Department of Mathematics

P.O. Box 400137
University of Virginia
Charlottesville, VA 22904-4137
Phone: (434) 924-4919
Fax: (434) 982-3084
www.virginia.edu/~woodson

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 and MRI 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 worldwide are at work on an unimaginably broad range of questions. Exciting recent advances include the proof of Fermat's Last Theorem, the classification of the finite simple groups, the proof of the Bieberbach conjecture, and the computer-assisted 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. The scope of mathematics courses offered at the University of Virginia allows majors to tailor their own programs. Students electing to major in mathematics should consult carefully with a faculty advisor to ensure the selection of a program of courses that 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 carry out high-level research on diverse problems in algebra, analysis, topology, probability, and statistics, mathematical physics, and the history of the discipline. Their research has been widely published in prestigious research journals and is recognized internationally. Members of the department have won 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 231 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 231 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 231 or its equivalent) which could be counted toward the major in mathematics, provided the student completes MATH 231 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 CS 101 or 120, 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 six concentrations. Completion of one of these concentrations is required. Each concentration contains a set of nine or ten 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 Concentration
This traditional program for the mathematics major provides an overview of key areas:

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

B. The Graduate Preparatory Concentration
This concentration 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:

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

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

C. The Probability and Statistics Concentration This concentration 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 Department of Statistics prior to the beginning of their fourth year.) The requirements for the concentration are the following:

D. The Financial Mathematics Concentration 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:

⁽¹⁾ Completing all four courses is recommended.

E. Actuarial Concentration This concentration offers some of the basic mathematics and statistics necessary for a successful career in actuarial science, and it provides some of the academic background needed to pass the first few actuarial exams.

Actuaries use mathematics, statistics, and financial theory to analyze future events, especially those related to insurance and pension programs. They may work for insurance companies, consulting firms, government, employee benefits departments of large organizations, banks, investment firms, or more generally, businesses that need to assess the financial consequences of risk.

To become an actuary, the student must pass a series of examinations administered by the professional actuarial societies: the Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS). The first few exams are jointly administered. Exams which correspond to various courses are indicated below.

The program consists of nine courses as follows:

(1) Part of exam 100, offered jointly by the SOA and CAS.
(2) Exam 110, offered jointly by the SOA and CAS.
(3) Required SOA exams 140, 150/CAS exam 4A.
(4) Required SOA exam 151/CAS exam 5A.
(5) Elective SOA exam 130.
(6) Elective SOA exam 135/required CAS exam 3C.
(7) One-third of required SOA exam 120/CAS exam 3A.
(8) One-half of required SOA exam 160.
(9) Two-thirds of required SOA exam 120/ CAS exam 3A.

It is highly advantageous for students interested in this concentration to take both MATH 310 and 312 in their second year. Actuarial Mathematics (MATH 517/STAT 540) and Actuarial Risk Theory (STAT 541) form the core of the actuarial program. Both of these courses are offered every year if there is sufficient student interest, and otherwise in alternate years. With sufficient early course preparation, a summer internship after the third year has been an integral part of the program for those students who wished to intern.

Other courses which are recommended but not required include:

(9) Required SOA exam 230/required CAS exam 5A.
(10) SOA exam 160.

F. 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 and is administered by the Curry School of Education. Required courses include:

The Curry School has additional requirements for this program.

Distinguished Majors Program in Mathematics  The department offers a Distinguished Majors Program (DMP) 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 DMP is the same as the graduate school preparatory concentration, except that in the fourth year the students also take the seminar course MATH 583 in which they give an hour lecture and prepare a written exposition of their work in the seminar under faculty guidance. Note that MATH 531 and 551 are prerequisites for the seminar. As with the concentrations, 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, as well as the student's College GPA.

Requirements for Minor in Mathematics  Students who wish to declare a minor in mathematics must complete the calculus sequence through MATH 231 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 the 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 300 or higher. Courses with the STAT mnemonic or from other departments or institutions can be taken if approved by the undergraduate committee.

