## Courses

### CPLX 701. Mathematical Foundations for Complex Systems (2)

*Prerequisite: Consent of the graduate advisor*. This course provides essential mathematical tools for the study of complex systems. Topics include vector calculus, linear algebra and aspects of differential equations and partial differential equations.

### CPLX 702. Physics Foundations for Complex Systems (2)

*Prerequisite: Consent of the graduate advisor*. This course introduces physical approaches to dealing with complex, many-body systems. Two distinct and complementary frameworks are presented: analytical mechanics and statistical physics. Core concepts include Lagrangian and Hamiltonian formulations of mechanics, phase space, non-linear dynamics, chaos, fractals, free energy, entropy, partition functions and statistical ensembles.

### CPLX 703. Chemistry Foundations for Complex Systems (2)

*Prerequisite: Consent of the graduate advisor*. The first part of the course will cover non-linear chemical kinetics, control of chemical reactions, self-assembly at microscopic and macroscopic levels, and development of new techniques for materials synthesis. In the second part, quantum chemical models for describing large systems and their interaction with the surroundings will be introduced.

### CPLX 704. Biology Foundations for Complex Systems (2)

*Prerequisite: Consent of the graduate advisor*. The course will explore how to analyze biology from a systems-level point of view. Students will explore design principles in biology, including plasticity, exploratory behavior, weak-linkage, constraints that deconstrain, robustness, (non)optimality and evolvability.

### CPLX 705. Computer Science Foundations for Complex Systems (2)

*Prerequisite: Consent of the graduate advisor*. Overview of computer science topics relevant to complex systems, including software development, design of algorithms, computer simulation, computer and sensor networks, social networks and agent-based systems.

### CPLX 706. Engineering Foundations for Complex Systems (2)

*Prerequisite: Consent of the graduate advisor*. This course introduces the principles and methods of complex systems engineering. The course is organized as a progression through the systems engineering processes of analysis, design, implementation and deployment with consideration of verification and validation throughout. Case studies and guest lectures in each phase present best practice in the field and both successes and failures are considered. Lectures will include modeling of engineering system performance and constraints; formulating systems of design rules; optimization algorithms; solver software. Students will work as an integrated conceptual design team. Complex systems engineering standards and selected journal articles will serve as a basis for readings. The course is relevant to all professions associated with the introduction of complex human-made systems.

### CPLX 710A. Complex Systems I (4)

*Prerequisites: Passing preliminary examination and graduate advisor consent*. This course provides an overview of complex systems and describes theoretical, numerical and computational approaches to defining, analyzing and solving applied problems in complex systems.

### CPLX 710B. Complex Systems II (4)

*Prerequisites: CPLX 710A, passing preliminary examination and graduate advisor consent*. Hands-on activities on complex systems topics. Examples of course project topics include complex networked systems (sensor networks and social networks) and agent-based modeling (genetic programming and evolutionary strategies).

### CPLX 791. Research Seminar (1)

*Prerequisites: Passing preliminary examination and graduate advisor consent*. Advanced studies in various subjects related to complex systems through special seminars, informal group studies of special problems, or group research on complete problems for analysis and experimentation.

### MATH 091A. Diagnostic for Mathematics (1)

*Corequisite: Enrollment in ESM 196QR or ESM 196S. * This is a self-paced diagnostic and preparation for General Education Mathematics/Quantitative Reasoning. Credit will not apply toward the baccalaureate degree but will apply as 1 unit of University credit. The course covers topics in basic arithmetic and elementary algebra. (Cross-listed with ESM 091A.)

### MATH 091B. Support Course for GE Mathematics (1)

*Corequisite: Enrollment in MATH 102, but only for students who qualify for MATH 102 by receiving a passing grade in MATH 196S or equivalent.* This is a credit/no credit pre-baccalaureate math class designed to support students in algebra intensive GE Math courses. It provides just in time remediation just prior to each of the linked GE Math lecture meetings, allowing students to have the prerequisites at their fingertips.

