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 faculty carries out high-level research on diverse problems in algebra, analysis, topology, probability and statistics, and mathematical physics.

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, the current president of Peru and former presidents of Ireland and France were mathematicians.) The breadth of mathematics courses offered at the University of Virginia allows each of its majors to tailor a program of study to his or her needs. Each student electing to major in mathematics should consult carefully with a faculty advisor to ensure the selection of a program of courses which will provide a solid grounding in the fundamentals of higher mathematics and is appropriate to his or her 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 90 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 (for example MATH 141, 142) 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 a student must complete at least eight courses approved by the department of mathematics with minimum grades of C in six of them and minimum grades of C- in the others. An approved course must carry at least three credits. Courses with the MATH mnemonic (Mathematics) numbered 225 or higher in the department of mathematics are approved. Courses with the STAT mnemonic (Statistics) are also approved, but no more than two such courses may be counted toward the major in mathematics. Courses from other departments or institutions may be offered if approved by the undergraduate committee.

Every major program must contain at least six credits from the algebra group and at least six credits from the analysis group, blocks 1 and 2 below. Within these blocks, the courses MATH 225 and 351 are required of all majors unless an appropriate (generally higher level) substitute is approved by the undergraduate committee.

Each major program should contain at least one closely related pair of one- semester courses. Allowable pairs are enclosed by parentheses at the end of each block.

Students enrolled in the five-year BA/MT teacher education program must satisfy the above requirements by completing the following: one algebra course beyond MATH 351; one analysis course beyond MATH 225 which, for students in this program only, may be MATH 300; MATH 311, 312 (probability and statistics); MATH 570 (geometry) or another approved geometry/topology substitute; and one additional 500-level course, such as MATH 503.

The mathematics program of a student is designed in consultation with an advisor. The advisor helps select and schedule courses giving a good background in mathematics. At the same time, the selection respects individual goals and abilities.

Course Blocks

  1. Algebra: (351, 354), (551, 552)
    (a) Linear algebra: 351
    (b) Abstract algebra: 354, 551, 552, 561
    (c) Applied algebra: 352, 355
    (d) Number theory: 353
  2. Analysis (225, 526), (521, 522), (330, 530), (531, 532)
    (a) Differential equations: 225, 515, 525, 526
    (b) Applied mathematics: 521, 522
    (c) Real and complex analysis: 332, 531, 532
    (d) Numerical Analysis: 330, 530
  3. Geometry and topology (570, 572)
    (a) Geometry: 570, 572
    (b) Topology: 577
  4. Discrete mathematics, algorithms, operations research: 304, 306, 321 (304, 306)
  5. Probability and statistics: (311, 312), (311, 511)
  6. History: 501, 503
  7. Foundations: 300
  8. Seminars, independent study: 493, 583
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).

As for the standard major program, at least eight courses are required, but the courses offered for the DMP must include MATH 531, 551, and 552. 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 MATH 583.

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 Scientific Computing Before declaring a minor in scientific computing a student must complete the calculus sequence through MATH 122 or MATH 132 or its equivalent with a 2.0 average, and show proficiency in FORTRAN or Pascal by successfully completing MATH 205 or 206, or their equivalents.

To graduate with a minor in scientific computing one must complete five courses 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 three core courses: MATH 306, 330, 351 or their equivalents. The other two courses must be chosen from MATH 321 (or SYS 321), MATH 352, 530, CS 455, 361. Except for MATH 351, courses which are being counted for the mathematics major or minor cannot also be counted for the minor in scientific computing. Students may minor both in scientific computing and in statistics.

Requirements for Minor in Statistics Before declaring a minor in statistics a student 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 one 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, 531, STAT 512, 513, 514, 516, 517, 518, 519. Students wishing to seek approval for courses not listed above may file a petition with the undergraduate committee. Courses will ordinarily be approved if they compare favorably in content and quality with the listed courses.

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

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 to choose from. 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.

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. A terminal one-year sequence intended for business and social science majors;
  2. MATH 131, 132, 221. 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;
  3. MATH 141, 142. Essentially an "honors" version of MATH 132, 221, intended primarily for well prepared entering students who place out of MATH 131 and anticipate a career in mathematics or the natural 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 either the MATH 131, 132, 221 sequence or the MATH 141, 142 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 who have had no previous calculus may elect MATH 121, MATH 131 (or its equivalent, MATH 131X). 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, 141, 142, 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).

The "honors calculus" sequence MATH 141, 142 is encouraged for those students who place out of MATH 131 and are contemplating a mathematics or physical science major. It is scheduled so as not to conflict with the corresponding upper echelon courses offered to incoming students in chemistry and physics (i.e., CHEM 161, 162, PHYS 151, 152).

