Department of Mathematics

Overview   In a world of increasing technological complexity, knowledge of mathematics is the gateway to the pursuit of many fields. Mathematics has long been the language of choice for expressing complex relationships and describing complicated patterns and processes. It is now true that many fields, in addition to mathematics and the sciences, rely on this in a fundamental way.

What was formerly "abstract" mathematics to many has become the concrete stuff of everyday life. "The unreasonable effectiveness of mathematics" manifests itself today in such familiar things as CAT scans, compact discs, satellite communications, and computer animation. These were all rendered possible by new discoveries made by mathematicians, within the last fifty years. Even the efficient operation of our financial markets is based, in part, on relatively recent theorems of mathematical analysis and probability theory.

Mathematics research today is a vibrant and dynamic enterprise. Thousands of mathematicians world-wide are at work on an unimaginably broad range of questions. Exciting recent advances include the solution of Fermat's Last Theorem, the classification of the finite simple groups, the solution of the Bieberbach conjecture, and the (controversial) proof of the four-color theorem.

The discipline and creativity required by the study of mathematics can be a formidable preparation for later life. Past students of mathematics have had successful careers in almost every sphere, including all the professions. (In fact, presidents of Peru, Ireland and France have been mathematicians.) The scope of mathematics courses offered at the University of Virginia allows each of its majors to tailor a program of study to meet their needs. Students electing to major in mathematics should consult carefully with a faculty advisor to ensure the selection of a program of courses which provides a solid grounding in the fundamentals of higher mathematics and is appropriate to future goals.

Faculty   The faculty of the Department of Mathematics is committed to excellence in teaching and research. Its members have been widely published in prestigious research jour-nals and are internationally recognized scholars. The faculty have held Sloan fellowships, Humboldt fellowships, and other scholarly honors, as well as, numerous research grants. Many are currently supported by grants from the National Science Foundation and other federal agencies. Most have held visiting professorships abroad. In addition, the department offerings and ambiance are enhanced each year by the presence of several internationally recognized visiting faculty.

Students   There are currently about 75 students majoring in mathematics. Class sizes vary from a few large introductory classes to an average class size of twenty students for upper level courses. This small class size affords students the opportunity to get individual attention.

Students who graduate with degrees in mathematics successfully pursue a variety of different careers. Many go directly into jobs in industry, insurance (as actuaries), government, finance, and other fields. Employers in the past have included Morgan Stanley, General Motors, MITRE Corp., the Census Bureau, the National Security Agency, and various consulting firms. Many find themselves well-equipped to go on to professional schools in law, medicine, and business. Some go directly into teaching. Others have gone on to graduate programs in mathematics, applied mathematics, statistics, engineering, systems engineering, economics, and computer science. Students who have combined the mathematics major with courses in computer programming, economics, and business have done exceptionally well in the job market.

Requirements for Major   Normally, the calculus sequence MATH 131, 132, and 221 or its equivalent must be completed before a student can declare a major in mathematics. At least a 2.2 average in the calculus sequence and a minimum grade of C in MATH 221 or its equivalent are required. However, the department may grant special permission to declare a major to a student who has only completed MATH 131 and 132, and at least one mathematics course (other than MATH 221 or its equivalent) which could be counted towards the major in mathematics, provided the student completes MATH 221 or its equivalent in the semester following the declaration of a mathematics major.

To graduate with a major in mathematics the student must show computer proficiency by completing CS120, CS101, or CS182, or an approved equivalent course. This should be done as early as possible.

To help guide the student through the major, the mathematics department offers five options. Completion of one of these options is required. Each option contains a set of nine required courses (approximately 28 credit hours). To graduate, a student must obtain minimum grades of C in seven of these courses and C- in the other two.

Certain substitutions are allowed in all options, for example, MATH 531 for MATH 331 and MATH 551 for MATH 354.

A. The basic option

This is a traditional program for the mathematics major which provides an overview of key areas of mathematics:

MATH 225 Ordinary Differential Equations
MATH 351 Elementary Linear Algebra
MATH 354 Survey of Algebra
Two from the following three:
MATH 311 Introduction to Mathematical Probability
MATH 331 Basic Real Analysis
MATH 332 Complex Variables with Applications
Four electives at the 300 level or higher

Students fulfilling the requirements for this option will have a wide range of career opportunities, from law to business to any field which requires deductive, logical reasoning skills.

