| Department of Mathematics, University of Utah | |||||||
|
|
|
|||||
| Department of Mathematics, University of Utah | |||||||
|
|
|
|
|||||
Professors. P. Alfeld, M. Bestvina, R. Brooks, C. E. Burgess, J. Carlson, A. Cherkaev, C H. Clemens, W. Coles, E. A. Davis, S. Ethier, P Fife, A Fogelson, E. Folias, S. Gersten, L. Glaser, K. Golden, F. Gross, G. Gustafson, H. Hecht, L. Horvath, J. Keener, J. Kollar, N. Korevaar, J. D. Mason, D. Milicic, G Milton, H. Othmer, P. Roberts, H. Rossi, T. B. Rushing, K Schmitt, J. Taylor, D. Toledo, A. Treibergs, P. Trombi, D. Tucker, D. Willett.
Professors Emeritus. E. A. Davis, C. Wilcox, J. Wolfe.
Associate Professors. A. Bertram, M. Kapovich, D. Khoshnevisan, M. Lewis, Y. Ruan, G. Savin, N. Smale.
Assistant Professors. F. Adler, A. Balk, R. McLaughlin, R. Morelli, W. Niziol, J. Zhu.
Instructors. D. Allcock, J. Amoros, D. Bottino, C. Chan, Y. Grabovsky, H. Kley, G. Muic, D. Sage, K. Solna.
Research Professor. Roger Horn.
Associate Research Professor. Elena Cherkaeva.
Adjunct Professors. M. Egger, J. Reading.
Adjunct Associate Professors. N. Beebe, D. Clark, L. Lewis, A. Roberts.
Adjunct Assistant Professors. S. Foresti, C. Johnson, J. Johnson, M. Pernice.
Initial placement in mathematics at the University of Utah is made on the basis of the student's school transcript, ACT scores, and CLEP or AP test scores. Students who receive a score of 3, 4, or 5 on either the AB or BC AP test in calculus are awarded 12 hours of mathematics credit Placement information follows:
| AP Test | Score | Placement |
| AB | 3 | MATH 211 |
| AB | 4 | MATH 212 |
| AB | 5 | MATH 213 |
| BC | 3 | MATH 213 |
| BC | 4 or 5 | MATH 221 or 251 |
Students who score less than 3 on one of the AP calculus examinations but believe they should be placed in a more advanced course than MATH 111, should consult a departmental adviser. MATH 211 is probably an appropriate course
Students who score above 50 on the CLEP college algebra or trigonometry test will have the corresponding course requirement (MATH 105 or 106) waived.
Students who have not taken AP or CLEP tests will be placed as follows:
| ACT Score | Placement |
| Below 17 | Take MATH 95 |
| 17 to 22 | Take MATH 101. |
| 23 or above | Take MATH 105, 106, 111, 128, or 129 according to high school preparation. |
Note: If the ACT was taken prior to October 1989, the minimum score required to take MATH 101 is
16 ; and the minimum score required to take MATH 105, 106, 111, or 128 is 25 .A mathematics placement test is given at the University Testing Center. It may be used to help determine placement if a recent ACT score is not available. Also available are an algebra and functions test for qualification to take MATH 111.
A student who scores above 25 on the ACT test, and has taken either college algebra or trigonometry in high school, and received a grade of A or B, will have the appropriate course requirement waived. A student who received a grade of C in one of these courses but would like that course requirement waived, should consult a mathematics adviser. Students with a strong background should consider MATH 211.
Transfer students and students resuming the study of mathematics after an interval of two years or more may be required to take appropriate placement tests. Consult the department for details.
Prerequisites for courses must be strictly observed, and a grade of at least C in prerequisite courses is required. Exceptions must be approved by the department. Students who enroll in a course without the prerequisites may have their registration canceled. If no prerequisite is announced for a course, departmental approval is required.
High school students who plan college programs that require calculus, and who wish to avoid delay in the completion of these programs, should complete trigonometry and college algebra in high school to permit registration for MATH 111 during their first quarter.
The required 45 hours of course work should include MATH 221, 222, 223; 325, 326, 327, and three sequences from: 504, 505, 507; 507, 508, 509; 511, 512, 513; 514, 515, 516; 521, 522, 523;
528, 529, 530 531, 532, 533; 541, 542, 543; 544, 545, 546; 547; 551, 552, 553; 561, 562, 563, 564, 565 ; 571, 572, 573 ,575 . In certain cases, a different course may, with prior approval, be substituted for the third quarter of one of the 500-level sequences. The credit hours in physics should normally include PHYCS 221, 222, and 223 or 321, 322 and 323. In a mathematics department with a large number of courses, some overlapping of topics is inevitable To minimize this duplication, the following courses may not be counted toward departmental majors without prior departmental approval: MATH 101, 103, 105, 106, 128, 129, 251, 252, 353, 354, 401, 402, 403, 404, 405, 406, 408, 409, 410, 411, 412, 413. Students planning a departmental and secondary teaching major normally include MATH 409 and 410 in the required hours. MATH 307, 310, and 330 are also required. Students in the Honors Program should consult regularly with the departmental Honors adviser.Specific course requirements in mathematics dictate less than half of the 183 hours required for the bachelor's degree. Options for majors can include emphasis in areas outside mathematics, and some students find it possible to pursue two degrees concurrently.
