
Mathematics
Manfred Stoll, Chair
Professors
Colin Bennett, Ph.D., University of Newcastle upon Tyne, 1971
Susanne C. Brenner, Ph.D., University of Michigan, 1988, Assistant Chair
Ronald A. DeVore, Ph.D., Ohio State University, 1967, Robert L. Sumwalt Professor of Mathematics
Stephen J. Dilworth, Ph.D., Cambridge University, 1985
Michael A. Filaseta, Ph.D., University of Illinois, 1984
Maria Girardi, Ph.D., University of Illinois, 1990
Jerrold R. Griggs, Ph.D., Massachusetts Institute of Technology, 1977
Ralph E. Howard, Ph.D., California Institute of Technology, 1982
Andrew R. Kustin, Ph.D., University of Illinois, 1979
George F. McNulty, Ph.D., University of California, Berkeley, 1972
Matthew Miller, Ph.D., University of Illinois, 1979
Peter J. Nyikos, Ph.D., CarnegieMellon University, 1971
Konstantin Oskolkov, Ph.D., Steklov Institute, 1978
Pencho Petrushev, Ph.D., University of Sofia, 1977
James W. Roberts, Ph.D., Rutgers University, 1970
Anton R. Schep, Ph.D., University of Leiden, 1977, Graduate Director
Robert C. Sharpley, Ph.D., University of Texas, 1972
Robert M. Stephenson Jr., Ph.D., Tulane University, 1967
Manfred Stoll, Ph.D., Pennsylvania State University, 1971
László A. Székely, Ph.D., Eötvös University, 1983
Vladimir Temlyakov, Ph.D., Steklov Institute, 1981
Hong Wang, Ph.D., University of Wyoming, 1992
Associate Professors
Daniel B. Dix, Ph.D., University of Chicago, 1988
Peter W. Harley III, Ph.D., University of Georgia, 1966
George W. Johnson III, Ph.D., University of Tennessee, 1971
Marek B. Kossowski, Ph.D., University of North Carolina, 1982
Douglas B. Meade, Ph.D., CarnegieMellon University, 1989, Undergraduate Director
David P. Sumner, Ph.D., University of Massachusetts, 1971
Liyeng Sung, Ph.D., State University of New York at Stony Brook, 1983
Ognian T. Trifonov, Ph.D., University of Sofia, 1990
Xian Wu, Ph.D., Harvard University, 1986
Assistant Professors
George Androulakis, Ph.D., University of Texas, 1996
Lili Ju, Ph.D., Iowa State University, 2002
Linyuan Lu, Ph.D., University of California, San Diego, 2002
Adela Vraciu, Ph.D., University of Michigan, 2000
Gang Yu, Ph.D., University of Georgia, 2000
Overview
The Department of Mathematics is fast evolving into one of the premier centers in the Southeast for mathematics research and education. Its master's and doctoral programs have been cited for excellence by the S.C. Commission on Higher Education. With its internationally renowned faculty and modern, inhouse library and computing facilities, the department provides a stimulating environment for graduate studies. A full range of graduate programs in pure and applied mathematics and mathematics education is available.
The department's degree programs provide the technical expertise and interdisciplinary skills required of the modern mathematician. We provide training for those who wish to pursue a career in teaching; those who plan mathematicsrelated careers in business, government, or industry; and those who wish to obtain the intensive training that will lead them to the frontiers of modern research in pure or applied mathematics.
The Department of Mathematics offers programs leading to the Master of Arts, Master of Science, and Doctor of Philosophy. The department also offers programs leading to the Master of Mathematics and, in cooperation with the College of Education, a program leading to the degree of Master of Arts in Teaching. A description of the basic M.A.T. requirements appears in the "College of Education" section of this bulletin.
Admission
For admission into the M.S., M.A., or Ph.D. degree programs, applicants must have a bachelor's degree from an approved institution and should have an undergraduate foundation in mathematics equivalent to that of a major in mathematics at the University of South Carolina. A minimum B average in all collegelevel math courses is required for full admission. Applicants who do not have this preparation may be admitted on probation and placed in such undergraduate courses as necessary to strengthen their backgrounds.
For admission to the M.M. or M.A.T. degree programs, applicants must have a bachelor's degree from an approved institution and are expected to have completed a minimum of 6 credit hours beyond multivariable calculus. Applicants with background deficiencies may be admitted on a conditional basis and placed in certain undergraduate courses to strengthen their foundation. Such course work may not be used toward the degree.
More specific admissions details accompany the discussion of degrees below.
Applicants should submit an official transcript from each school or college previously attended, at least two letters of recommendation from persons familiar with their abilities in mathematics, and a report of scores achieved on the GRE.