B. The graduate preparatory option

This option is for the student who plans to attend graduate school in mathematics or an allied field. The program emphasizes the fundamental ideas of mathematics with substantial work in proving and understanding the basic theorems. It consists of:

MATH 225 Ordinary Differential Equations
MATH 332 Complex Variables with Applications
MATH 351 Elementary Linear Algebra
MATH 531 Introduction to Real Analysis I
MATH 551 Introduction to Abstract Algebra I
MATH 552 Introduction to Abstract Algebra II

Three electives at the 300 level or higher (Students may wish to take MATH 354 and/or MATH 331 in preparation for MATH 551 and/or MATH 531.)

This constitutes a minimum expected of an incoming graduate student in most programs nationwide. We strongly recommend MATH 532 (Real Analysis in Several Variables), as well as courses in differential geometry and topology (MATH 572 and MATH 577). Many of our graduate school bound students take additional courses including 700-level graduate courses.

C. The probability and statistics option

This option is designed to give the student a good theoretical underpinning in probability and statistics, as well as the opportunity to go deeper in these fields. The program can lead to a Master of Science in Statistics with one additional year of course work, if additional courses in Statistics are taken in the fourth year. (Those interested in the M.S. in Statistics should contact the graduate advisor in the Statistics Division prior to the beginning of their fourth year.) The requirements for the option are the following:

MATH 225 Ordinary Differential Equations
MATH 311 Introduction to Mathematical Probability
MATH 312 Introduction to Mathematical Statistics
Either MATH 331 Basic Real Analysis
or MATH 332 Complex Variables with Applications
MATH 351 Elementary Linear Algebra
MATH 511 Stochastic Processes
STAT 512 Applied Linear Models
One additional course chosen from MATH 430, MATH 531, STAT 313, STAT 316, STAT 513, STAT 516, STAT 517, or STAT 519

D. The financial mathematics option

This program provides the student with a broad background of basic mathematics which is essential for an understanding of the mathematical models used in the financial markets. The mathematics of modern finance includes, but is not limited to, probability, statistics, regression, time series, partial differential equations, stochastic processes, stochastic calculus, numerical methods, and analysis. Probability and statistics and some acquaintance with numerical methods are essential as is some knowledge of economics/accounting and some computing experience. Additional background in statistics, optimization, and stochastic processes is also desirable. The program consists of:

MATH 225 Ordinary Differential Equations
MATH 311 Introduction to Mathematical Probability
MATH 312 Introduction to Mathematical Statistics
Either MATH 331 Basic Real Analysis
or MATH 332 Complex Variables with Applications
MATH 351 Elementary Linear Algebra
MATH 354 Survey of Algebra
MATH 414 Mathematics of Derivative Securities
MATH 430 Elementary Numerical Analysis
One additional course chosen from MATH 408, MATH 511, STAT 512, or STAT 517

Other requirements: Two courses chosen from ECON 201, ECON 202, COMM 201, or COMM 202. It is recommended, however, that the student complete all four of these courses.

E. Five-year teacher education program

This option leads to both Bachelor of Arts and Master of Teaching degrees after five years. The program is for both elementary and secondary teachers; it is administered by the Curry School of Education. The requirements are:

MATH 225 Ordinary Differential Equations
MATH 311 Introduction to Mathematical Probability
MATH 312 Introduction to Mathematical Statistics
Either MATH 331 Basic Real Analysis or
MATH 332 Complex Variables with Applications
MATH 351 Elementary Linear Algebra
MATH 354 Survey of Algebra
Either MATH 501 History of Calculus or
MATH 503 History of Mathematics
MATH 570 Introduction to Geometry
One elective at the 300 level or higher

The Education School has additional requirements for this program.

Distinguished Majors Program   The department offers a Distinguished Majors Program to qualified majors in mathematics. Admission to the program is granted by the departmental committee for the DMP, usually at the end of the student's fourth semester. Criteria for acceptance into the program are based on the GPA in mathematics, letters of recommendation from mathematics instructors, and the cumulative GPA in the College (which should be near 3.4 or higher).