All department majors are urged to gain computer experience, through MATH 216, 217, 218, 219; or 561 and two courses from 562-567. In addition, acquiring a reading knowledge of at least two foreign languages (French, German, or Russian) during undergraduate study may be advisable for students in a pregraduate program.
Students who plan to do graduate work in mathematics should inform their adviser and are strongly urged to take (in addition to the other departmental requirements) MATH 521, 522, 523, either 531, 532, 533, or 551, 552 553, and either 541, 542, 543, or 561,
and two quarters from 562, 563, 564 and 565, and, if possible, at least one sequence at a more advanced levelThe following special programs are offered. Any student with an interest in one of these programs should consult the departmental adviser.
Scientific-Computing Specialization. Students who elect the scientific-computing specialization must take the following: MATH 111, 112, 113, or 211, 212, 213; 221, 222, 223;
325, 326, 327; four of the courses 561-567; one of the following sequences 507, 508, 509, 510; 511, 512, 513; 541, 542, 543; 544, 545, 546; and the computer project, MATH 596. Other courses are also recommended for this emphasis. Interested students should contact the departmental adviser for further details.Statistics Emphasis. Students who wish to develop an emphasis in statistics should take (in addition to other requirements) 12 approved credit hours in statistical methodology courses offered by other departments. The mathematics hours should include 307, 308; 507, 508, 509; and some work with computers.
Double Majors. Mathematics can be combined into a double major for those with majors in science and engineering. For example, students can satisfy requirements for degrees in mathematics and geophysics by a careful choice of elective hours. Programs approved by both departments include (in addition to other departmental requirements) 561, 562, 563; either 507, 508, 509, or 507, 504, 505, plus a third senior sequence. It may be advantageous to substitute a course for 563, 509, or 505. Interested students should see Geology and Geophysics in that section of this catalog.
Teaching Major. Required are a minimum of 45 approved credit hours in mathematics including (1) MATH 111, 112, 113 with 216, 217; and (2) MATH 307; 310; 221, 222, 223; 325, 326, 327; 330; 409, 410.
One of PHYCS 211, 221, or 321 must be taken, and students are advised to complete the sequence started.Teaching Minor. Required are at least 29 approved credit hours in mathematics including MATH 409 and four additional courses taken in residence from MATH 307 or 507, 310, 221,
or 330.Mathematics Internships. The Department of Mathematics participates in the University's Cooperative Education Program (Coop), which provides internship opportunities for students in business, industry, and government. The program involves either full-time employment during a quarter when the student is not enrolled in school, or part-time employment during a quarter in which the student is enrolled part-time.
While exposing students to mathematics in nonacademic settings, the internship enables them to defray part of their education costs. Students also become known by potential employers.
Students who want to participate in the program through the Mathematics Department should be mathematics majors who have completed two-thirds of each of the sequences MATH 221, 222, 223 and 325, 326, 327. They should contact the departmental undergraduate adviser to get an application form and two faculty recommendation forms. Once approved by the Mathematics Department, the student is assisted by the Cooperative Education Center in preparing a resume and applying for a coop position. Placements are decided by the employer.
Interns register for MATH 491 during the quarters they are involved with the program. The course carries variable credit, decided by the undergraduate adviser once the hours and duties of the internship are known. At the end of the quarter, interns write a report describing the completed work and present an evaluation written by their supervisor during the internship. The course, which may be taken a maximum of two times, is graded CR or NC (credit/no credit).
Degrees. M.A., M.S, M.Phil., Ph D. in mathematics; M Stat. in statistics. For additional information, see the Graduate Information section of this catalog.
Areas of Specialization. Algebra, analysis, applied mathematics, differential equations, numerical analysis, probability, statistics, and topology. Detailed information is available in "Graduate Mathematics;", available from the department office.
Master's Degree. The equivalent of an undergraduate mathematics major at the University of Utah is required for admission to the master's programs.
Degree requirements include 45 credit hours of course work beyond the prerequisites; (1) two one-year sequences numbered above 500, one of which must be MATH 521, 522, 523 unless this was used to fulfill the prerequisite; (2) one one-year sequence numbered above 600; and (3) a project. Students whose primary interest is secondary education may, with the approval of the graduate committee, replace requirement (2) with a 500-level sequence. Students should consult their supervisory committees about a project and specific programs for a master's degree.
The College of Science offers a special option leading to a master's degree for secondary-school teachers with at least three years' teaching experience in science or mathematics. For details see Science in the Colleges section of this catalog. M. Stat. Degree. Applicants to the M.Stat. degree program must have the equivalent of a bachelor's degree in mathematics at the University of Utah plus MATH 307, 308, 309, or equivalent. Applications for this degree are made to the University Statistics Committee.