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.

"Computer calculus", MATH 131X, 132X, a computer assisted version of MATH 131, 132 is also offered. It incorporates Mathematica, a computer system for doing mathematics. Similar computer aspects of mathematics are routinely incorporated into MATH 141, 142.

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.

The courses MATH 205 (FORTRAN) and MATH 206 (Pascal) develop computer programming skills for the purpose of solving mathematical problems. A strong mathematical interest and a previous course in calculus are needed for these courses.

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)
Foundations for Elementary Mathematics
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)
Introduction to 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 131X, 132X - (4) (Y)
Computer Calculus I
Prerequisite: Same as MATH 131, 132
A version of MATH 131, 132 incorporating computer problem solving with the Mathematica system for doing mathematics. For description of course contents, see MATH 131, 132.

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 141, 142 - (4) (SI)
Principles of Calculus I, II
Prerequisite: Success in a strong high school calculus course (as indicated, for example, by a score of 4 or 5 on the calculus (AB) Advanced Placement Examination) and permission of instructor
A course in multivariate calculus, vector spaces, and infinite series, incorporating differential forms, power series, and Fourier series, emphasizing classical problems in the natural sciences from which calculus has developed-for example, from Newtonian mechanics, thermodynamics, competing species, fluid flow, electricity and magnetism. An "honors sequence" intended for exceptionally well-prepared students, especially those planning to major in mathematics or the natural sciences. This sequence is offered as an alternative to MATH 132, 221. After passing MATH 142, students would normally take any of MATH 225, 300, or 351.

MATH 170, 171 - (1-3) (IR)
Liberal Arts Seminar
This course may not be used to satisfy the mathematics/science area requirement.

MATH 200 - (3) (IR)
Introduction to Computer Techniques in Mathematics (Basic)
Prerequisite or corequisite: A semester of college mathematics or its equivalent
Various problems drawn from first-year courses will be discussed, and students will be shown how to devise computer programs to obtain numerical solutions for these problems using the BASIC language.

MATH 205 - (3) (IR)
Computer Techniques in Mathematics (Fortran)
Prerequisite: MATH 121 or 131
Introduction to computer methods and numerical analysis. Computer programming in FORTRAN, with applications chosen from mathematics and statistics. Numerical solution of systems of equations, numerical integration, simulation, iteration. Evaluation of functions, including interpolation techniques. Elementary sorting and searching algorithms and their implementation as computer programs.

MATH 206 - (3) (IR)
Computer Techniques in Mathematics (Pascal)
Prerequisite: MATH 121 or 131
Introduction to computer methods and numerical analysis. Computer programming in PASCAL, with applications chosen from mathematics. Topics from numerical integration, sorting and searching, recursion, and binary trees.

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 With Linear Algebra
Prerequisite: MATH 132 or its equivalent
A study of differential equations of the first order, linear differential equations, their applications, and linear algebra with applications to systems of ordinary differential equations.

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

MATH 304 - (3) (Y)
Discrete Mathematics
Prerequisite: MATH 132 or its equivalent
Combinatorial principles, the binomial and multinomial theorems, partitions, discrete probability. Algebraic structures, trees, graphs, symmetry groups, Polya's enumeration formula. Linear recursions, generating functions.

MATH 306 - (3) (Y)
Algorithms
Prerequisite: MATH 132 and 206 or a knowledge of Pascal
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 321 - (3) (Y)
Operations Research
Prerequisite: MATH 132, 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 330 - (3) (S)
Elementary Numerical Analysis
Prerequisite: MATH 132 and proficiency in FORTRAN or PASCAL
Taylor's theorem, solution of nonlinear equations, interpolation and approximation by polynomials, numerical quadrature; further topics selected from numerical solutions of ordinary differential equations, Fourier series, least-square approximation.

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. Computations are stressed.

MATH 352 - (3) (Y)
Algebraic Coding Theory
Prerequisite: MATH 351
Basic introduction to algebraic structures such as finite fields, polynomial rings, linear algebra and groups by means of their application to coding theory.

MATH 353 - (3) (IR)
Number Theory
Prerequisite: Permission of instructor
Congruences, quadratic reciprocity, Diophantine equations, number-theoretic functions, and similar topics.

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. Emphasis is placed on the connection with standard secondary school algebra.

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 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)
The History of the Calculus
Prerequisite: MATH 221
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
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 515 - (3) (IR)
Calculus of Variations
Prerequisite: MATH 221
Single-integral minimum problems with fixed end points: necessary conditions and sufficient conditions; isoperimetric problems, variable endpoint problems, and multiple integral problems.