Applicants whose native language is not English are also required to submit a satisfactory score on the TOEFL or the University of Cambridge's IELTS Academic Course Type 2 exam. The minimum acceptable score on the TOEFL is 230 (computerbased) or 570 (paperbased). The minimum acceptable overall band score on the IELTS Academic Course Type 2 exam is 6.5. A minimum TOEFL score of 600 (250 computerbased score), or comparable IELTS score, is required for consideration for a teaching assistantship.
Application materials should be sent to The Graduate School, University of South Carolina, Columbia, SC 29208. Inquiries concerning the graduate program and graduate assistantships in mathematics should be directed to: Director of Graduate Studies, Department of Mathematics, University of South Carolina, Columbia, SC 29208.
Email: graddir@math.sc.edu
Degree Requirements
All master's degrees require 30 approved semester hours of course work, at least half of which (excluding the thesis) must be taken at the 700 level or above. In addition, a comprehensive examination taken upon conclusion of the program is required. Both the M.S. and the M.A. degrees require a thesis. No foreign language proficiency is required for any of the master's degrees.
Each candidate for the Ph.D. degree is required to complete a minimum of 60 hours of course work beyond the baccalaureate degree, including 12 semester hours of graduate course work separate from the course work covered by the admissiontocandidacy and comprehensive examinations (see below). Students pursuing the Ph.D. degree in mathematics are required to take three examinations: the admissiontocandidacy, comprehensive, and dissertation examinations. Demonstrated reading proficiency is also required in one of the following foreign languages: French, German, or Russian. The student must also show proficiency in a computer programming language.
Master of Science in Mathematics
The M.S. is designed primarily for students who seek broad and intensive preparation for teaching in a junior college or working in industry.
For admission into the M.S. program, applicants must have a bachelor's degree from an approved institution and should have an undergraduate foundation in mathematics equivalent to that of a major in mathematics at the University of South Carolina, which at a minimum should include a course in abstract algebra (equivalent to MATH 546) or advanced calculus (equivalent to MATH 554). A minimum B average in all collegelevel math courses is required for full admission.
The M.S. degree requires a thesis and 30 approved semester hours of graduate course work, including satisfactory completion of MATH 700 and 703, the threesemesterhour thesis course MATH 799, and MATH 790. MATH 790 is a onesemesterhour seminar designed for firstyear graduate students. The courses in the student's program should be numbered above 699. However, in special circumstances some 500level courses may be approved for a student's program if the courses supplement 700level course work. In general, a student's M.S. program should be fairly broad in scope and should include courses of both a pure and applied nature.
The thesis for this degree is generally a short monograph (to be bound and placed in the University library and in the department), the content of which is drawn from several research papers in an area of interest to the student.
Upon conclusion of the program, each M.S. degree candidate either undergoes an oral examination (which includes an oral presentation of the thesis) administered by a committee chaired by the student's thesis advisor, or obtains a master's pass on the Ph.D. admissiontocandidacy examination. Students who earn a master's pass are invited to present the thesis in a colloquium address to the department.
Master of Arts in Mathematics
The M.A. is designed primarily for students who wish to enter the Ph.D. program in mathematics. A student's program of study for this degree is usually narrow in scope but intense in content. Course work for the degree is generally regarded as preparatory for the Ph.D.
For admission into the M.A. degree program, applicants must have a bachelor's degree from an approved institution and should have an undergraduate foundation in mathematics equivalent to that of a major in mathematics at the University of South Carolina, which at a minimum should include a course in abstract algebra (equivalent to MATH 546) or advanced calculus (equivalent to MATH 554). A minimum B average in all collegelevel math courses is required for full admission.
The M.A. degree requires a master's pass on the admissiontocandidacy examination as well as a thesis and 30 approved semester hours of graduate mathematics course work, including MATH 790 and the threecredit thesis course, MATH 799. All courses in the student's program must be numbered 700 and above and must include a oneyear sequence in linear algebra/algebra (MATH 700, and one of 701, 706) and the analysis sequence (MATH 703, 704). These courses form the core of the student’s program and provide the topics upon which the master’s examination is based.
The thesis for this degree is generally a short monograph (to be bound and placed in the University library and in the department), the content of which is drawn from several current research papers (possibly including the student's original contributions) in an area of interest to the student, which could lead to topics and issues of suitable depth for a Ph.D. dissertation. Upon conclusion of the program, the student is invited to present the thesis to the department in a colloquium address.
Doctor of Philosophy Degree
The Ph.D. is designed to produce a skilled, professional mathematician who is trained to conduct research in mathematics, function effectively as a classroom teacher at the college level, or become a professional practitioner in an industrial setting.
For admission into the Ph.D. degree program in mathematics, applicants must have a bachelor's degree from an approved institution and should have an undergraduate foundation in mathematics equivalent to that of a major in mathematics at the University of South Carolina, which at a minimum should include a course in abstract algebra (equivalent to MATH 546) and advanced calculus (equivalent to MATH 554). A minimum B average in all collegelevel math courses is required for full admission. The subject GRE in mathematics is strongly recommended for applicants to the Ph.D. program. Applicants who have taken the subject GRE will get preferential treatment for assistantship awards.