The distinguished majors program is the same as the Graduate School preparatory option, except that in the fourth year the student will also take the seminar course MATH 583 in which the student will give an hour lecture and prepare a written exposition of his or her work in the seminar, under faculty guidance. Note that MATH 531 and 551 are prerequisites for the seminar). As with the options, the DMP must consist of at least nine courses.

Three levels of distinction are possible: distinction, high distinction, or highest distinction. The departmental recommendation for the level of distinction to be awarded is based on the quality of the student's seminar presentations, the overall work in the DMP and the entire major program of the student, as well as the student's College GPA.

Requirements for Minor in Mathematics   Students not majoring in mathematics but who wish to declare a minor in mathematics must complete the calculus sequence through MATH 221 or its equivalent with at least a 2.0 average.

To graduate with a minor in mathematics a student must complete five courses approved by the department of mathematics with minimum grades of C in three of these courses and minimum grades of C- in the other two. An approved course must carry at least three credits. Currently, the approved courses are those from the College department of mathematics with the MATH mnemonic numbered 225 or higher. Courses with the STAT mnemonic or from other departments or institutions can be offered if approved by the undergraduate committee.

Courses which are being counted for a major or another minor cannot also be counted for the minor in mathematics.

Requirements for Minor in Statistics   Before declaring a minor in statistics, students must complete the calculus sequence through MATH 221 or its equivalent with at least a 2.0 average. To graduate with a minor in statistics, students must complete five courses approved by the Department of Mathematics with minimum grades of C in three of these courses and minimum grades of C- in the other two.

The program must include the following two core courses: MATH 311-312 or their equivalents. The other three courses must be chosen from MATH 511, MATH 531, STAT 512, STAT 513, STAT 514, STAT 516, STAT 517, STAT 518, or STAT 519. Students wishing approval for courses not listed above may file a petition with the undergraduate committee. Courses are ordinarily approved if they compare favorably in content and quality with the listed courses.

Except for MATH 531, courses which are being counted for the mathematics major or minor may not also be counted for the minor in statistics.

Echols Mathematics Club   is an undergraduate club for mathematics students which sponsors lectures, mathematics films, problem solving sessions for the Putnam Mathematical Competition and other similar activities.

Additional Information   For more information, contact:

Charles Dunkl, Lower Division Advisor, Room 223, (804) 924-4939
or Ira Herbst, Upper Division Advisor, Room 211, (804) 924-4933
Kerchof Hall
Charlottesville, VA 22903-3199
Mathematics World Wide Web site
Mathematics faculty


Courses

Mathematics

The entering College student has a variety of courses in mathematics from which to choose. Among those which may be counted towards the College area requirement in natural science and/or mathematics, there are several options in calculus, elementary (non-calculus based) courses in probability and in statistics, and courses dealing with computer techniques in mathematics.

The courses MATH 100 (algebra and trigonometry) and MATH 103 (precalculus) are available for students who need to improve basic skills that are required in other courses such as calculus, chemistry, psychology, economics, and statistics. However, neither MATH 100 nor MATH 103 may be counted toward the area requirement in natural science and/or mathematics. Courses equivalent to MATH 100 may not be transferred for College credit.

Students planning to major in the social sciences, arts, or humanities who wish to take a mathematics course but omit the study of calculus may choose from elementary probability theory, MATH 111, or elementary statistics, MATH 112. Even though it is not a prerequisite for MATH 112, MATH 111 is frequently taken prior to MATH 112. Note that both MATH 111 and MATH 112 may be counted toward the area requirement in natural science and/or mathematics.

The study of calculus is the foundation of college mathematics for students planning to major in mathematics or the physical sciences or anticipating a career or graduate study in any of the natural sciences, engineering, or applied social sciences (such as economics). There are essentially three programs of study available in the calculus:

1. MATH 121, 122 is a terminal one-year sequence intended for business and social science majors;

2. MATH 131, 132, 221 is the traditional calculus sequence intended for students of mathematics and the natural sciences, as well as for students intending to pursue graduate work in the applied social sciences;

The MATH 121, 122 sequence is unacceptable as a prerequisite for mathematics coursed numbered 221 and above. Students anticipating the need for higher mathematics courses such as MATH 225 (Differential Equations) or MATH 311, 312 (Probability and Statistics) should instead elect the MATH 131, 132, 221 sequence. Note that credit is not allowed for both MATH 121 and MATH 131 (or its equivalent). Students who begin their calculus with MATH 121 but wish to transfer into the traditional calculus sequence may follow MATH 121 with MATH 132A, which is equivalent to MATH 132.