### MATH 102. Pre-Calculus I (3)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared must have credit for or concurrently enroll in MATH 102L.* A preparation for the algebra necessary for calculus. This course is intended for computer science, engineering, mathematics, and natural science majors. It builds on student’s familiarity with linear, quadratic, and rational expressions to achieve fluent proficiency in analyzing the local and global behavior of functions involving such expressions. Not open to students who have credit in MATH 105. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 102L. Pre-Calculus I Lab (1)

*Prerequisite: Required for all conditionally prepared students enrolled in MATH 102.* All students in MATH 102 are encouraged to enroll in this course. This is a Credit/No Credit hybrid enrichment laboratory for students in MATH 102. This course will include a self-paced, modular online component. 2 hours lab per week. (Credit/No Credit only)

### MATH 103. Mathematical Methods for Business (3)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared must have credit for or concurrently enroll in MATH 103L.* Concepts and applications of algebra and calculus to business. Topics include functions, systems of equations, matrices, the derivative and business-related topics in calculus. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 103L. Mathematics for Business Laboratory (1)

*Prerequisite: Conditionally prepared for MATH 103. Corequisite: MATH 103. *This self-paced, module-based laboratory is designed to give students additional exposure to the applications of college algebra to business and economics beyond what can be done in lecture. The additional hands-on problem-solving skills learned in this class enhance the lecture experience and strengthen the skills necessary for success in MATH 103 and subsequent courses in business majors. The lab environment allows students to both work at their own pace and receive small-group instruction with the laboratory instructor on all modules. 2 hours lab per week. (Credit/No Credit only)

### MATH 104. Trigonometry and Analytic Geometry (3)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared must have credit for or concurrently enroll in MATH 104L.* Rectangular and polar coordinates; trigonometric functions, identities and equations; inverse trigonometric functions; conic sections; complex numbers. Not open to students who have credit in MATH 103, MATH 105 or MATH 106.

### MATH 104L. Trigonometry Lab (1)

*Prerequisite: Required for all conditionally prepared students enrolled in MATH 104.* All students in MATH 104 are encouraged to enroll in this course. This is a Credit/No Credit hybrid enrichment laboratory for students in MATH 104. This course will include a self-paced, modular online component. 2 hours lab per week. (Credit/No Credit only)

### MATH 105. Pre-Calculus II (5)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared must have credit for or concurrently enroll in MATH 105L.* A preparation for the trigonometric, exponential, and logarithmic functions used in calculus. This course is intended for computer science, engineering, mathematics, and natural science majors. This course builds on student’s familiarity with exponential, logarithmic, and trigonometric expressions to achieve proficiency in analyzing the local and global behavior of functions involving such expressions. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 105L. Pre-Calculus II Lab (1)

*Prerequisite: Required for all conditionally prepared students enrolled in MATH 105.* All students in MATH 105 are encouraged to enroll in this course. This is a Credit/No Credit hybrid enrichment laboratory for students in MATH 105. This course will include a self-paced, modular online component. 3 hours lab per week. (Credit/No Credit only)

### MATH 106. Mathematical Foundations for Non-Calculus Physics (5)

*Prerequisites: Listed in Table 1.* Mathematics applicable to problems in non-calculus based physics. Sets, inequalities; functions and graphs: polynomial, rational, exponential, logarithmic, trigonometric; introduction to vectors, angular velocity, and parametric equations. This course is not open to students who have credit in MATH 105 or MATH 255A. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 131. Mathematical Ideas (3)

*Prerequisite: Multiple Measures Placement in GE-level Mathematics, or credit in MATH 093 or MATH 196QR or MATH 196S or equivalent. *General Education course intended to acquaint the student with basic mathematical ideas. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 140. Introductory Statistics (4)

*Prerequisite: Multiple Measures Placement in GE-level Mathematics, or credit in MATH 093 or MATH 196QR or MATH 196S or equivalent. *Methods for displaying, describing and producing data. Normal distribution. Correlation and regression. Sampling distributions and probability. Statistical inference for means and proportions. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 150A. Calculus I (5)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared or who transfer the equivalent of MATH 105 or both MATH 102 and MATH 104 must have credit for or concurrently enroll in MATH 150AL.* Limits, derivatives, and applications of differentiation. Definite and indefinite integrals. The fundamental theorem of calculus and applications of integration. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 150AL. Calculus I Laboratory (1)