Each candidate for the Ph.D. degree is required to complete a minimum of 60 hours of course work beyond the baccalaureate degree, including 12 semester hours of graduate course work separate from the course work covered by the admissiontocandidacy and comprehensive examinations (see below). These 12 hours are in addition to directed reading courses. The student's doctoral committee decides which courses are appropriate to fulfill this requirement. In addition, the Ph.D. candidate is required to take three semester hours of the doctoral seminar MATH 890, the content of which must include research. Students may earn these doctoral seminar credits by taking and participating in a seminar in their research area. Demonstrated reading proficiency is also required in one of the following foreign languages: French, German, or Russian. In addition, the student must show proficiency in a computer programming language.
Students pursuing the Ph.D. degree in mathematics are required to take three examinations: the admissiontocandidacy, comprehensive, and dissertation examinations.
The admissiontocandidacy examination in mathematics is administered in two versions. The first version consists of two threehour written examinations, each of which is based primarily (but not exclusively) on the subject matter of the two oneyear sequences in algebra (MATH 700,701) and analysis (MATH 703, 704). The second version consists of two threehour written examinations, each of which is based primarily (but not exclusively) on the subject matter of the two oneyear sequences in linear and numerical linear algebra (MATH 700, 706) and analysis (MATH 703, 704). Two attempts of the admissiontocandidacy examination are allowed. The first attempt should occur after the first year of graduate study and within the first two years of graduate study. The second attempt must be made at the next scheduled examination. Exceptions to the time constraint for unusual cases may be petitioned to the graduate advisory council.
The Ph.D. comprehensive is an indepth examination consisting of a written part administered in three, threetofour hour sessions, and an oral component. The written portion of the examination must include the subject matter of two oneyear sequences numbered 710 or higher from two of the eight areas listed below and also test in depth the subject matter of the student’s research area. The oral portion of the comprehensive will be based on the student's program of study and may include topics not covered by either the admissiontocandidacy examination or the written portion of the comprehensive examination.
The student's research specialization for the comprehensive examination should be selected from one of the following areas: algebra, analysis, applied mathematics, discrete mathematics, geometry, logic, number theory, and topology.
The comprehensive examination may be repeated only once. All portions of the examination must be completed within three weeks. As a general rule, the exam is offered twice each year, once in August and again in January, and should be taken after candidates have completed all courses required in their program. In special cases, the examination may be scheduled any time during the year with permission from the examination committee. The examination must be completed at least 60 days prior to the date in which the student expects to receive the degree.
To complete the program, the student must write a dissertation (to be bound and placed in the University library and in the department), under the direction of a member of the graduate faculty, and defend the content of the dissertation in a final examination before the doctoral committee. It is expected that the content of the student's dissertation will be a significant contribution to the body of current research and will be published in a reputable journal.
To ensure diversity, each student is required to satisfactorily complete (B or better) 12 semester hours of course work in subject areas not covered by the admissiontocandidacy and comprehensive examinations. Additional course requirements may be stipulated by the student's doctoral committee. The Graduate School requires a doctoral candidate to register for at least 12 hours of dissertation preparation (MATH 899).
Mathematics Education
The department offers two degrees for students who wish to pursue graduate programs emphasizing secondary mathematics educationthe M.M. and the M.A.T. Courses specifically designed for these programs are designated by the letter I adjoined to the course number. These courses are generally offered in the late afternoon during the academic year and during the summer to provide area teachers the opportunity to work toward a degree on a parttime basis.
For admission to the M.M. or M.A.T. degree programs, applicants must have a bachelor’s degree from an approved institution and are expected to have completed a minimum of six credit hours beyond multivariable calculus (i.e., six credit hours in mathematics beyond three semesters of calculus). Applicants with background deficiencies may be admitted on a conditional basis and placed in certain undergraduate courses to strengthen their foundation. Such course work may not be used toward the degree. Typically, a minimum of 1000 is required on the combined scores from the verbal and quantitative portions of the GRE.
Master of Mathematics Degree
The Master of Mathematics degree is designed primarily for students who seek a broad, thorough training in mathematics which includes course work specifically designed to meet the needs of secondaryschool teachers.
The M.M. degree requires 30 approved semester hours of graduate course work, up to 6 hours of which may be outside the departments of mathematics, computer science, and statistics. A core of four courses is required of all students:
MATH 701I, 702I, 703I, and 704I
In addition, students must include in their program (if similar courses have not been taken previously) a course in geometry (chosen from MATH 531, 532, or 736I) and a course in linear algebra (MATH 526 or 544). To ensure breadth in the program of study, the remaining course work should include courses in probability and statistics and computer science.