Students with no previous calculus may elect MATH 121, MATH 131. An alternative to MATH 121 is MATH 121S, which places greater emphasis on problem solving and giving more individual attention to the student.

Students who have previously passed a calculus course in high school may elect MATH 122, 131, 132, or 221 as their first course, depending upon placement, preparation, and interest. A strong high school calculus course is generally adequate preparation for MATH 132 as a first calculus course, even if advanced placement credit has not been awarded for MATH 131. Students planning to take any advanced course in mathematics should not take MATH 122, because credit in MATH 122 must be forfeited if one takes MATH 132 (or its equivalent).

Exceptionally well prepared students (who place out of both MATH 131 and 132) may choose either MATH 221 or MATH 225 (Differential Equations) as their first course.

Advanced placement credit in the calculus sequence is granted on the basis of the College Entrance Examination Board Advanced Placement Test (either AB or BC). A score of 4 or 5 on the AB test or a score of 3 on the BC test will give the student credit for MATH 131. A score of 4 or 5 on the BC test will give the student credit for both MATH 131 and 132. Students who wish to enter the calculus sequence but who have not received advanced placement credit should consult the First-Year Handbook for placement guidelines based on grades and achievement test scores. The Department of Mathematics offers short advisory placement tests during fall orientation.

Pre-commerce students are required to take a statistics course, usually MATH 112, and one other mathematics course, usually MATH 111, 121, 122, or MATH 131.

Warning: There are numerous instances of equivalent courses offered by the Department of Mathematics as well as by the Department of Applied Mathematics in the School of Engineering. A student may not offer for degree credit two equivalent courses, e.g., MATH 131 and APMA 101, or MATH 131 and MATH 121.

MATH 100 - (3) (Y)
Algebra and Trigonometry
Computational and algebraic skills, patterns of quantitative problem solving and mathematical thought. Computations, linear and quadratic equations, functions and graphs, trigonometry of triangles. Prerequisite to MATH 103. (Credit/No Credit; chargeable against allowable non-College credits.)

MATH 103 - (4) (S)
Precalculus
Computational skills, patterns of quantitative problem solving and mathematical thought. Linear and quadratic equations, polynomials, inverse functions, logarithms, arithmetic and geometric sequences, trigonometric functions, and linear systems. Prerequisite to MATH 121. (Does not satisfy College science requirement.)

MATH 108 - (3) (Y)
Modes of Mathematical Thinking
Logic, number systems, functions, analytic geometry, equations, matrices, enumeration, computer algebra systems. Intended for liberal arts students and emphasizes the connection between analytic-algebraic and geometric reasoning in the understanding of mathematics. Facilitated by the use of a modern computer algebra system, such as Maple.

MATH 110 - (3) (Y)
Mathematics for Elementary Teachers
Prerequisite or corequisite: EDHS 201
A study of numbers, operations, and their properties; geometric figures and their properties; and introductory probability and statistics.

MATH 111 - (3) (S)
Probability/Finite Mathematics
Finite probability theory including combinatorics, equiprobable models, conditional probability and Bayes' theorem, expectation and variance, Markow chains.

MATH 112 - (3) (S)
Introduction to Statistics
Probability distributions, tests of hypotheses, chi-square tests, sampling, regression and correlation.

MATH 121 - (4) (S)
Applied Calculus I
Limits and continuity. Differentiation and integration of algebraic and elementary transcendental functions. Applications to maximum-minimum problems, curve sketching and exponential growth. Credit is not given for both MATH 121 and MATH 131.

MATH 121S - (4) (Y)
Introduction to Calculus
Prerequisite: Permission of instructor
Limits and continuity. Differentiation and integration of algebraic and elementary transcendental functions. Applications to maximum-minimum problems, curve sketching and exponential growth.

