functions. Sufficiency and completeness. Estimation theory. Rao Blackwell and the Cramer Rao theorems. Graduate standing.

**Prerequisite: **MATH 4/52002.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 60062 MATHEMATICAL STATISTICS II 3 Credit Hours**

Tests of statistical hypothesis. Neyman Pearson Lemma. Exponential families and invariance. Sequential tests. Non-parametric procedures. Graduate standing.

**Prerequisite: **MATH 6/70061.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 60070 FINANCIAL MATHEMATICS 3 Credit Hours**

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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 60091 SEMINAR IN STATISTICS PROBABILITY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in statistics and probability.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 60093 VARIABLE TITLE WORKSHOP IN MATHEMATICS 1-3 Credit Hours**

(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 and graduate standing.

**Schedule Type: **Workshop

**Contact Hours: **1-3 other

**Grade Mode: **Satisfactory/Unsatisfactory

** **

**MATH 60094 COLLEGE TEACHING OF MATHEMATICS 1 Credit Hour**

(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.

**Schedule Type: **Lecture

**Contact Hours: **1 lecture

**Grade Mode: **Standard Letter

** **

**MATH 61051 ABSTRACT ALGEBRA I 4 Credit Hours**

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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 61052 ABSTRACT ALGEBRA II 3 Credit Hours**

Advanced topics, commutative ring theory, and field theory including polynomial rings, unique factorization domains, matrix rings, Galois theory.

**Prerequisite: **MATH 6/71051 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 61091 SEMINAR IN ALGEBRA 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in algebra.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 62041 METHODS OF APPLIED MATHEMATICS I 3 Credit Hours**

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 MATH 52041 and MATH 52045 and MATH 52048 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62042 METHODS OF APPLIED MATHEMATICS II 3 Credit Hours**

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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62051 FUNCTIONS OF A REAL VARIABLE I 4 Credit Hours**

Introduction to modern concepts of real analysis including metric spaces, measure and integration theory.

**Prerequisite: **MATH 4/52002 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62052 FUNCTIONS OF A REAL VARIABLE II 3 Credit Hours**

A continuation of MATH 62051/72051. Included are basic topics in functional analysis and Hilbert space theory.

**Prerequisite: **MATH 6/72051 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62091 SEMINAR IN REAL ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in real analysis.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 62151 FUNCTIONS OF A COMPLEX VARIABLE I 4 Credit Hours**

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

**Prerequisite: **MATH 52002 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62152 FUNCTIONS OF A COMPLEX VARIABLE II 3 Credit Hours**

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

**Prerequisite: **MATH 62151 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62191 SEMINAR IN COMPLEX ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in complex analysis.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 62202 NUMERICAL COMPUTING II 3 Credit Hours**

(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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62203 COMPUTATIONAL FINANCE 3 Credit Hours**

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 MATH 21001 and MATH 32044 and CS 10061 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62251 NUMERICAL ANALYSIS I 4 Credit Hours**

(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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62252 NUMERICAL ANALYSIS II 3 Credit Hours**

(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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62261 NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 3 Credit Hours**

(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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62262 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 3 Credit Hours**

(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.

**Prerequisite: **MATH 42045 or 52045; and MATH 42202 or 52202; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62263 NUMERICAL SOLUTION OF LARGE SPARSE LINEAR SYSTEMS 3 Credit Hours**

(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 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62264 NUMERICAL SOLUTION OF NONLINEAR SYSTEMS 3 Credit Hours**

(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.

**Prerequisite: **MATH 42041 or MATH 52041; and MATH 42202 or MATH 52202; and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 62291 SEMINAR IN NUMERICAL ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in numerical analysis.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 62391 SEMINAR IN MEASURE THEORY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in measure theory.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 62491 SEMINAR IN NONLINEAR ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in nonlinear analysis.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 62591 SEMINAR IN OPERATOR THEORY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in operator theory.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 62691 SEMINAR IN HARMONIC ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in harmonic analysis.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 63291 SEMINAR IN APPLIED MATHEMATICS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in applied mathematics.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 64091 SEMINAR IN MATHEMATICS EDUCATION 3 Credit Hours**

(Repeatable for credit)Studies in geometry, algebra, mathematics for junior and senior high school. Mathematics content professionalized for teachers.

**Prerequisite: **MATH 34001 and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **3 other

**Grade Mode: **Standard Letter

** **

**MATH 66051 INTRODUCTION TO TOPOLOGY I 4 Credit Hours**

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

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 66052 INTRODUCTION TO TOPOLOGY II 3 Credit Hours**

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

**Prerequisite: **MATH 6/76051 and graduate standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 66091 SEMINAR IN TOPOLOGY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in topology.