*Prerequisite: Required for all conditionally prepared students enrolled in MATH 150A or MATH 255A.* All students in MATH 150A and MATH 255A are encouraged to enroll in this course. This is a Credit/No Credit hybrid enrichment laboratory for students in MATH 150A and MATH 255A. This course will include a self-paced, modular online component. 3 hours lab per week. (Credit/No Credit only)

### MATH 150B. Calculus II (5)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared must have credit for or concurrently enroll in MATH 150BL.* Techniques of integration and improper integrals. Sequences and series. Power series and Taylor polynomials. Parametric and polar coordinates. Vectors and solid geometry.

### MATH 150BL. Calculus II Laboratory (1)

*Prerequisite: Required for all conditionally prepared students enrolled in MATH 150B or MATH 255B.* All students in MATH 150B and MATH 255B are encouraged to enroll in this course. This is a Credit/No Credit hybrid enrichment laboratory for students in MATH 150B and MATH 255B. This course will include a self-paced, modular online component. 3 hours lab per week. (Credit/No Credit only)

### MATH 210. Basic Number Concepts (3)

*Prerequisites: Multiple Measures Placement in GE-level Mathematics, or credit in MATH 093 or MATH 196QR or MATH 196S or equivalent.* Language of sets, systems of numeration, nature of numbers and fundamentals of operations, relations and functions, domain of integers, and field of rational and real numbers. Designed primarily for students intending to teach in elementary or junior high school. Not available for credit toward the major or minor in Mathematics.

### MATH 211. Statistics and Probability for Elementary and Middle School Teachers (3)

*Prerequisite: Completion of MATH 210 with a grade of “C” or better.* Univariate and bivariate data analysis; probability and probability distributions; design of studies and concepts of inferential statistics; applications to procedures used to evaluate teaching and learning. Not available for credit toward a Mathematics major or minor.

### MATH 250. Calculus III (3)

*Prerequisite: Completion of MATH 150B with a grade of “C” or better. *Continuation of MATH 150B. Solid analytic geometry and space curves. Partial differentiation and applications. Multiple integrals. Line integrals and independence of path. Vector calculus including the divergence theorem and Stokes’ theorem.

### MATH 255A. Calculus for the Life Sciences I (3)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared must have credit for or concurrently enroll in MATH 255AL. Knowledge of trigonometry is assumed.* First semester of a brief course in calculus. Topics include calculus of functions of one real variable, techniques of differentiation, applications to graphing, optimization problems, and an introduction to integration. Applications to life sciences are emphasized. Not open for credit to students who have successfully completed MATH 150A. (Available for General Education, Basic Skills B4 Mathematics and Quantitative Reasoning.)

### MATH 255AL. Calculus for the Life Sciences I Lab (1)

*Required for all conditionally prepared students enrolled in MATH 255A*. All students in MATH 255A are encouraged to enroll in this lab. This course will include a self-paced, modular outline component. 3 lab hours per week. (Credit/No Credit only)

### MATH 255B. Calculus for the Life Sciences II (3)

*Prerequisites: Listed in Table 1. Students who are conditionally prepared must have credit for or concurrently enroll in MATH 255BL.* Second semester of a brief course in calculus. Topics include techniques of integration, introduction to differential equations, applications of calculus in probability, elements of multivariable calculus, and linear algebra. Applications to life sciences are emphasized. Not open for students who have successfully completed MATH 150AB.

### MATH 255BL. Calculus for the Life Sciences II Lab (1)

*Required for all conditionally prepared students enrolled in MATH 255B*. All students in MATH 255B are encouraged to enroll in this course. This course will include a self-paced, modular online component. 3 lab hours per week. (Credit/No Credit only)

### MATH 262. Introduction to Linear Algebra (3)

*Prerequisite: MATH 150B.* Systems of linear equations, matrices, determinants, eigenvalues, vector spaces and linear transformations, as well as introduction to inner products on Rn and spectral theorem for symmetric matrices.