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

Echols Mathematics Club is an undergraduate club for mathematics students that 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, 924-4939, or Thomas Kriete, Upper Division Advisor, Room 205, 924-4932, Kerchof Hall, Charlottesville, VA 22904-4137; www.math.virginia.edu.

Mathematics

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

MATH 103 (precalculus) is available for students who need to improve basic skills that are required in other courses such as calculus, chemistry, psychology, economics, and statistics. However, it may not be counted toward the area requirement in natural science and 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 MATH 108 (Modes of Mathematical Thinking) or MATH 111 (Elementary Probability Theory). Even though it is not a prerequisite for STAT 112, MATH 111 is frequently taken prior to STAT 112. MATH 115 and 116 are introductory courses that investigate familiar areas of elementary mathematics at a profound level and are intended for first- and second-year non-majors, especially those preparing to teach in elementary and middle schools.

In MATH 114, the students study the mathematics needed to understand and answer a variety of questions that arise in everyday financial dealings. The emphasis in this course will be on applications, including simple and compound interest, valuation of bonds, rates of return on investments, and more. Although the topics in this course are drawn primarily from business and economics, students of all majors are welcome and should find the applications interesting and relevant.

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 two programs of study available in calculus:

MATH 121, 122 is a terminal one-year sequence intended for business, biology, and social science majors; MATH 131, 132, 231 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 courses numbered 231 and above. Students anticipating the need for higher mathematics courses such as MATH 325 (Differential Equations) or MATH 310, 312 (Probability and Statistics) should instead elect the MATH 131, 132, 231 sequence. Credit is not allowed for both MATH 121 and 131 (or its equivalent).

Students who have previously passed a calculus course in high school may elect MATH 122, 131, 132, or 231 as their first course, depending on 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 for that course must be forfeited if the student takes MATH 132 (or its equivalent).

MATH 133 and 134 is a two semester calculus workshop sequence taken in conjunction with specific sections of MATH 131 and 132. Participants in the calculus workshop meet for six hours per week to work in small groups on challenging problem sets related to material covered in MATH 131 and 132. They typically enjoy getting to work closely with fellow calculus students, and find that their performance in MATH 131 and 132 is significantly improved. Permission is required to sign up for the calculus workshop. For more information, contact Professor Jeffrey Holt, Calculus Workshop Coordinator; 924-4927; jjh2b@virginia.edu.

Exceptionally well prepared students (who place out of both MATH 131 and 132) may choose either MATH 231 or 325 (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 on the AB subscore of the BC test gives the student credit for MATH 131. A score of 4 or 5 on the BC test gives 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 Student 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 STAT 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 and Applied Science. 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 103 - (3) (Y)
Precalculus
Prerequisite: High school algebra II and geometry.
Studies computational skills, patterns of quantitative problem solving, and mathematical thought. Includes linear and quadratic equations, polynomials, inverse functions, logarithms, arithmetic and geometric sequences, trigonometric functions, and linear systems. (Does not satisfy the College natural science and mathematics requirement.)

MATH 108 - (3) (IR)
Modes of Mathematical Thinking
Studies 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 111 - (3) (S)
Probability/Finite Mathematics
Studies finite probability theory including combinatorics, equiprobable models, conditional probability and Bayes' theorem, expectation and variance, and Markov chains.

MATH 114 - (3) (Y)
Financial Mathematics
The study of the mathematics needed to understand and answer a variety of questions that arise in everyday financial dealings. The emphasis is on applications, including simple and compound interest, valuation of bonds, amortization, sinking funds, and rates of return on investments. A solid understanding of algebra is assumed.

MATH 115 - (3) (IR)
The Shape of Space
Provides an activity and project-based exploration of informal geometry in two and three dimensions. Emphasizes visualization skill, fundamental geometric concepts, and the analysis of shapes and patterns. Topics include concepts of measurement, geometric analysis, transformations, similarity, tessellations, flat and curved spaces, and topology.