A minimum of 45 credit hours of course work beyond the prerequisites is required, including: (1) MATH 507, 508, 509 (if taken as an undergraduate, then either two courses from (2), following, or another mathematics sequence approved by the supervisory committee is required); (2) one of the sequences MATH 604, 605, 606, or 607, 608, 609, or 601, 602, 603; (3) three sequences in applications as approved by the supervisory committee; (4) a written competency examination in applied statistics; and (5) a project in applied statistics, from three to six credit hours.
For additional information, see the Statistics section in this catalog.
M. Phil. Degree. The M.Phil. degree has the same requirements as the Ph.D. degree except that no doctoral dissertation is required. Consult the Bulletin of the University of Utah, The Graduate School, the Graduate Information section of this catalog and the departmental director of graduate studies for details.
Ph. D. Degree. The student must pass both written and oral qualifying examinations in order to be admitted to candidacy for the Ph.D. degree in mathematics. The written portion is in three of the following fields: applied mathematics, differential equations, modern algebra, numerical analysis, real and complex analysis, probability and statistics, and topology and geometry. Graduate students, in consultation with the supervisory committee, choose areas in which they will be examined. If the supervisory committee unanimously agrees, an examination in some other field may be substituted for one of the three tests. The oral portion of the qualifying examination must be scheduled within nine months after the candidate has passed the written examinations and is confined to an area of specialization and allied topics.
Requirements. Course requirements for the Ph.D. degree consist of at least seven sequences numbered 600 or above, or an equivalent requirement approved by the graduate committee. The student's committee may require additional and/or specific courses. The department requires resident students to perform teaching duties. Students should consult their advisers and read "Graduate Mathematics", available from the department office, for information about other requirements.
101 Intermediate Algebra (5) Qtr.: A W S Su. Prereq.: ACT score of at least 17 or satisfactory score on the MATH 101 qualifying examination. Rapid review of elementary algebra; exponents and radicals; linear functions, equations, inequalities; complex numbers; quadratic functions and equations: logarithm and exponential functions
103 Finite Mathematics (5) Qtr.: A W S Su Prereq.: MATH 101 or equivalent or adequate placement test score. Satisfies the mathematics requirement for the B.S. degree. Does not replace precalculus (MATH 105) when that course is listed as a prerequisite. Introduction to matrices, linear programming, probability and statistics with applications to Markov chains, game theory, and the mathematics of finance.
105,106 Precalculus (5,5) Qtr.: A W S Su. Prereq.: MATH 101 or equivalent, or adequate placement test score, or ACT score of at least 23. Precalculus 1 focuses on algebraic functions and considers also some elementary probability and combinatorics. Precalculus II is concerned with transcendental functions (exponential, logarithm, and trigonometric) and geometry of the plane (conic sections and vectors)
107 Elementary Statistics (4) Qtr.: A W S Su Prereq.: MATH 101. Students who have completed one quarter of calculus should take MATH 307 instead of 107. Modern statistics, including summarization of data, introduction to probability, elementary methods of estimation, and statistical tests.
111 Calculus I (4) Qtr.: A W S Su Prereq.: MATH 105 and 106, or ACT score of at least 26 with equivalent high school preparation as noted earlier. Functions and their graphs, differentiation of polynomial, rational, and trigonometric functions, velocity and acceleration, geometric applications of the derivative, minimization and maximization problems.
Note: The course 111 described above will start the new 1-variable Calculus I sequence. 112. Autumn Quarter and 113 Autumn and Winter Quarter will phase out the Calculus I sequence as taught during 1996-97.112 Calculus 1(4) Qtr.: A W S Su Prereq.: MATH 111 or 211. Autumn Quarter: Formal integration procedures, applications of integration, conic sections, polar coordinates, geometry and vectors in the plane. Winter and Spring quarters: Indefinite integrals and an introduction to differential equations, the definite integral and the fundamental theorem of calculus. Applications of the integral, transcendental functions, techniques of integration.
113 Calculus I (4) Qtr.: A W S Su. Prereq.: MATH 112 or 212. Autumn and Winter Quarters: Infinite series, geometry and vectors in space, introduction to several variable calculus (differential and integral). Spring Quarter: Indeterminate forms and improper integrals, numerical methods and approximations. Infinite series and power series, conics and polar coordinates (a brief review only), differential equations.
121 Mathematics for Life Scientists I (4) Qtr.: A Prereq.: MATH 105, 106 or ACT score of at least
23 with equivalent high school preparation as noted earlier. Differential calculus applied to biological problems Derivation and analysis of discrete time dynamical systems for growth, diffusion, and selection. Concepts of differentiation, stability, and approximation of functions. Analysis of functions with methods of maximization and limits. Computer Lab using Maple122 Mathematics for Life Scientists II (4) Qtr.: W. Prereq.: MATH 121 or MATH 111 and permission of instructor. Integral calculus applied to biological problems Derivation and analysis of differential equations describing growth, diffusion, and selection. Application of integration to differential equations and other problems. Introduction to analysis in two dimensions, including the phase plane, vectors, and matrices. Computer lab using Maple.
