Department of Mathematics

216 Kerchof Hall
University of Virginia
P.O. Box 400137
Charlottesville, VA 22904-4137
(434) 924-4919 Fax: (434) 982-3084
www.math.virginia.edu

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 100 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 2.200 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, CS 120 or PHY 254 or an approved equivalent course with a grade of C- or higher. This should be done as early as possible.

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

Up to two courses that are being counted for another College major or another College minor can also be counted for the major in mathematics.

Up to two courses that are taken from outside the University and which are equivalent to College Mathematics courses may be offered for the College mathematics major.

Certain substitutions are allowed in all options, for example, MATH 531 for MATH 331, MATH 551 for MATH 351, MATH 552 for MATH 354, and PHYS 553 for MATH 430. PHIL 542 Symbolic Logic is an approved elective for both the major and minor in Mathematics.

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:

Four electives at the 300 level or higher, of which at least two are MATH courses. (You may wish to take MATH 331 in preparation for MATH 531, MATH 351 in preparation for MATH 551, and MATH 354 in preparation for MATH 552).

This constitutes the minimum expected of an incoming graduate student in most programs nationwide. The department strongly recommends MATH 533 (Advanced Multivariate Calculus), as well as courses in differential geometry (MATH 572) or topology (MATH 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 probability, statistics, regression, time series, partial differential equations, stochastic processes, stochastic calculus, numerical methods, and analysis. The program consists of:

⁽¹⁾ Completing all four courses is recommended.

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 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. The departmental committee for the DMP grants admission to the program, 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 3.400 or higher).

The DMP is the same as the graduate school preparatory concentration, except that in the fourth year the students 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. 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 2.000 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. Either Math 331 or Math 354 should be one of the five approved courses. Courses with the STAT mnemonic or from other departments or institutions can be taken if approved by the undergraduate committee.

Up to two courses that are being counted for another College major or another College minor can also be counted for the minor in mathematics.

Up to two courses that are taken from outside the University and which are equivalent to College Mathematics courses may be offered for the College mathematics minor.

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 John Faulkner, Lower Division Advisor, Room 325, 924-4942, or Thomas Kriete, Upper Division Advisor, Room 205, 924-4932, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904-4137; www.math.virginia.edu.

Course Descriptions

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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 111 (Elementary Probability Theory) and MATH 114 (Financial Mathematics). 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:

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. The Department of Mathematics offers short advisory online 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.

Substitutions 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 111, or MATH 131 and MATH 121). Up to two courses, taken from outside of the University and which are equivalent to College Mathematics courses, may be offered for the College Mathematics major or minor. The following are equivalent courses from the School of Engineering and Applied Sciences:

• APMA 213 and MATH 325 Ordinary Differential Equations
• APMA 302 and MATH 404 Discrete Mathematics
• APMA 308 and MATH 351 Elementary Linear Algebra
• APMA 310 and MATH 310 Introduction to Mathematical Probability
• APMA 507 and MATH 430 Elementary Numerical Analysis
• SYS 321 and MATH 408 Operations Research

Course Descriptions

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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 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 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 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 - (3) (Y)
Complex Variables with Applications
Prerequisite: MATH 231.
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 - (3) (E)
Discrete Mathematics
Prerequisite: MATH 354 or instructor permission.
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 475 - (3) (IR)
Introduction to Knot Theory
Prerequisite: MATH 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 504 - (3) (E)
Discrete Mathematics
Prerequisite: graduate standing.
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 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 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 - (3) (Y)
Introduction to Real Analysis
Prerequisite: MATH 231, 351.
Includes the basic topology of Euclidean spaces; continuity, and differentiation of functions of a single variable; Riemann-Stieltjes integration; and convergence of sequences and series.

MATH 533 - (3) (Y)
Advanced Multivariate Calculus
Prerequisite: MATH 531.
Differential and Integral Calculus in Euclidean spaces; implicit and inverse function theorems, differential forms and Stokes’ Theorem.

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

MATH 551 - (3) (Y)
Advanced Linear Algebra
Prerequisite: MATH 351 or instructor permission.
This course includes a systematic review of the material usually considered in MATH 351 such as matrices, determinants, systems of linear equations, vector spaces, and linear operators. However, these concepts will be developed over general fields and more theoretical aspects will be emphasized. The centerpiece of the course is the theory of canonical forms, including the Jordan canonical form and the rational canonical form. Another important topic is general bilinear forms on vector spaces. Time permitting, some applications of linear algebra in differential equations, probability, etc. are considered.

MATH 552 - (3) (Y)
Introduction to Abstract Algebra
Prerequisite: MATH 351 or instructor permission.
Focuses on structural properties of basic algebraic systems such as groups, rings and fields. A special emphasis is made on polynomials in one and several variables, including irreducible polynomials, unique factorization and symmetric polynomials. Time permitting, such topics as group representations or algebras over a field may be included.

MATH 553 - (3) (O)
Number Theory
Prerequisite: MATH 354 or instructor permission.
Includes congruences, quadratic reciprocity, Diophantine equations, and number-theoretic functions, among others.

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) (O)
Introduction to Geometry
Topics selected from analytic geometry, affine geometry, projective geometry, and hyperbolic and non-Euclidean geometry.

MATH 572 - (3) (E)
Introduction to Differential Geometry
Prerequisite: MATH 231 and 351, or instructor permission.
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.
Topological spaces and continuous functions, connectedness, compactness, countability and separation axioms, and function spaces. Time permitting, more advanced examples of topological spaces, such as projectives spaces, as well as an introduction to the fundamental group will be covered.

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.