**Prerequisite: **Special approval and graduate standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 67091 SEMINAR IN NUMBER THEORY 1-3 Credit Hours**

(Repeatable for credit) (Slashed with MATH 77091) Seminar on current research in number theory.

**Prerequisite: **Graduate standing and special approval.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 67098 RESEARCH 1-15 Credit Hours**

(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.

**Schedule Type: **Research

**Contact Hours: **1-15 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 67199 THESIS I 2-6 Credit Hours**

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.

**Schedule Type: **Masters Thesis

**Contact Hours: **2-6 other

**Grade Mode: **Satisfactory/Unsatisfactory-IP

** **

**MATH 67299 THESIS II 2 Credit Hours**

Thesis students must continue registration each semester until all degree requirements are met.

**Prerequisite: **MATH 67199 and graduate standing.

**Schedule Type: **Masters Thesis

**Contact Hours: **2 other

**Grade Mode: **Satisfactory/Unsatisfactory-IP

** **

**MATH 70051 PROBABILITY I 4 Credit Hours**

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

**Prerequisite: **MATH 4/50011 or 4/52002; and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 70052 PROBABILITY II 3 Credit Hours**

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

**Prerequisite: **MATH 6/70051 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 70061 MATHEMATICAL STATISTICS I 4 Credit Hours**

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.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 70062 MATHEMATICAL STATISTICS II 3 Credit Hours**

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

**Prerequisite: **MATH 6/70061 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 70070 FINANCIAL MATHEMATICS 3 Credit Hours**

Topics from replication of trading strategies, arbitrage completeness, Martingale epresentation theorem, fundamental theorem of finance, stochastic differential equations, Black and Scholes formula of option pricing.

**Prerequisite: **MATH 50051 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 70091 SEMINAR IN STATISTICS PROBABILITY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in statistics and probability.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 70094 COLLEGE TEACHING OF MATHEMATICS 1 Credit Hour**

(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.

**Schedule Type: **Lecture

**Contact Hours: **1 lecture

**Grade Mode: **Standard Letter

** **

**MATH 70095 SELECTED TOPICS IN STATISTICS AND PROBABILITY 1-3 Credit Hours**

(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: **Doctoral standing and special approval.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 71001 STRUCTURE OF RINGS AND ALGEBRAS I 3 Credit Hours**

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

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 71002 STRUCTURE OF RINGS AND ALGEBRAS II 3 Credit Hours**

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

**Prerequisite: **MATH 71001 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 71011 ADVANCED GROUP THEORY 3 Credit Hours**

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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 71012 CHARACTERS OF FINITE GROUPS 3 Credit Hours**

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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 71051 ABSTRACT ALGEBRA I 4 Credit Hours**

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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 71052 ABSTRACT ALGEBRA II 3 Credit Hours**

Advanced topics, commutative ring theory, and field theory including polynomial rings, unique factorization, domains, matrix rings, Galois theory.

**Prerequisite: **MATH 6/71051 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 71091 SEMINAR IN ALGEBRA 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in algebra.

**Prerequisite: **Permission and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 71095 SELECTED TOPICS IN ALGEBRA 1-3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72001 FUNCTIONAL ANALYSIS I 3 Credit Hours**

A study of principles of linear analysis in the setting of normed linear spaces and topological vector spaces.

**Prerequisite: **MATH 72052 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72002 FUNCTIONAL ANALYSIS II 3 Credit Hours**

A study of principles of linear analysis in the setting of normed linear spaces and topological vector spaces.

**Prerequisite: **MATH 72001 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72041 METHODS OF APPLIED MATHEMATICS I 3 Credit Hours**

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 MATH 52041 and MATH 52045 and MATH 52048 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72042 METHODS OF APPLIED MATHEMATICS II 3 Credit Hours**

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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72051 FUNCTIONS OF A REAL VARIABLE I 4 Credit Hours**

Introduction to modern concepts of real analysis including metric spaces, measure and integration theory.

**Prerequisite: **MATH 4/52002 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72052 FUNCTIONS OF A REAL VARIABLE II 3 Credit Hours**

A continuation of MATH 62051/72051. Included are basic topics in functional analysis and Hilbert space theory.

**Prerequisite: **MATH 6/72051 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72091 SEMINAR IN REAL ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in real analysis.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 72095 SELECTED TOPICS IN REAL ANALYSIS 1-3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72151 FUNCTIONS OF A COMPLEX VARIABLE I 4 Credit Hours**

(Slashed with MATH 62151) Topological properties of the complex plane; analytic, entire, meromorphic functions; analytic continuation; conformal mappings; Picard's Theorem; Riemann surfaces.