123 Mathematics for Life Scientists III (4) Qtr.: S Prereq.: MATH 122 or MATH 112 and permission of instructor. Probability and statistics applied to biological problems. Derivation and analysis of stochastic dynamical systems describing growth, diffusion, and selection. Probability theory, distributions, random variables, probability density functions. Applications of binomial, exponential Poisson and normal distributions. Introduction to statistics, parameter estimation, maximum likelihood, hypothesis testing, correlation. Computer lab using Maple.
128 Algebra for Business, Social Sciences (5) Qtr.: A W S Su. Prereq.: MATH 101 or equivalent, adequate placement test score, or ACT score of at least 23. Functions and graphs, linear and quadratic functions, matrices, Gaussian elimination, Leontieff models, exponential and logarithmic functions, growth, periodic and continuously compounded interest, arithmetic and geometric series, annuities and loans.
129 Quantitative Analysis (4) Qtr.: A W S Su. MATH 128. Not for students who have completed more than one quarter of calculus. Differentiation, maximization and minimization of functions, marginal analysis and the optimization of constrained functions, integration and applications.
211 Calculus for Advanced Placement Students (4) Qtr.: A. Prereq.: Consent of department. For freshmen with strong high school background in college algebra and trigonometry,
or score of 3 on AB calculus test .212 Calculus for Advanced Placement Students (4) Qtr.: A W. Prereq.: MATH 211, departmental consent, or score of 4 on AB calculus test.
213 Calculus for Advanced Placement Students (4) Qtr.: A W S. Prereq.: MATH 212, departmental consent, or score of 5 on AB calculus test, or score of 3 on BC calculus test.
216, 217 Computational Mathematics Using C (3,3) Qtr.: W. S Prereq.: MATH 11 1, MATH 108, or instructor's consent. A short introduction to those aspects of C essential for mathematics, followed by extensive work with mathematics problems in which computation plays an important role. Students will learn series, numerical integration, numerical solution of ordinary differential equations, matrix computations. The second quarter of the course is topical and consists of several major individual student projects, e.g. discrete dynamical systems, Numerical solution of partial differential equations (via difference equations), deterministic, and probabilistic models.
221, 222, 223 Calculus II (4,4,4) Qtr.: A W S Su Prereq.: MATH 113 or MATH 213. Linear algebra, transformations and matrices, eigenvalues and applications; differential equations, systems of differential equations; initial and boundary value problems.
251 Ordinary Differential Equations for Science and Engineering (3) Qtr.: A W S Su. Prereq.: MATH 113. Coreq.: PHYCS 301. First-order equations, second- and higher-order linear equations, linear systems of differential equations, Laplace transform methods, applications to mechanics and electrical Circuits.
252 Matrices and Vector Analysis for Science and Engineering (3) Qtr.: A W S Su Prereq.: MATH 113 Coreq.: PHYCS 301. Vector algebra, matrices, and systems of linear algebraic equations; vector differential calculus, divergence, curl; vector-integral calculus: line, surface, and volume integrals, Gauss', Green's, and Stokes' theorems.
304 Mathematical Logic (4) Qtr.: S. Formal systems for propositional calculus, with applications to logical problems in ordinary language.
305 Set Theory (3) Qtr.: W. Algebra of sets, algebra of relations and functions, arithmetic of transfinite cardinals and ordinals, axioms for set theory.
307 Applied Statistics I(5) Qtr.: A W S Su. Prereq.: MATH 111. An introduction to basic probability theory, sampling from normal populations, large-sample problems, sampling from one or two populations, estimation of parameters including population proportions, using the SAS programming language to perform statistical analyses.
308 Applied Statistics II(4) Qtr.: W S Su. Prereq.: MATH 307. Introduction to regression analysis, introduction to analysis of variances, correlation analysis, contingency tables, introduction to nonparametric techniques, continued use of SAS programming language.
309 Applied Statistics III(4) Qtr.: S Prereq.: MATH 308. Introduction to the design of experiments, multiple regression, factorial and nested designs, use of the SAS programming language for computations.
310 Foundations of Geometry (3) Qtr.: A S. Prereq.: MATH 113. Modern axiomatic development of plane geometry and related systems.
316 Scientific Computing in C (3) Qtr.: A. Prereq.: MATH 111 and knowledge of Fortran or Pascal. Knowledge of C not required. Second course in science-oriented programming using the language C.
325, 326, 327 Advanced Calculus (4,4,4) Qtr.: A W S Su Prereq.: MATH 113 or 213. A rigorous treatment of the main ideas of one-variable calculus. Calculus in n-space, including the basic topological properties of R n , Frechet differentials of real-and vector-valued functions, mapping theorems and applications, Riemann integration, curves and surfaces in R n and integrals over them, Green's and Stokes' theorems.
330 Foundations of Algebra (3) Qtr.: S. Prereq.: MATH 113. Natural numbers and extensions of this system
353 Partial Differential Equations (3) Qtr.: A W S Su Prereq.: MATH 251 and 252. Fourier series and boundary-value problems for the wave, heat, and Laplace equations; separation of variables; Sturm-Liouville problems and orthogonal expansions; Bessel functions and Legendre polynomials.