MATH 122 - (3) (Y)
Applied Calculus II
Prerequisite: MATH 121 or equivalent
A second calculus course for business, biology, and social science students. Functions of several variables, their graphs, partial derivatives and optimization; multiple integrals. Includes a review of basic single variable calculus (MATH 121 or equivalent) and an introduction to differential equations and infinite series. Credit is not given for both MATH 122 and MATH 132.

MATH 131 - (4) (S)
Calculus I
Prerequisite: Background in algebra, trigonometry, exponentials, logarithms, and analytic geometry
Introductory calculus with emphasis on techniques and applications. Recommended for natural science majors and students planning additional work in mathematics. The differential and integral calculus for functions of a single variable is developed through the fundamental theorem of calculus. Credit is not given for both MATH 121 and MATH 131.

MATH 132 - (4) (S)
Calculus II
Prerequisite: MATH 131 or permission of instructor
Continuation of 131. Applications of the integral, techniques of integration, infinite series, vectors. Credit is not given for both MATH 122 and MATH 132.

MATH 132A - (5) (Y)
Calculus II
Prerequisite: MATH 121 or permission of instructor
Continuation of MATH 121 for students who wish to cover the material of MATH 132.

MATH 221 - (4) (S)
Calculus III
Prerequisite: MATH 132 or its equivalent
A study of functions of several variables including lines and planes in space, differentiation of functions of several variables, maxima and minima, multiple integration, line integrals, and volume.

MATH 225 - (4) (S)
Ordinary Differential Equations
Prerequisite: MATH 132 or its equivalent
Introduction to the methods, theory and applications of differential equations. Topics: first-order, second and higher-order linear equations, series solutions, linear systems of first-order differential equations and the associated matrix theory. Additional topics may include numerical methods, non-linear systems, boundary value problems, and additional applications. Section 1 of MATH 225 in the spring covers, in addition to the basic topics, an introduction to Sturm-Liouville theory, Fourier series and boundary value problems and their connection with partial differential equations. Physics majors should enroll in section one. No knowledge of physics is assumed in MATH 225, section 1.

MATH 300 - (3) (Y)
Foundations of Analysis
Prerequisite: MATH 132 or equivalent
Selection of topics from logic and the construction of mathematical proofs, basic set theory, number systems, continuity of functions and foundations of analysis. Introduces at an intermediate level the standards of mathematical rigor and abstraction that are encountered in advanced mathematics, based on the material of the calculus and other basic mathematics.

MATH 306 - (3) (Y)
Algorithms
Prerequisite: MATH 132 and computer proficiency
The study of abstract algorithms to solve mathematical problems and their implementation in a high-level language. Topics from sorting problems, recursive algorithms, and dynamic data structures.

MATH 311 - (3) (Y)
Introduction to Mathematical Probability
Prerequisite: MATH 132. A knowledge of double integrals is recommended
Sample spaces, combinatorial analysis, discrete and continuous random variables, classical distributions, expectation, Chebyshev theorem, independence, central limit theorem, conditional probability, generating functions.

MATH 312 - (3) (Y)
Introduction to Mathematical Statistics
Prerequisite: MATH 311
Sampling theory, point estimation, interval estimation, testing hypothesis (including the Neyman-Pearson lemma and likelihood ratio tests), regression and correlation.

MATH 331 - (3) (S)
Basic Real Analysis
Prerequisite: MATH 132
Concentrates on proving the basic theorems of calculus, with due attention to the beginner with little or no experience in the techniques of proof. Topics include limits, continuity, differentiability, the Bolzano-Weierstrass theorem, Taylor's theorem, integrability of continuous functions, and uniform convergence.

MATH 332 - (3) (Y)
Complex Variables With Applications
Prerequisite: MATH 221
Analytic functions, Cauchy formulas, power series, residue theorem, conformal mapping, Laplace transforms.

MATH 351 - (3) (S)
Elementary Linear Algebra
Prerequisite: MATH 132
Topics include matrices, elementary row operations, inverses, vector spaces and bases, inner products and Gram-Schmidt orthogonalization, orthogonal matrices, linear transformations and change of basis, eigenvalues, eigenvectors, and symmetric matrices.