Each candidate for the M.M. degree is required to pass a written comprehensive examination, which is based primarily on the four core courses. The examination will consist of two, twotothree hour written examinations that will be offered by the department three times each year, in May, August, and December. Students should take the comprehensive examination immediately upon completion of the core courses. Foreign language proficiency is not required.
Master of Arts in Teaching
The M.A.T. in mathematics is offered by the Department of Mathematics jointly with the College of Education. This degree program is designed specifically for students who wish to obtain certification in mathematics at the secondary level.
The M.A.T. degree requires 30 approved semester hours of graduatelevel course work in mathematics and education (exclusive of student teaching), no less than 6 and no more than 15 of which may be in education. The individual student's program is planned according to that student's background and goals. At least half of the student's course work must be numbered 700 or higher.
Each student's program of study must include a course in geometry (one from MATH 531, 532, or 736I), algebraic structures (MATH 546 or 701I), analysis (MATH 554 or 703I), probability and statistics (MATH 712I or STAT 509), and two approved math electives at the 500 or 700 level. If similar courses have already been taken, appropriate substitutions will be made.
Unless previously taken, the student must also take a course in linear algebra (MATH 526 or 544), a course in discrete mathematics (MATH 574), and a course in computer science using a highlevel programming language (CSCE 500 or 145).
Course work in education must include human growth and development (EDPY 705), philosophy and education (EDFN 744 or 749), a curriculum course (one of EDSE 770, EDSE 783, or EDLP 725), and methods of teaching (EDSE 764).
The student must also complete an undergraduate mathematics methods course (EDSE 550) and a program of directed teaching in mathematics (EDSE 778A and 778B), which is administered through the College of Education. Students must apply for admission to directed teaching through the College of Education’s Office of Field Experience early in the fall or spring semester prior to the semester of directed teaching.
Upon admission to the M.A.T. program, the student is assigned a faculty advisor in mathematics to assist in the development of the mathematics portion of the program. Approval of the candidate's program will be granted by a committee of three faculty members, consisting of the faculty advisor in mathematics, the faculty advisor in education, and a faculty member from either mathematics or education.
Each student must maintain a B average on all graduatelevel course work in mathematics and a B average on all graduatelevel course work in education.
Candidates for the M.A.T. degree are required to pass a written comprehensive examination covering their program of study and emphasizing calculus, algebra (MATH 546 or 701I), and analysis (MATH 554 or 703I). The M.A.T. requires neither a foreign language nor a thesis.
Course Descriptions (MATH)
 511  Probability. {=STAT 511} (3) (Prereq: MATH 241 with a grade of C or higher) Probability and independence; discrete and continuous random variables; joint, marginal, and conditional densities, moment generating functions; laws of large numbers; binomial, Poisson, gamma, univariate, and bivariate normal distributions.
 520  Ordinary Differential Equations. (3) (Prereq: MATH 544 or 526; or consent of department) Differential equations of the first order, linear systems of ordinary differential equations, elementary qualitative properties of nonlinear systems.
 521  Boundary Value Problems and Partial Differential Equations. (3) (Prereq: MATH 520 or 241 and 242) Laplace transforms, twopoint boundary value problems and Green's functions, boundary value problems in partial differential equations, eigenfunction expansions and separation of variables, transform methods for solving PDE's, Green's functions for PDE's, and the method of characteristics.
 522  Wavelets. (3) (Prereq: MATH 544 or 526 or consent of instructor) Basic principles and methods of Fourier transforms, wavelets, and multiresolution analysis; applications to differential equations, data compression, signal and image processing; development of numerical algorithms. Computer implementation.
 524  Nonlinear Optimization. (3) (Prereq: MATH 526 or 544 or consent of department) Descent methods, conjugate direction methods, and QuasiNewton algorithms for unconstrained optimization; globally convergent hybrid algorithm; primal, penalty, and barrier methods for constrained optimization. Computer implementation of algorithms.
 525  Mathematical Game Theory. (3) (Prereq: MATH 526 or 544) Twoperson zerosum games, minimax theorem, utility theory, nperson games, market games, stability.
 526  Numerical Linear Algebra. (4) (Prereq: MATH 241) Matrix algebra, Gauss elimination, iterative methods; overdetermined systems and least squares; eigenvalues, eigenvectors; numerical software. Computer implementation. Three lectures and one laboratory hour per week. Credit may not be received for both MATH 526 and MATH 544.
 527  Numerical Analysis. {=CSCE 561} (3) (Prereq: MATH 242 or 520) Interpolation and approximation of functions; solution of algebraic equations; numerical differentiation and integration; numerical solutions of ordinary differential equations and boundary value problems; computer implementation of algorithms.
 531  Foundations of Geometry. (3) (Prereq: MATH 241) The study of geometry as a logical system based upon postulates and undefined terms. The fundamental concepts and relations of Euclidean geometry developed rigorously on the basis of a set of postulates. Some topics from nonEuclidean geometry.