354 Complex Variables (3) Qtr.: A W S Su. Prereq.: MATH 353 or 323. Analytic functions, conformal mapping, Taylor and Laurent series, complex integration and Cauchy integral theorem, residue theorem, applications to electrostatic, heat, and fluid flows.
373 Applied Linear Algebra (3) Qtr.: W Su. Prereq.: MATH 221 or 252. Vector spaces, linear transformations, systems of linear equations, eigenvalues and eigenvectors, applications to Markov chains, linear difference and differential equations, numerical methods.
401 Real Numbers (4) Qtr.: A S Prereq. MATH 105 (with grade of C or better). Introduction to real number arithmetic; communicating concepts to children in grades K-6.
402 Elementary Geometry (4) Qtr.: A W. Prereq.: MATH 105 (with grade of C or better). Intuitive and computational geometry; communicating concepts to children in grades K-6
403 Equations and Algorithms (4) Qtr.: W S. Prereq.: MATH 105 (with grade of C or better). Develops a more mature view of arithmetic skills, including use of computers in teaching problem-solving skills.
404 Discrete Mathematics (4) Qtr.: A. Prereq.: MATH 105 (with grade of B or better). Elementary probability and statistics, networks, and graphs.
405 Geometry and Trigonometry (4) Qtr.: W. Prereq.: MATH 105 (with grade of B or better). Intuitive and computational geometry (builds on MATH 402); introduction to trigonometry.
406 Intuitive Calculus (4) Qtr.: S. Prereq.: MATH 105 (with grade of B or better). Introduction to ideas of differential and integral calculus.
408 Teaching of Middle School Mathematics (4) Qtr.: A Prereq.: MATH 111 or 406. Presentation of materials appropriate for middle school students with special emphasis on the transition from arithmetic to algebra and the appropriate use of technology.
409 Teaching of Secondary School Mathematics 1(3) Qtr.: W Prereq.: MATH 113. Presentation of mathematical material at the appropriate level, remedial instruction methods, curriculum development.
410 Teaching of Secondary School Mathematics 11(3) Qtr.: S. Prereq.: MATH 409. A continuation of MATH 409.
411, 412, 413 Mathematics for Secondary School Teachers (Arr,4,4)
415 Topics in the Modern Secondary School Mathematics Curriculum. (1 to 6)
416 Developing the Secondary School Mathematics Curriculum. (1 to 6)
491 Internship in Mathematics (Arr.) Qtr.: A W S Su. Prereq.: Consent of departmental adviser and completion of two-thirds of each of the sequences MATH 221, 222, 223 and 325, 326, 327. Mathematics-related work in industry, business, or government.
500 Undergraduate Problem Seminar ( 1 ) Qtr.: A Prereq.: MATH 111. Repeatable for credit. Requires mathematics at undergraduate level only. Difficult problems presented for solution; sharpens skills and develops problem-solving techniques Prepares students for Putnam Examination (given annually by the Mathematical Association of America).
501 Nonparametric Statistics (4) Qtr.: S. Prereq.: MATH 307. Ranking methods in statistics, order statistics, Mann-Whitney test, Wilcoxon tests, rank correlation, contingency tables, analysis of runs, regression analysis, analysis of variance.
504, 505 Applied Stochastic Processes (3,3) Qtr.: W S. Prereq.: MATH 507. Basic ideas for finite and denumerable state models, Markov chains, Poisson processes, the Ergodic Theorem, Brownian motion, time-series analysis with applications.
507 Introduction to Probability (4) Qtr.: A W S Prereq.: MATH 221 or 252. Combinatorial problems, random variables, independence and dependence, conditional probability, moments, binomial, Poisson and normal laws, law of large numbers, and central-limit theorems.
508, 509 Statistical Inference (3,3) Qtr.: W S Prereq.: MATH 507. Sampling, sampling distributions, point and interval estimation, tests of hypotheses, regression, correlation, distribution free methods, sequential analysis.
510 Stochastic Simulation (3) Qtr.: S Prereq.: MATH 507. Required for students in scientific computing emphasis who wish to count MATH 507, 508, 509 as one of their 500 level sequences. Computer simulations of stochastic processes.
511, 512, 513 Mathematical Biology (3, 3, 3) Qtr.: A W S. Prereq.: MATH 327 or 353 or instructor's consent. Cross-listed as BIOL 608, 609, 610 Topics from population Biology, physiology, and developmental biology.
514, 515, 516 Topics in Mathematical Chemistry (3,3,3) Qtr.: A W S Prereq.: MATH 353 or equivalent. Techniques for studying chemically reacting systems, including mass-action kinetics, diffusion processes, combustion phenomena, heterogeneous catalysis, chromatography
521, 522 Introduction to Real Analysis (3,3) Qtr.: A W. Prereq.: MATH 327. Consequences of completeness of the reals elementary topology of metric spaces, rigorous treatment of the topics of calculus, consequences of uniform convergence, several variable calculus, introduction to Lebesque integration.