**Prerequisite: **MATH 52002; and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72152 FUNCTIONS OF A COMPLEX VARIABLE II 3 Credit Hours**

(Slashed with MATH 62152) Topological properties of the complex plane; analytic, entire, meromorphic functions; analytic continuation; conformal mappings; Picard's theorem; Riemann surfaces.

**Prerequisite: **MATH 72151; and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72191 SEMINAR IN COMPLEX ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in complex analysis.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 72195 SELECTED TOPICS IN COMPLEX ANALYSIS 1-3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72202 NUMERICAL COMPUTING II 3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72203 COMPUTATIONAL FINANCE 3 Credit Hours**

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 MATH 21001 and MATH 32044 and CS 10061 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72251 NUMERICAL ANALYSIS I 4 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72252 NUMERICAL ANALYSIS II 3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72261 NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72262 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 3 Credit Hours**

(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.

**Prerequisite: **MATH 42045 or 52045; and MATH 42202 or 52202; and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72263 NUMERICAL SOLUTION OF LARGE SPARSE LINEAR SYSTEMS 3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72264 NUMERICAL SOLUTION OF NONLINEAR SYSTEMS 3 Credit Hours**

(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.

**Prerequisite: **MATH 4/52041 and MATH 4/52202 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72291 SEMINAR IN NUMERICAL ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in numerical analysis.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 72295 SELECTED TOPICS IN NUMERICAL ANALYSIS 1-3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 72391 SEMINAR IN MEASURE THEORY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in measure theory.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 72491 SEMINAR IN NONLINEAR ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in nonlinear analysis.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 72591 SEMINAR IN OPERATOR THEORY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in operator theory.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 72691 SEMINAR IN HARMONIC ANALYSIS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in harmonic analysis.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 73291 SEMINAR IN APPLIED MATHEMATICS 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in applied mathematics.

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 76051 INTRODUCTION TO TOPOLOGY I 4 Credit Hours**

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

**Prerequisite: **Special approval and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **4 lecture

**Grade Mode: **Standard Letter

** **

**MATH 76052 INTRODUCTION TO TOPOLOGY II 3 Credit Hours**

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

**Prerequisite: **MATH 6/76051 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 76091 SEMINAR IN TOPOLOGY 1-3 Credit Hours**

(Repeatable for credit)Seminar on current research in topology.

**Prerequisite: **Permission and doctoral standing.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 76095 SELECTED TOPICS IN TOPOLOGY 1-3 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 77011 ALGEBRAIC NUMBER THEORY 3 Credit Hours**

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 thoerem on units.

**Prerequisite: **MATH 57011 and MATH 6/71052 and doctoral standing.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 77012 ANALYTIC NUMBER THEORY 3 Credit Hours**

Multiplicative functions and summatory functions, Riemann's Zeta function and the prime number theorem, L-functions and Dirichlet's theorem on primes in artithmetic progressions, asymptotic formula for partitions.

**Prerequisite: **MATH 57011 and MATH 6/72151.

**Schedule Type: **Lecture

**Contact Hours: **3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 77091 SEMINAR IN NUMBER THEORY 1-3 Credit Hours**

(Repeatable for credit) (Slashed with MATH 67091) Seminar on current research in number theory.

**Prerequisite: **Doctoral standing and special approval.

**Schedule Type: **Seminar

**Contact Hours: **1-3 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 77095 SELECTED TOPICS IN MATHEMATICS 1-3 Credit Hours**

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

**Prerequisite: **Permission.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 77098 RESEARCH 1-15 Credit Hours**

(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.

**Schedule Type: **Research

**Contact Hours: **1-15 other

**Grade Mode: **Standard Letter-S/U

** **

**MATH 77195 SELECTED TOPICS IN NUMBER THEORY 1-3 Credit Hours**

(Repeatable for credit)Content varies with each offering and complements topics covered in MATH 77011 and MATH 77012.

**Prerequisite: **Doctoral standing and special approval.

**Schedule Type: **Lecture

**Contact Hours: **1-3 lecture

**Grade Mode: **Standard Letter

** **

**MATH 87098 RESEARCH 1-15 Credit Hours**

(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.

**Schedule Type: **Research

**Contact Hours: **1-15 other

**Grade Mode: **Standard Letter

** **

**MATH 87199 DISSERTATION I 15 Credit Hours**

(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 and doctoral standing.

**Schedule Type: **Dissertation

**Contact Hours: **15 other

**Grade Mode: **Satisfactory/Unsatisfactory-IP

** **

**MATH 87299 DISSERTATION II 15 Credit Hours**

(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.

**Schedule Type: **Dissertation

**Contact Hours: **15 other

**Grade Mode: **Satisfactory/Unsatisfactory-IP