MATH 116 - (3) (IR)
Algebra, Number Systems, and Number Theory
Studies basic concepts, operations, and structures occurring in number systems, number theory, and algebra. Inquiry-based student investigations explore historical developments and conceptual transitions in the development of number and algebraic systems.

MATH 121 - (3) (S)
Applied Calculus I
Topics include limits and continuity; differentiation and integration of algebraic and elementary transcendental functions; and applications to maximum-minimum problems, curve sketching and exponential growth. Credit is not given for both MATH 121 and 131.

MATH 121S - (4) (IR)
Introduction to Calculus
Prerequisite: Instructor permission.
Includes limits and continuity; differentiation and integration of algebraic and elementary transcendental functions; and applications to maximum-minimum problems, curve sketching and exponential growth.

MATH 122 - (3) (S)
Applied Calculus II
Prerequisite: MATH 121 or equivalent.
A second calculus course for business, biology, and social science students. Analyzes functions of several variables, their graphs, partial derivatives and optimization; multiple integrals. Reviews basic single variable calculus and introduces differential equations and infinite series. Credit is not given for both MATH 122 and 132.

MATH 131 - (4) (S)
Calculus I
Prerequisite: Background in algebra, trigonometry, exponentials, logarithms, and analytic geometry.
Introduces 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 131.

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

MATH 133 - (2) (Y)
Calculus Workshop I
Prerequisite: Instructor permission; corequisite: MATH 131.
Intensive calculus problem-solving workshop with topics drawn from MATH 131.

MATH 134 - (2) (Y)
Calculus Workshop II
Prerequisite: Instructor permission; corequisite: MATH 132.
Intensive calculus problem-solving workshop with topics drawn from MATH 132.

MATH 231 - (4) (S)
Calculus III
Prerequisite: MATH 132 or its equivalent.
Studies 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 300 - (3) (IR)
Foundations of Analysis
Prerequisite: MATH 132 or equivalent.
Topics from logic and the construction of mathematical proofs, basic set theory, number systems, continuity of functions, and foundations of analysis. Intermediate introduction of 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 310 - (3) (Y)
Introduction to Mathematical Probability
Prerequisite: MATH 132. A knowledge of double integrals is recommended. Includes sample spaces, combinatorial analysis, discrete and continuous random variables, classical distributions, expectation, Chebyshev theorem, independence, central limit theorem, conditional probability, and generating functions.

MATH 312 - (3) (Y)
Introduction to Mathematical Statistics
Prerequisite: MATH 310.
Includes sampling theory, point estimation, interval estimation, testing hypotheses (including the Neyman-Pearson lemma and likelihood ratio tests), and regression and correlation.

MATH 325 - (4) (S)
Ordinary Differential Equations
Prerequisite: MATH 132 or its equivalent.
Introduces the methods, theory, and applications of differential equations. Includes first-order, second and higher-order linear equations, series solutions, linear systems of first-order differential equations, and the associated matrix theory. May include numerical methods, non-linear systems, boundary value problems, and additional applications.

MATH 325P - (4) (S)
Ordinary Differential Equations
Prerequisite: MATH 132 or its equivalent.
Usually offered in the spring, this course covers the same material as MATH 325 with some additional topics, including an introduction to Sturm-Liouville theory, Fourier series and boundary value problems, and their connection with partial differential equations. Physics majors should enroll in MATH 325P, although no knowledge of physics is assumed.

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. Includes limits, continuity, differentiability, the Bolzano-Weierstrass theorem, Taylor's theorem, integrability of continuous functions, and uniform convergence.

MATH 334/534 - (3) (Y)
Complex Variables With Applications
Prerequisite: MATH 231 and graduate standing for MATH 534. Topics include analytic functions, Cauchy formulas, power series, residue theorem, conformal mapping, and Laplace transforms.