523 Introduction to Complex Variables (3) Qtr.: S Prereq.: MATH 522 or instructor's consent. Analytic functions, Cauchy integral formula and its consequences, calculation of important integrals by residues and steepest descent, harmonic functions, conformal mapping. Applications to potential theory.
528, 529, 530 Introduction to Number Theory (3,3,3) Qtr.: A W S Prereq.: one quarter of I linear algebra. An overview of algebraic number theory covering factorization and primes, modular arithmetic, quadratic residues, continued fractions, quadratic forms, and diophantine equations.
531, 532, 533 Introduction to Modern Algebra (3,3,3) Qtr.: A W S Prereq.: MATH 325, 326, 327, or instructor's consent. An introduction to groups, rings, and fields.
541, 542, 543 Ordinary Differential Equations (3, 3, 3) Qtr.: A W S. Prereq.: MATH 325, 326, 327, or instructor's consent. Linear differential equations, systems of differential equations, discrete and continuous dynamical systems, stability theory, chaotic behavior, strange attractors, numerical methods, boundary value problems.
544, 545, 546 Partial Differential Equations (3,3,3) Qtr.: A W S. Prereq.: MATH 222; or 251, 252. Existence, uniqueness, perturbation theorems Methods of characteristics, Fourier analysis, Green's functions. Problems of Cauchy, Neuman, Dirichlet, and Goursat Classical wave, Laplace, and heat equations. Integral equations and calculus of variations with applications to numerical analysis.
547 Applied Dynamical Systems (3) Qtr.: S. Prereq.: MATH 221, 223; or engineering math. Introduction to chaotic motions, strange attractors, fractal geometry. Models from fluid dynamics and mechanical and electrical oscillators.
551, 552 Introduction to Topology (3,3) Qtr.: A W S. Prereq.: MATH 326. Introduction to algebraic topology and topological spaces and their properties.
553 Curves and Surfaces in Euclidean Space (3) Qtr.: S Prereq.: MATH 326; 552 recommended. Frenet theory, Gaussian and mean curvatures Gauss-Bonnet theorem, minimal surfaces, and surfaces of constant curvature.
560 Survey of Numerical Analysis (5) Qtr.: A W S Su. Prereq.: MATH 113, MATH 221 or 252, and programming ability. Numerical techniques in linear algebra, nonlinear equations, interpolation, integration, and ordinary and partial differential equations. This is a survey of topics covered in MATH 561-565.
561 Basic Numerical Analysis (5) Qtr.: A. Prereq.: MATH 113, MATH 221 or 252, and programming ability. Numerical linear algebra, interpolation and numerical integration in one or more variables. This course covers subjects that are used widely within numerical analysis as well as in scientific and engineering applications.
562 Approximation Theory(3) Qtr.: W Prereq.: MATH 561. Discrete and continuous least squares, Fourier series, fast Fourier Transform, wavelets, including problems in several variables.
563 Nonlinear Equations and Optimization (3) Qtr.: S. Prereq.: MATH 561. Numerical techniques for solving nonlinear systems and optimization problems.
564 Numerical Solution of Initial Value Problems (3) Qtr.: W. Prereq.: MATH 560 or 561. Numerical techniques for solving initial value problems in ordinary and partial differential equations, with emphasis on modern computational techniques.
565 Numerical Solution of Boundary Value Problems (3) Qtr.: S. Prereq.: MATH 561. Numerical techniques for solving boundary value problems in ordinary and partial differential equations, with emphasis on modern computational techniques.
566 Parallel Numerical Linear Algebra (5) Qtr.: W Prereq.: MATH 560 or 561. An introduction to parallel computing (hardware, software, programming environments, algorithm design, and performance evaluation) in the context of numerical linear algebra. Note: This course may also be taken as MATH 666 with the addition of an extra project agreed to by the instructor.
567 Parallel Numerical Techniques for Partial Differential Equations (3) Qtr.: S Prereq.: MATH 566 and either MATH 564 or 565. Parallel algorithms for solving partial differential equations, including sparse iterative techniques, preconditioners, and domain decomposition strategies. Note: This course may also be taken as MATH 667 with the addition of an extra project agreed to by the instructor.
571, 572, 573 Introduction to Applied Mathematics (3,3,3) Qtr.: A W S. Prereq.: MATH 221, 222 or MATH 251, 252, 353 or instructor's consent. Mathematical and computer modeling of physical, biological, and engineering problems. Development and application of analytic and computational methods to these problems. Linear algebra, differential and integral equations, calculus of variations, tensor methods, asymptotic series, and perturbation theory.
574 Mathematical Modeling (3) Qtr.: S Prereq.: MATH 564 or CP SC 522. Description of physical, biological, and engineering problems and phenomena using mathematical models which involve ordinary and partial differential equations. Development of analytical and numerical tools suitable for analysis and visualization of the solutions to these problems. Use of packages such as LINPACK, EISPACK, Maple, Matlab, and Explorer will be included.
575 Topics in Applied Mathematics. (3 to 6) Prereq.: MATH 353 or instructor's consent. Repeatable for credit when topics vary. Consult quarterly Class Schedule for specific offering. Possible topics include integral equations, calculus of variations, control theory, continuum mechanics, applied matrix theory, vector and tensor analysis.