 532  Modern Geometry. (3) (Prereq: MATH 241) Projective geometry, theorem of Desargues, conics, transformation theory, affine geometry, Euclidean geometry, nonEuclidean geometries, and topology.
 533  Elementary Geometric Topology. (3) (Prereq: MATH 241) Topology of the line, plane, and space, Jordan curve theorem, Brouwer fixed point theorem, Euler characteristic of polyhedra, orientable and nonorientable surfaces, classification of surfaces, network topology.
 534  Elements of General Topology. (3) (Prereq: MATH 241) Elementary properties of sets, functions, spaces, maps, separation axioms, compactness, completeness, convergence, connectedness, path connectedness, embedding and extension theorems, metric spaces, and compactification.
 540  Modern Applied Algebra. (3) (Prereq: MATH 241) Finite structures useful in applied areas. Binary relations, Boolean algebras, applications to optimization, and realization of finite state machines.
 541  Algebraic Coding Theory. (3) (Prereq: MATH 526 or MATH 544 or consent of department) Errorcorrecting codes, polynomial rings, cyclic codes, finite fields, BCH codes.
 544  Linear Algebra. (3) (Prereq: MATH 241) Matrix algebra, solution of linear systems; notions of vector space, independence, basis, dimension; linear transformations, change of basis; eigenvalues, eigenvectors, Hermitian matrices, diagonalization; CayleyHamilton theorem. Credit may not be received for both MATH 526 and MATH 544.
 546  Algebraic Structures I. (3) (Prereq: MATH 241) Permutation groups; abstract groups; introduction to algebraic structures through study of subgroups, quotient groups, homomorphisms, isomorphisms, direct product; decompositions; introduction to rings and fields.
 547  Algebraic Structures II. (3) (Prereq: MATH 546) Rings, ideals, polynomial rings, unique factorization domains; structure of finite groups; topics from: fields, field extensions, Euclidean constructions, modules over principal ideal domains (canonical forms).
 550  Vector Analysis. (3) (Prereq: MATH 241) Vector fields, line and path integrals, orientation and parametrization of lines and surfaces, change of variables and Jacobians, oriented surface integrals, theorems of Green, Gauss, and Stokes; introduction to tensor analysis.
 551  Introduction to Differential Geometry. (3) (Prereq: MATH 241) Parametrized curves, regular curves and surfaces, change of parameters, tangent planes, the differential of a map, the Gauss map, first and second fundamental forms, vector fields, geodesics, and the exponential map.
 552  Applied Complex Variables. (3) (Prereq: MATH 241) Complex integration, calculus of residues, conformal mapping, Taylor and Laurent Series expansions, applications.
 554  Analysis I. (3) (Prereq: MATH 241) Least upper bound axiom, the real numbers, compactness, sequences, continuity, uniform continuity, differentiation, Riemann integral and fundamental theorem of calculus.
 555  Analysis II. (3) (Prereq: MATH 554 or consent of department) RiemannStieltjes integral, infinite series, sequences and series of functions, uniform convergence, Weierstrass approximation theorem, selected topics from Fourier series or Lebesgue integration.
 561  Introduction to Mathematical Logic. (3) (Prereq: MATH 241) Syntax and semantics of formal languages; sentential logic, proofs in first order logic; Godel's completeness theorem; compactness theorem and applications; cardinals and ordinals; the LowenheimSkolemTarski theorem; Beth's definability theorem; effectively computable functions; Godel's incompleteness theorem; undecidable theories.
 562  Theory of Computation. {=CSCE 551} (3) (Prereq: CSCE 350 or MATH 526 or 544 or 574) Basic theoretical principles of computer science as modeled by formal languages and automata; computability and computational complexity. Major credit may not be received for both CSCE 355 and CSCE 551.
 570  Discrete Optimization. (3) (Prereq: MATH 526 or 544) Discrete mathematical models. Applications to such problems as resource allocation and transportation. Topics include linear programming, integer programming, network analysis, and dynamic programming.
 574  Discrete Mathematics I. (3) (Prereq: MATH 142) Mathematical models; mathematical reasoning; enumeration; induction and recursion; tree structures; networks and graphs; analysis of algorithms.
 575  Discrete Mathematics II. (3) (Prereq: MATH 574) A continuation of MATH 574. Inversion formulas; Polya counting; combinatorial designs; minimax theorems; probabilistic methods; Ramsey theory; other topics.
 576  Combinatorial Game Theory. (3) (Prereq: MATH 526, 544, or 574) Winning in certain combinatorial games such as Nim, Hackenbush, and Domineering. Equalities and inequalities among games, SpragueGrundy theory of impartial games, games which are numbers.
 580  Elementary Number Theory. (3) (Prereq: MATH 241) Divisibility, primes, congruences, quadratic residues, numerical functions. Diophantine equations.