MATH 354 - (3) (Y)
Survey of Algebra
Prerequisite: MATH 132 or equivalent
An introductory survey of the major topics of modern algebra: groups, rings, and fields. Applications to other areas of mathematics, such as geometry and number theory are presented. The rational, real, and complex number systems are developed, and the algebra of polynomials explored.

MATH 355 - (3) (IR)
Algebraic Automata Theory
Prerequisite: MATH 351
An introduction to the theory of sequential machines, including an introduction to the theory of finite permutation groups and transformation semigroups. Examples from biological and electronic systems as well as computer science. The Krohn-Rhodes decomposition of a state machine. Mealy machines.

MATH 404 - (3) (Y)
Discrete Mathematics
Prerequisite: MATH 354 or permission of instructor
Topics include combinatorial principles, the binomial and multinomial theorems, partitions, discrete probability, algebraic structures, trees, graphs, symmetry groups, Polya's enumeration formula, linear recursions, and generating functions.

MATH 408 - (3) (Y)
Operations Research
Prerequisite: MATH 132 and MATH 351
Development of mathematical models and their solutions, including linear programming, the simplex algorithm, dual programming, parametric programming, integer programming, transportation models, assignment models, and network analysis.

MATH 414/514 - (3) (Y)
Mathematics of Derivative Securities
Prerequisite: MATH 221 or MATH 122 or its equivalent, and a knowledge of probability and statistics. MATH 311 or its equivalent is recommended.
Topics include arbitrage arguments, valuation of futures, forwards and swaps, hedging, option-pricing theory, and sensitivity analysis.

MATH 430 - (3) (S)
Elementary Numerical Analysis
Prerequisite: MATH 225 and computer proficiency
Topics include Taylor's theorem, solution of nonlinear equations, interpolation and approximation by polynomials, numerical quadrature. Further topics may be selected from numerical solutions of ordinary differential equations, Fourier series, and least-square approximation.

MATH 452 - (3) (IR)
Algebraic Coding Theory
Prerequisite: MATH 351 and MATH 354 or permission of instructor
An introduction to the use of algebraic techniques for communicating information in the presence of noise. Topics include the basic concepts of coding theory: linear codes, bounds for codes, BCH codes and their decoding algorithms. Other topics may include quadratic residue codes, Reed-Muller codes, algebraic geometry codes, and connections with groups, designs, and lattices.

MATH 453 - (3) (IR)
Number Theory
Prerequisite: MATH 354 or permission of instructor
Topics include congruences, quadratic reciprocity, Diophantine equations, and number-theoretic functions, among others.

MATH 493 - (3) (IR)
Independent Study
Reading and study programs in areas of interest to the individual student. This course is primarily for juniors and seniors who have developed an interest in a branch of mathematics not covered in a regular course. It is the responsibility of the student to obtain a faculty advisor to approve and direct the program.

MATH 501 - (3) (Y)
Prerequisite: MATH 221 and MATH 351 or permission of instructor
Evolution of the various mathematical ideas leading up to the development of the calculus in the seventeenth century, and how those ideas were perfected and extended by succeeding generations of mathematicians. Special emphasis placed, wherever possible, on primary source materials.

MATH 503 - (3) (Y)
The History of Mathematics
Prerequisite: MATH 221 and MATH 351 or permission of instructor
The development of mathematics from classical antiquity through the end of the nineteenth century, focusing on the critical periods in the evolution of such areas as geometry, number theory, algebra, probability and set theory. Special emphasis placed, wherever possible, on primary source materials.

MATH 509 - (3) (Y)
Mathematical Probability
Prerequisite: Three semesters of calculus, and graduate standing. Students who have received credit for MATH 311 may not take MATH 509 for credit.
The development and analysis of probability models through the basic concepts of sample spaces, random variables, probability distributions, expectations, and conditional probability. Additional topics covered include distributions of transformed variables, moment generating functions, and the central limit theorem.

MATH 510 - (3) (Y)
Mathematical Statistics
Prerequisite: MATH 509 and graduate standing. Students who have received credit for MATH 312 may not take MATH 510 for credit.
Methods of estimation, general concepts of hypothesis testing, linear models and estimation by least squares, categorical data, nonparametric statistics.