591 Supervised Reading (Arr.)
596 Undergraduate Special Projects (4) Qtr.: A W S Su. Prereq.: Instructor's consent Special computer project to serve as "senior thesis" for students in scientific computing emphasis.
601, 602 Analysis of Variance (3,3) Qtr.: A W. Prereq.: MATH 221 or 252; and 509. Distribution theory, general linear hypotheses, analysis of covariance, components of variances, regression analysis, experimental design models, transformations, and mixed models. Computer analysis included.
603 Multivariate Analysis (3) Qtr.: S. Prereq.: MATH 602. Principal component analysis, cluster analysis, multivariate normal distribution, multivariate analysis of variance, discriminant analysis, and applications to diverse fields. Computer analysis included.
604, 605, 606 Mathematical Probability (3,3,3) Qtr. A W S Prereq. or coreq.: MATH 621. Analytical approach to probability theory, random variables and their distributions, limit theorems for sums of independent random variables.
607, 608, 609 Mathematical Statistics (3,3,3) Qtr.: A W S Prereq.: MATH 221, 508. Topics from distribution theory, decision theory, estimation, testing hypotheses, confidence intervals, Neyman-Pearson theory, large-sample theory, nonparametric inference, and sequential analysis from advanced viewpoint.
621, 622, 623. Real and Complex Analysis (3,3,3) Qtr.: A W S Prereq.: MATH 523.. Recommended: MATH 552. Measures and integrals, LP-spaces, Hilbert spaces, Banach spaces, analytic functions, complex integration, conformal mapping.
624, 625, 626 Lie Groups and Lie Algebras (3,3,3) Qtr. A W S. Prereq.: MATH 621, 622, 623. An introduction to the study of Lie groups and Lie algebras.
631, 632, 633 Modern Algebra (3,3,3) Qtr.: A W S. Prereq.: MATH 533. Groups, rings, modules, homological algebra, fields, and Galois theory.
634, 635, 636 Group Theory (3,3,3) Qtr.: A W S Prereq.: MATH 633. Solvable and nilpotent groups, simple groups, permutation groups, linear groups, and representation theory.
637, 638, 639 Rings and Modules (3,3,3) Qtr.: A W S. Prereq.: MATH 633. Topics vary from year to year.
641, 642, 643 Theory of Differential Equations (3,3,3) Qtr.: A W S Prereq.: MATH 523 or instructor's consent. Existence and uniqueness, linear equations, stability, oscillation theory, Sturm-Liouville boundary value problems, equations of classical physics, Green's functions, Cauchy problems.
644, 645, 646 Partial Differential Equations (3,3,3) Qtr.: A W S. Prereq.: MATH 521, 522, 523 or instructor's consent. The partial differential equations of classical physics; study of Sobolev spaces; variational methods; weak solutions of semilinear equations. Nonlinear functional analysis methods in the study of nonlinear problems; semigroup theory and evolution equations.
651, 652, 653 Topology and Geometry (3,3,3) Qtr.: A W S Prereq.: MATH 553. General topology, homotopy and covering spaces, bundles, complexes, manifolds, homology and cohomology, De Rham theorem, Riemannian geometry.
654, 655, 656 Homological Algebra (3,3,3) Qtr.: A W S. Prereq.: MATH 633. Algebraic construction of homology and cohomology theories, with applications to topology, geometry, groups, rings and sheaves. Projective, injective and flat modules, exact sequences, chain complexes. Derived functors, derived categories, and spectral categories.
661, 662, 663 Analysis of Numerical Methods (3,3,3) Qtr.: A W S. Prereq.: MATH 561 or equivalent. Numerical linear algebra; interpolation and approximation; solutions of nonlinear equations; numerical solutions of differential equations.
666 Parallel Numerical Linear Algebra (5) Qtr.: W. Prereq.: MATH 560 or 561. An introduction to parallel computing (hardware, software, programming environments, algorithm design, and performance evaluation) in the context of numerical linear algebra. Note: This course is the same as MATH 566, but requires completion of an extra project agreed to by the instructor.
667 Parallel Numerical Techniques for Partial Differential Equations (3) Qtr.: S. Prereq.: MATH 566 and either MATH 564 or 565. Parallel algorithms for solving partial differential equations, including sparse iterative techniques, preconditioners, and domain decomposition strategies Note: This course is the same as MATH 567, but requires completion of an extra project agreed to by the instructor.
671, 672, 673 Methods of Applied Mathematics (3,3,3) Qtr.: A W S Prereq.: MATH 354; and 521 or 541. Theory of linear equations applied to matrix, differential, and integral equations; Fredholm alternative, spectral theory, generalized inverses, Green's functions, eigenfunction expansions, Fourier and Laplace transforms, asymptotic expansions, and complex variable techniques.
674 Case Studies in Computational Engineering and Science (3) Qtr.: W. Prereq. MATH 574. Two to five faculty members from various disciplines will describe in detail a project in which they are engaged that involves all ingredients of computational engineering and science: a scientific or engineering problem, a mathematical problem leading to mathematical questions, and the solution and interpretation of these questions obtained by the use of modern computing techniques. Participating faculty will vary from year to year.