 587  Introduction to Cryptography. {=CSCE 557} (3) (Prereq: CSCE 145, MATH 241, and either CSCE 355 or MATH 574) Design of secret codes for secure communication, including encryption and integrity verification: ciphers, cryptographic hashing, and public key cryptosystems such as RSA. Mathematical principles underlying encryption. Codebreaking techniques. Cryptographic protocols.
 599  Topics in Mathematics. (13) Recent developments in pure and applied mathematics selected to meet current faculty and student interest.
 650  AP Calculus for Teachers. (3) (Prereq: current secondary highschool teacher certification in mathematics and at least 6 hours of calculus) A thorough study of the topics to be presented in AP calculus, including limits of functions, differentiation, integration, infinite series, and applications. (Not intended for degree programs in mathematics.)
 700  Linear Algebra. (3) Vector spaces, linear transformations, dual spaces, decompositions of spaces, and canonical forms.
 701  Algebra I. (3) (Prereq: MATH 700) Algebraic structures, substructures, products, homomorphisms, and quotient structures of groups, rings, and modules.
 701I  Foundations of Algebra I. (3) (Prereq: MATH 241 or equivalent) An introduction to algebraic structures; group theory including subgroups, quotient groups, homomorphisms, isomorphisms, decomposition; introduction to rings and fields.
 702I  Foundations of Algebra II. (3) (Prereq: MATH 701I or equivalent) Theory of rings including ideals, polynomial rings, and unique factorization domains; structure of finite groups; fields; modules.
 703, 704  Analysis I, II. (3 each) Compactness, completeness, continuous functions. Outer measures, measurable sets, extension theorem and LebesgueStieltjes measure. Integration and convergence theorems. Product measures and Fubini's theorem. Differentiation theory. Theorems of Egorov and Lusin. Lpspaces. Analytic functions: CauchyRiemann equations, elementary special functions. Conformal mappings. Cauchy's integral theorem and formula. Classification of singularities, Laurent series, the Argument Principle. Residue theorem, evaluation of integrals and series.
 703I  Foundations of Analysis I. (3) (Prereq: MATH 241 or equivalent) The real numbers and least upper bound axiom; sequences and limits of sequences; infinite series; continuity; differentiation; the Riemann integral.
 704I  Foundations of Analysis II. (3) (Prereq: MATH 703I or equivalent) Sequences and series of functions; power series, uniform convergence; interchange of limits; limits and continuity in several variables.
 705  Analysis III. (3) (Prereq: MATH 703, 704) Signed and complex measures, RadonNikodym theorem, decomposition theorems. Metric spaces and topology, Baire category, StoneWeierstrass theorem, ArzelaAscoli theorem. Introduction to Banach and Hilbert spaces, Riesz representation theorems.
 706  Numerical Linear Algebra. (3) (Prereq: MATH 700 or consent of the department) Matrix factorization; iterative methods including preconditioning, iterative methods of eigenvalue problems, singular value decomposition, least squares. Includes theoritical development of concepts and practical algorithm implementation.
 710  Probability Theory I. {=STAT 710} (3) (Prereq: STAT 511, 512, or MATH 703) Probability spaces, random variables and distributions, expectations, characteristic functions, laws of large numbers, and the central limit theorem.
 711  Probability Theory II. {=STAT 711} (3) (Prereq: MATH 710) More about distributions, limit theorems, Poisson approximations, conditioning, martingales, and random walks.
 712I  Probability and Statistics. (3) This course will include a study of permutations and combinations; probability and its application to statistical inferences; elementary descriptive statistics of a sample of measurements; the binomial, Poisson, and normal distributions; correlation and regression.
 716  Selected Topics in Probability. {=STAT 716} (3) Fields of study to be individually assigned. Primarily for doctoral candidates.
 720  Applied Mathematics I. (3) (Prereq: MATH 555 or equivalent) Methods for solving equations from applied mathematics and the natural sciences, including a study of boundary value problems, integral equations, and eigenvalue problems using transform techniques, Green's functions, and variational principles.
 721  Applied Mathematics II. (3) (Prereq: MATH 720) Topics in partial differential equations with emphasis on the equations of the natural sciences; includes classifications of higher order equations, existence and uniqueness of solutions, theory of characteristics, basic properties of elliptic and parabolic equations, Dirichlet and Neumann problems, and the Cauchy problem for hyperbolic equations.
 722  Numerical Optimization. (3) (Prereq: graduate standing or consent of the department) Topics in optimization; includes linear programming, integer programming, gradient methods, least squares techniques, and discussion of existing mathematical software.
 723  Advanced Differential Equations. (3) (Prereq: MATH 721 or consent of instructor) Advanced topics in ordinary and partial differential equations.
 724  Numerical Differential Equations. (3) Techniques for numerically solving differential equations; includes finite difference methods, Galerkin methods, finite element method, and collocation.