MATH 351 - (3) (S)
Elementary Linear Algebra
Prerequisite: MATH 132. Includes 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.
Surveys major topics of modern algebra: groups, rings, and fields. Presents applications to areas such as geometry and number theory; explores rational, real, and complex number systems, and the algebra of polynomials.

MATH 404/504 - (3) (E)
Discrete Mathematics
Prerequisite: MATH 354 or instructor permission, and graduate standing for MATH 504.
Includes 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 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 430 - (3) (IR)
Elementary Numerical Analysis
Prerequisite: MATH 325 and computer proficiency. Includes Taylor's theorem, solution of nonlinear equations, interpolation and approximation by polynomials, numerical quadrature. May also cover numerical solutions of ordinary differential equations, Fourier series, or least-square approximation.

MATH 452 - (3) (IR)
Algebraic Coding Theory
Prerequisite: MATH 351 and 354, or instructor permission.
Introduces algebraic techniques for communicating information in the presence of noise. Includes linear codes, bounds for codes, BCH codes and their decoding algorithms. May also include quadratic residue codes, Reed-Muller codes, algebraic geometry codes, and connections with groups, designs, and lattices.

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

MATH 475 - (3) (IR)
Introduction to Knot Theory
Prerequisite: MATH 331, 354, or instructor permission. Examines the knotting and linking of curves in space. Studies equivalence of knots via knot diagrams and Reidemeister moves in order to define certain invariants for distinguishing among knots. Also considers knots as boundaries of surfaces and via algebraic structures arising from knots.

MATH 493 - (3) (IR)
Independent Study
Reading and study programs in areas of interest to individual students. For third- and fourth- years interested in topics not covered in regular courses. Students must obtain a faculty advisor to approve and direct the program.

MATH 495 - (3) (IR)
Undergraduate Research Seminar
Prerequisite: Instructor permission.
Emphasizes direct contact with advanced mathematical ideas, communication of these ideas, the discovery of new results and connections among them, and the experience of mathematics as a collaborative venture among researchers at all levels. Students work collaboratively and individually on research projects, and present their results to the class.

MATH 501 - (3) (E)
The History of the Calculus
Prerequisite: MATH 231 and 351 or instructor permission.
Studies the evolution of the various mathematical ideas leading up to the development of calculus in the 17th century, and how those ideas were perfected and extended by succeeding generations of mathematicians. Emphasizes primary source materials when possible.

MATH 503 - (3) (O)
The History of Mathematics
Prerequisite: MATH 231 and 351 or instructor permission.
Studies the development of mathematics from classical antiquity to the end of the 19th century, focusing on critical periods in the evolution of geometry, number theory, algebra, probability, and set theory. Emphasizes primary source materials when possible.

MATH 506 - (3) (IR)
Algorithms
Prerequisite: MATH 132 and computer proficiency.
Studies abstract algorithms to solve mathematical problems and their implementation in a high-level language. Includes sorting problems, recursive algorithms, and dynamic data structures.

MATH 510 - (3) (Y)
Mathematical Probability
Prerequisite: Graduate standing and MATH 132, or equivalent.
Those who have received credit for MATH 310 may not take 510 for credit. Studies the development and analysis of probability models through the basic concepts of sample spaces, random variables, probability distributions, expectations, and conditional probability. Also includes distributions of transformed variables, moment generating functions, and the central limit theorem.

MATH 511 - (3) (Y)
Stochastic Processes
Prerequisite: MATH 310 or instructor permission.
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 512 - (3) (Y)
Mathematical Statistics
Prerequisite: MATH 510 and graduate standing.
Topics include methods of estimation, general concepts of hypothesis testing, linear models and estimation by least squares, categorical data, and nonparametric statistics. Those who have received credit for MATH 312 may not take 512 for credit.