MATH 511 - (3) (Y)
Stochastic Processes
Prerequisite: MATH 311 or permission of instructor
Topics in probability theory selected from: Random walks, Markov processes, Brownian motion, Poisson processes, branching processes, stationary time series, linear filtering and prediction, queuing process, and renewal theory.

MATH 521 - (3) (Y)
Advanced Calculus with Applied Mathematics
Prerequisite: MATH 221, 225
Vector analysis, Green's, Stokes', divergence theorems, conservation of energy, potential energy functions. Emphasis on physical interpretation. Sturm-Liouville problems, Fourier series, special functions, orthogonal polynomials, Green's functions.

MATH 522 - (3) (Y)
Partial Differential Equations and Applied Mathematics
Prerequisites: MATH 521 (MATH 351 recommended)
Introduction to complex variables and partial differential equations. Analytic functions, complex integration, power series, residues, conformal mapping; separation of variables, boundary value problems, Laplace's equation, wave equation, heat equation.

MATH 525 - (3) (IR)
Advanced Ordinary Differential Equations
Prerequisites: MATH 221, 225, 351 or permission of instructor
Emphasis on the qualitative geometrical theory of ordinary differential equations. Topics include all or most of the following: Picard's method and basic existence and uniqueness theorems; linear systems; the phase plane and Sturm's theorems; the Poincaré-Bendixon theorem; Lyapunov's method and stability. Other topics presented as time permits.

MATH 526 - (3) (IR)
Partial Differential Equations
Prerequisite: MATH 221 225 and 351or permission of instructor
A theoretical introduction from a classical viewpoint. Topics include: Harmonic and subharmonic functions, wave and heat equations, Cauchy-Kowalewski and Holmgren theorems, characteristics and Hamilton- Jacobi theory.

MATH 530 - (3) (IR)
Computer Methods in Numerical Analysis
Prerequisites: MATH 351, 430 and computer proficiency
A study of the underlying mathematical principles, and the use of sophisticated software for numerical problems such as spline interpolation, ordinary differential equations, nonlinear equations, optimization, and singular-value decomposition of a matrix.

MATH 531, 532 - (3) (Y)
Introduction to Real Analysis I, II
Prerequisite: MATH 221, 351
Basic topology of Euclidean spaces, continuity and differentiation of functions on Euclidean spaces. Riemann-Stieltjes integration, convergence of sequences and series of functions. Equicontinuous families of functions, Weierstrass theorem, inverse function theorem and implicit function theorem, integration of differential forms and Stokes' Theorem.

MATH 551, 552 - (3) (Y)
Introduction to Abstract Algebra I, II
Prerequisite: MATH 351 or permission of instructor
Introduction to algebraic systems: groups, rings, fields, vector spaces and their general properties; subsystems, quotient systems, homomorphisms. Basic examples, such as permutation groups, polynomial rings, and groups and rings of matrices. Additional topics may include applications to linear algebra and number theory.

MATH 570 - (3) (Y)
Introduction to Geometry
Prerequisite: MATH 221 and 351 or permission of instructor
Topics selected from analytic geometry, affine geometry, projective geometry, hyperbolic and non-Euclidean geometry.

MATH 572 - (3) (Y)
Introduction to Differential Geometry
Prerequisite: MATH 221
Topics selected by the instructor from the theory of curves and surfaces in Euclidean space and the theory of manifolds.

MATH 577- (3) (Y)
General Topology
Prerequisite: MATH 221; corequisite: MATH 551 or equivalent
Topological spaces and continuous functions; product and quotient topologies; compactness and connectedness; separation and metrication; the fundamental group and covering spaces.

MATH 583 - (3) (IR)
Seminar
Prerequisite: Permission of instructor
Presentation of selected topics in mathematics.

MATH 596 - (3) (S)
Supervised Study in Mathematics
Prerequisite: Permission of instructor and graduate standing
In exceptional circumstances, a student may undertake a rigorous program of supervised study designed to expose the student to a particular area of mathematics. Regular homework assignments and scheduled examinations are required.

Courses in Statistics   Courses at the 300-500 levels offered by the Division of Statistics may be found later in this chapter.