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)
Partial Differential Equations
Prerequisite: MATH 521 or equivalent
A theoretical introduction from a classical viewpoint. Harmonic and subharmonic functions. Wave and heat equations. Cauchy- Kowalewski and Holmgren theorems. Characteristics. Hamilton- Jacobi theory.

MATH 526 - (3) (IR)
Advanced Ordinary Differential Equations
Prerequisites: MATH 221, 225, 351
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 Poincare-Bendixon theorem; Lyapunov's method and stability. Other topics presented as time permits.

MATH 530 - (3) (Y)
Computer Methods in Numerical Analysis
Prerequisites: MATH 330, 351, and a knowledge of FORTRAN
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 and polynomial rings. Structure of linear transformations and matrices; eigenvectors and eigenvalues; diagonal forms for symmetric and hermitian matrices.

MATH 554 - (3) (Y)
Survey of Algebra
Prerequisite: Permission of instructor
This is the same as MATH 354 and is open only to students in the five year BA/MT teacher education program.

MATH 561 - (3) (IR)
Topics in Algebra
Prerequisite: Permission of instructor
Development in some depth of selected topics.

MATH 570 - (3) (Y)
Introduction to Geometry
Prerequisite: MATH 221 with knowledge of matrices recommended
Topics selected from analytic geometry, affine geometry, projective geometry, hyperbolic and non-Euclidean geometries.

MATH 572 - (3) (Y)
Introduction to Differential Geometry
Prerequisite: MATH 221
Differential geometry of curves and surfaces in Euclidean space.

MATH 577- (3) (Y)
Elementary Topology
Prerequisite: MATH 221
General topology, naive set theory, topological and metric spaces, connectedness and compactness.

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


Statistics

Courses with STAT numbers at the 500 level are offered by the Statistics Division of the Department of Mathematics. The aim of these courses is to supplement courses MATH 312 or 510 by providing students with an opportunity of studying Statistics as a branch of applied mathematics directed towards the description and analysis of situations where uncertainty and variability play a dominant role. The requirements of a minor in Statistics are listed earlier in this section.

STAT 512 - (3) (Y)
Applied Linear Models
Prerequisite: MATH 312 or 510 or permission of instructor
Linear regression models, inferences in regression analysis, model validation, selection of independent variables, multicollinearity, influential observations, auto correlation in time series data, polynomial regression, nonlinear regression, and other topics in regression analysis.

STAT 513 - (3) (O)
Applied Multivariate Statistics
Prerequisites: MATH 351 and MATH 312 or 510 or permission of instructor
Matrix algebra, random sampling, multivariate normal distributions, multivariate regression, MANOVA, principal components, factor analysis, discriminant analysis. Statistical software will be used.

STAT 514 - (3) (SI)
Statistical Modeling
Prerequisite: MATH 312 or 510 or permission of instructor
Introduces new and modern information-theoretic model selection procedures in developing and selecting statistical models among a class of competing alternative models for a given finite data set. Topics include basic concepts and principles of statistical modeling, the likelihood and the method of maximum likelihood, information-based model selection criteria, choosing the optimal functional form of a model, regression and polynomial models, model choice and subset selection in generalized linear and related regression models, selecting log-linear models, econometric, and time series models.

STAT 516 - (3) (E)
Experimental Design
Prerequisite: MATH 312 or 510 or permission of instructor
Introduction to the basic concepts in experimental design, analysis of variance, multiple comparison tests, completely randomized design, general linear model approach to ANOVA randomized block designs, latin square and related designs, completely randomized factorial design with two or more treatments, hierarchical designs split-plot and confounded factorial designs, and analysis of covariance.

STAT 517 - (3) (E)
Applied Time Series
Prerequisite: MATH 312 or 510 or permission of instructor
Development of basic time series models in both the time domain (ARMA models) and the frequency domain (spectral models). Emphasis is on application to real data sets.

STAT 518 - (3) (Y)
Numerical Methods in Statistics
Prerequisite: MATH 351 and knowledge of a programming language suitable for scientific computation, or permission of instructor
Selected topics in linear algebra and related numerical algorithms of special importance in statistics: linear least squares, eigenvalues and eigenvectors, QR decomposition, singular value decomposition, generalized inverses.

STAT 519 - (3) (Y)
Introduction to Mathematical Statistics
Prerequisite: MATH 312 or 510 or permission of instructor
Covers the fundamentals of statistical distribution theory, moments, transformations of random variables, point estimation, hypothesis testing, confidence regions.

STAT 599 - (3) (IR)
Special Topics in Statistics
Prerequisite: Permission of instructor
A study of topics in statistics that are not part of the regular course offerings.