 725  Approximation Theory. (3) (Prereq or coreq: MATH 703) Approximation of functions; existence, uniqueness and characterization of best approximants; Chebyshev's theorem; Chebyshev polynomials; degree of approximation; Jackson and Bernstein theorems; Bsplines; approximation by splines; quasiinterpolants; spline interpolation.
 726  Numerical Analysis I. (3) (Prereq: MATH 554 or equivalent and MATH 706) Error analysis; approximation of functions by algebraic polynomials, splines, and trigonometric polynomials; divided differences; numerical differentiation; quadrature including Gaussian and Romberg integration; a thorough study of numerical ODEs.
 727  Numerical Analysis II. {=CSCE 761} (3) (Prereq: MATH 726) Continuation of MATH 726.
 728  Selected Topics in Applied Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
 730, 731  General Topology I, II. (3 each) Topological spaces, filters, compact spaces, connected spaces, uniform spaces, complete spaces, topological groups, function spaces.
 732, 733  Algebraic Topology I, II. (3 each) (Prereq: MATH 730 or 705, and 701) The fundamental group, homological algebra, simplicial complexes, homology and cohomology groups, cupproduct, triangulable spaces.
 734  Differential Geometry. (3) (Prereq: MATH 550) Differentiable manifolds; classical theory of surfaces and hypersurfaces in Euclidean space; tensors, forms and integration of forms; connections and covariant differentiation; Riemannian manifolds; geodesics and the exponential map; curvature; Jacobi fields and comparison theorems, generalized GaussBonnet theorem.
 735  Lie Groups. (3) (Prereq: MATH 705 or 730) Manifolds; topological groups, coverings and covering groups; Lie groups and their Lie algebras; closed subgroups of Lie groups; automorphism groups and representations; elementary theory of Lie algebras; simply connected Lie groups; semisimple Lie groups and their Lie algebras.
 736I  Modern Geometry. (3) (Prereq: MATH 241 or equivalent) Synthetic and analytic projective geometry, homothetic transformations, Euclidean geometry, nonEuclidean geometries, and topology.
 738  Selected Topics in Geometry and Topology. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
 740  Algebra II. (3) (Prereq: MATH 701) Fields and field extensions Galois theory, topics from: transcendental field extensions, algebraically closed fields, finite fields.
 741, 742  Algebra III, IV. (3 each) (Prereq: MATH 740) Theory of rings, modules, fields, bilinear forms, and advanced topics in matrix theory.
 743  Lattice Theory. (3) (Prereq: MATH 740) Sublattices, homomorphisms and direct products of lattices; freely generated lattices; modular lattices and projective geometries; the Priestley and Stone dualities for distributive and Boolean lattices; congruence relations on lattices.
 744  Matrix Theory. (3) (Prereq: MATH 700) Extremal properties of positive definite and hermitian matrices, doubly stochastic matrices, totally nonnegative matrices, eigenvalue monotonicity, HadamardFisher determinantal inequalities.
 746  Communtative Algebra. (3) (Prereq: MATH 701) Prime spectrum and Zariski topology; finite, integral, and flat extensions; dimension; depth; homological techniques, normal and regular rings.
 747  Algebraic Geometry. (3) (Prereq: MATH 701) Properties of affine and projective varieties defined over algebraically closed fields, rational mappings, birational geometry and divisors especially on curves and surfaces, Bezout's theorem, RiemannRoch theorem for curves.
 748  Selected Topics in Algebra. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
 750  Fourier Analysis. (3) (Prereq: MATH 703 and 704) The Fourier transform on the circle and line, convergence of Fejer means; Parseval's relation and the square summable theory, convergence and divergence at a point; conjugate Fourier series, the conjugate function and the Hilbert transform, the HardyLittlewood maximal operator and Hardy spaces.
 751  The Mathematical Theory of Wavelets. (3) (Prereq: MATH 750) The L^{1} and L^{2} theory of the Fourier transform on the line, bandlimited functions and the PaleyWeiner theorem, ShannonWhittacker Sampling Theorem, Riesz systems, MallatMeyer multiresolution analysis in Lebesgue spaces, scaling functions, wavelet constructions, wavelet representation and unconditional bases, nonlinear approximation, Riesz's factorization lemma, and Daubechies' compactly supported wavelets.
 752  Complex Analysis. (3) (Prereq: MATH 703, 704) Normal families, meromorphic functions, Weierstrass product theorem, conformal maps and the Riemann mapping theorem, analytic continuation and Riemann surfaces, harmonic and subharmonic functions.
 752I  Complex Variables. (3) (Prereq: MATH 241 or equivalent) Properties of analytic functions, complex integration, calculus of residues, Taylor and Laurent series expansions, conformal mappings.