MATH 514 - (3) (Y)
Mathematics of Derivative Securities
Prerequisite: MATH 231 or 122 and a knowledge of probability and statistics. MATH 310 or its equivalent is recommended.
Topics include arbitrage arguments, valuation of futures, forwards and swaps, hedging, option-pricing theory, and sensitivity analysis.

MATH 517 - (3) (IR)
Actuarial Mathematics
Prerequisite: MATH 312 or 512, instructor permission.
Covers the main topics required by students preparing for the examinations in actuarial statistics, set by the American Society of Actuaries. Topics include life tables, life insurance and annuities, survival distributions, net premiums and premium reserves, multiple life functions and decrement models, valuation of pension plans, insurance models, benefits, and dividends.

MATH 521 - (3) (Y)
Advanced Calculus with Applied Mathematics
Prerequisite: MATH 231, 325 (351 recommended).
Topics include vector analysis, Green's, Stokes', divergence theorems, conservation of energy, potential energy functions. Emphasis on physical interpretation. Also includes Sturm-Liouville problems, Fourier series, special functions, orthogonal polynomials, and Green's functions.

MATH 522 - (3) (Y)
Partial Differential Equations and Applied Mathematics
Prerequisite: MATH 521. Introduces complex variables and partial differential equations.
Topics include analytic functions, complex integration, power series, residues, conformal mapping; separation of variables, boundary value problems, Laplace's equation, wave equation, and heat equation.

MATH 525 - (3) (IR)
Dynamical Systems
Prerequisite: MATH 231, 325, 351 or instructor permission.
Studies the qualitative geometrical theory of ordinary differential equations. Topics include basic well-posedness (existence, uniqueness, continuation of solutions, dependence on parameters, comparison theory); linear and periodic systems (Floquet theory); stability theory (Lyapunov's method and invariance theory, domain of attraction, comparison principle); perturbation of linear systems; center manifold theorem; periodic solutions and Poincare´-Bendixson theory; Hopf bifurcation; introduction to chaotic dynamics; control theoretic questions; and differential-geometric methods (Lie theory).

MATH 531, 532 - (3) (Y)
Introduction to Real Analysis I, II
Prerequisite: MATH 231, 351.
Includes the basic topology of Euclidean spaces, continuity, and differentiation of functions on Euclidean spaces; Riemann-Stieltjes integration, convergence of sequences and series of functions; and 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 instructor permission.
Introduces algebraic systems: groups, rings, fields, vector spaces and their general properties; subsystems, quotient systems, homomorphisms. Includes permutation groups, polynomial rings, and groups and rings of matrices. Additional topics may include applications to linear algebra and number theory.

MATH 554 - (3) (Y)
Survey of Algebra
Prerequisite: MATH 132 or equivalent and graduate standing.
Surveys groups, rings, and fields, and presents applications to other areas of mathematics, such as geometry and number theory. Explores the rational, real, and complex number systems, and the algebra of polynomials.

MATH 555 - (3) (IR)
Algebraic Automata Theory
Prerequisite: MATH 351.
Introduces the theory of sequential machines, including an introduction to the theory of finite permutation groups and transformation semigroups. Includes examples from biological and electronic systems as well as computer science, the Krohn-Rhodes decomposition of a state machine, and Mealy machines.

MATH 570 - (3) (Y)
Introduction to Geometry
Prerequisite: MATH 231 and 351 or instructor permission.
Topics selected from analytic geometry, affine geometry, projective geometry, hyperbolic and non-Euclidean geometry.

MATH 572 - (3) (IR)
Introduction to Differential Geometry
Prerequisite: MATH 231.
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 231; corequisite: MATH 551 or equivalent.
Includes topological spaces and continuous functions; product and quotient topologies; compactness and connectedness; separation and metrication; and the fundamental group and covering spaces.

MATH 583 - (3) (IR)
Seminar
Prerequisite: Instructor permission.
Presentation of selected topics in mathematics usually for DMP students.

MATH 596 - (3) (S)
Supervised Study in Mathematics
Prerequisite: Instructor permission 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.