675 Advanced Topics in Applied Mathematics (3 to 6) Qtr.: A W S Prereq.: MATH 673 or instructor's consent. Repeatable for credit when topics vary. Consult quarterly Class Schedule for specific offerings. Possible topics include mathematical biology, mathematical chemistry, direct and inverse scattering theory, theories of imaging, stochastic differential equations, and mathematics of phase transitions.
676 Seminar in Computational Engineering and Science (1 to 3) Qtr.: S. Prereq.: MATH 674. Crosslisted as CP SC 676. Students in the final quarter of the Computational Engineering and Science Program will present their own CES-related research.
677, 678, 679 Topics in Mathematical Biology (3,3,3) Qtr.: A W S Prereq.: Graduate level courses in ordinary and partial differential equations and numerical analysis or instructor's consent. Repeatable for credit when topics vary. Topics drawn from the areas of biology and medicine, ecology and population biology.
681, 682, 683 Nonlinear Oscillations and Perturbation Theory (3,3,3) Qtr. : A W S. Prereq, : MATH 543 or instructor's consent. Dynamical systems theory (phase plane analysis, weakly nonlinear systems, nonlinear resonance and chaos discrete maps) Regular perturbation theory (implicit-function theorem, Hopf bifurcation, Melnikov's integral, higher co-dimensional bifurcations). Singular perturbation theory (matched asymptotic expansions, relaxation oscillations, multiscale, and averaging techniques).
684, 685, 686 Bifurcation Theory and Its Applications (3,3,3) Qtr.: A W S Prereq.: MATH 543 or instructor's consent. Liapunov-Schmidt reduction, degree theory and fixed-point theorems, constructive approach to bifurcating solutions, Hopf bifurcation theorem for flows and maps, stability of branching solutions, normal forms, center manifold theorem, singularity theory for functions and maps, applications to nonlinear ordinary and partial differential equations.
687, 688, 689 Methods of Nonlinear Wave Propagation (3,3,3) Qtr.: A W S. Prereq.: MATH 522 or equivalent First-order hyperbolic systems, breaking and shock fitting, geometrical optics, reaction diffusion waves, solitary waves, and inverse scattering transform.
691 Supervised Reading (Arr.)696 Special Projects (Arr.)
697 Thesis Research: Master's (Arr.)698 Faculty Consultation (3)
Usually only one sequence from each decade (e.g., 711 to 719) is offered in any year.
711, 712, 713 Riemann Surfaces (3, 3, 3) Qtr.: A W S. Prereq.: MATH 653.
714, 715, 716 Complex Manifolds (3, 3, 3) Qtr.: A W S. Prereq.: MATH 653.
717, 718, 719 Differential and Riemannian Geometry (3, 3, 3) Qtr.: A W S. Prereq.: MATH 653.
721, 722, 723 Lie Group Representation (3, 3, 3) Qtr.: A W S. Prereq.: MATH 624, 625, 626.
724, 725, 726 Several Complex Variables (3, 3, 3) Qtr.: A W S. Prereq.: MATH 623.
727, 728, 729 Functional Analysis (3, 3, 3) Qtr.: A W S. Prereq.: MATH 623.
731, 732, 733 Algebraic Groups (3, 3, 3) Qtr.: A W S. Prereq.: MATH 624, 625, 626. Basic results in the areas of algebraic groups and homogeneous spaces.
737, 738, 739 Algebraic Geometry (3, 3, 3) Qtr.: A W S. Prereq.: MATH 633.
741, 742, 743 Nonlinear Functional Analysis (3, 3, 3) Qtr.: A W S. Prereq.: MATH 643.
751, 752, 753 Algebraic Topology (3, 3, 3) Qtr.: A W S. Prereq.: MATH 653.
754, 755, 756 Geometric Topology (3, 3, 3) Qtr.: A W S. Prereq.: MATH 653.
761, 762, 763 Approximation and Optimization (3, 3, 3) Qtr.: A W S. Prereq.: MATH 561-2-3 or equivalent.
764, 765, 766 Numerical Solution of Differential and Integral Equations (3, 3, 3) Qtr.: A W S. Prereq.: MATH 561-4-5 or equivalent.
Seminars in specialized topics are scheduled according to availability of instructors and needs of advanced students.
780 Seminar in Algebraic Geometry (Arr.) 781 Seminar in Differential Geometry (Arr.) 782 Seminar in Analysis (Arr.) 783 Seminar in Algebra (Arr.) 784 Seminar in Differential Equations (Arr.) 785 Seminar in Topology (Arr.) 786 Seminar in Numerical Analysis (Arr.) 787 Seminar in Applied Mathematics (Arr.) 788 Seminar in Probability and Statistics (Arr.) 789 Seminar in Lie Groups (Arr.) Qtr.: A W S. 797 Thesis Research: Ph. D. (Arr.) 798 Faculty Consultation (3)799 Continuing Registration: Ph. D. (0)