 754  Several Complex Variables. (3) (Prereq: MATH 703 and 704) Properties of holomorphic functions of several variables, holomorphic mappings, plurisubharmonic functions, domains of convergence of power series, domains of holomorphy and pseudoconvex domains, harmonic analysis in several variables.
 755  Applied Functional Analysis. (3) (Prereq: MATH 703) Banach spaces, Hilbert spaces, spectral theory of bounded linear operators, Fredholm alternatives, integral equations, fixed point theorems with applications, least square approximation.
 756, 757  Functional Analysis I, II. (3 each) (Prereq: MATH 704) Linear topological spaces; HahnBanach theorem; closed graph theorem; uniform boundedness principle; operator theory; spectral theory; topics from linear differential operators or Banach algebras.
 758  Selected Topics in Analysis. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
 760  Set Theory. (3) An axiomatic development of set theory: sets and classes; recursive definitions and inductive proofs; the axiom of choice and its consequences; ordinals; infinite cardinal arithmetic; combinatorial set theory.
 761  The Theory of Computable Functions. (3) Models of computation; recursive functions, random access machines, Turing machines, and Markov algorithms; Church's Thesis; universal machines and recursively unsolvable problems; recursively enumerable sets; the recursion theorem; the undecidability of elementary arithmetic.
 762  Model Theory. (3) First order predicate calculus; elementary theories; models, satisfaction, and truth; the completeness, compactness, and omitting types theorems; countable models of complete theories; elementary extensions; interpolation and definability; preservation theorems; ultraproducts.
 768  Selected Topics in Foundations of Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
 770  Discrete Optimization. (3) The application and analysis of algorithms for linear programming problems, including the simplex algorithm, algorithms and complexity, network flows, and shortest path algorithms. No computer programming experience required.
 774  Discrete Mathematics I. (3) An introduction to the theory and applications of discrete mathematics. Topics include enumeration techniques, combinatorial identities, matching theory, basic graph theory, and combinatorial designs.
 775  Discrete Mathematics II. (3) (Prereq: MATH 774 or consent of the instructor) A continuation of MATH 774. Additional topics will be selected from: the structure and extremal properties of partially ordered sets, matroids, combinatorical algorithms, matrices of zeros and ones, and coding theory.
 776  Graph Theory I. (3) The study of the structure and extremal properties of graphs, including Eulerian and Hamiltonian paths, connectivity, trees, Ramsey theory, graph coloring, and graph algorithms.
 777  Graph Theory II. (3) (Prereq: MATH 776 or consent of instructor) Continuation of MATH 776. Additional topics will be selected from: reconstruction problems, independence, genus, hypergraphs, perfect graphs, interval representations, and graphtheoretical models.
 778  Selected Topics in Discrete Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
 780  Elementary Number Theory. (3) Diophantine equations, distribution of primes, factoring algorithms, higher power reciprocity, Schnirelmann density, and sieve methods.
 780I  Theory of Numbers. (3) (Prereq: MATH 241 or equivalent) Elementary properties of integers, Diophantine equations, prime numbers, arithmetic functions, congruences, and the quadratic reciprocity law.
 782  Analytic Number Theory I. (3) (Prereq: MATH 580 and 552) The prime number theorem, Dirichlet's theorem, the Riemann zeta function, Dirichlet's Lfunctions, exponential sums, Dirichlet series, HardyLittlewood method partitions, and Waring's problem.
 783  Analytic Number Theory II. (3) (Prereq: MATH 580 and 552) The prime number theorem, Dirichlet's theorem, the Riemann zeta function, Dirichlet's Lfunctions, exponential sums, Dirichlet series, HardyLittlewood method partitions, and Waring's problem.
 784  Algebraic Number Theory. (3) (Prereq: MATH 546 and 580) Algebraic integers, unique factorization of ideals, the ideal class group, Dirichlet's unit theorem, application to Diophantine equations.
 785  Transcendental Number Theory. (3) (Prereq: MATH 580) ThueSiegelRoth theorem, Hilbert's seventh problem, diophantine approximation.
 788  Selected Topics in Number Theory. (3) Course content varies and will be announced in the schedule of classes by suffix and title.
 790  Graduate Seminar. (1) (Although this course is required of all candidates for the master's degree it is not included in the total credit hours in the master's program.)
 797  Mathematics into Print. (3) The exposition of advanced mathematics emphasizing the organization of proofs and the formulation of concepts; computer typesetting systems for producing mathematical theses, books, and articles.
 798  Directed Readings and Research. (16) (Prereq: full admission to graduate study in mathematics).
 799  Thesis Preparation. (19) For master's candidates.
 890  Graduate Seminar. (13) (Prereq: consent of instructor) A review of current literature in specified subject areas involving student presentations. Content varies and will be announced in the schedule of classes by suffix and title. Minimum of 3 credit hours required of all doctoral students. (PassFail grading)
 899  Dissertation Preparation. (112) For doctoral candidates.
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