### MATH 280. Applied Differential Equations (3)

*Prerequisite: Completion of MATH 150B with a grade of “C” or better. Recommended Corequisite or Preparatory: MATH 250. *First order equations. Explicit solution methods, existence and uniqueness for initial value problems. Higher order linear equations. Undetermined coefficients and variation of parameters. Laplace transforms and transform solution methods. Linear first-order systems. Emphasis on engineering applications. Not available for students majoring in Mathematics.

### MATH 310. Basic Concepts of Geometry (3)

*Prerequisite: Completion of MATH 210 with a grade of “C” or better. Second course for students intending to teach in elementary or junior high school.* Geometry as a system; congruence and similarity through construction with straightedge and compass; transformational geometry; the nature of measurement, precision and accuracy. Not available for credit toward the major or minor in Mathematics.

### MATH 310L. Basic Concepts of Geometry Lab (1)

*Recommended Corequisite or Preparatory: MATH 310.* Problem solving using models and simulation in mathematics appropriate for the elementary-school classroom. 2 hours of activities per week. (Credit/No Credit only)

### MATH 311. Basic Geometric Concepts (3)

*Prerequisites: Completion of MATH 210 and MATH 310 with grades of “C” or better, or instructor consent.* Continuation of the investigation of elementary geometry begun in MATH 310. Topics selected from topology, motion geometry, metric geometry, geometry as a mathematical system, absolute geometry, Euclidean geometry and non-Euclidean geometry. Not available for credit toward the Mathematics major or minor.

### MATH 312. Basic Algebraic Concepts (3)

*Prerequisites: Completion of MATH 210 and MATH 310 with grades of “C” or better, or instructor consent.* Topics selected from: abstract algebra and applied algebra using elementary mathematical models. Not available for credit toward the Mathematics major or minor.

### MATH 320. Foundations of Higher Mathematics (3)

*Prerequisite: MATH 150B.* The goal of this course is to help students transition from a primarily computational mode of doing mathematics to a more conceptual mode. The emphasis will be on proofs, which are taught in the context of elementary number theory, combinatorics and analysis; the language of sets, relations, order, equivalence classes, functions and cardinality is introduced. Students are expected to write large numbers of proofs and communicate mathematical ideas clearly.

### MATH 326. Discrete Mathematics (3)

*Prerequisites: ECE 320 or PHIL 230; MATH 150B.* Propositional calculus, predicate calculus, set algebra, relations, functions, mappings, fields and number systems.

### MATH 331. Mathematical Explorations (3)

*Prerequisites: Multiple Measures Placement in GE-level Mathematics, or credit in MATH 093 or MATH 196QR or MATH 196S. Completion of the lower division writing requirement; Upper division standing.* A course designed to give students an appreciation of the diversity of mathematics and the spirit in which it is employed in various applications. The character and origin of key topics from different branches of mathematics are explored. The contributions of various cultures to the field are studied, along with the use of mathematical models for physical problems. The development is conceptual rather than axiomatic, and includes several supervised reading and writing assignments. One significant writing assignment is required. Strongly recommended for prospective teachers in all fields. Not available for credit toward the major or minor in Mathematics. (Available for General Education, B5 Scientific Inquiry and Quantitative Reasoning.)

### MATH 340. Introductory Probability (3)

*Prerequisite: MATH 150B.* Sample spaces, probability rules, independence, conditional probability, Bayes Theorem, discrete and continuous random variables and distributions (e.g. binomial, Poisson, geometric, normal, exponential, uniform), expectation, moment generating functions, joint distributions and central limit theorem.

### MATH 341. Applied Statistics I (3)

*Prerequisite: MATH 150B.* Introduction to the practice of statistics, emphasizing the role of probability. Includes basic probability, discrete and continuous probability distributions, expectation and variance, sample surveys and experiments, displaying and summarizing data, sampling distributions, central limit theorem, inference for proportions, chi-square test and least squares regression. Mathematics majors who are not in the Secondary Teaching option may not receive credit for both MATH 340 and MATH 341.

### MATH 351. Differential Equations (3)

*Prerequisites: Completion of MATH 250 and MATH 262 with a grade of “C” or better. Not open to students who have credit for MATH 280.* First-order equations and explicit solution methods. The Picard-Lindelöf existence and uniqueness theorem. Higher order linear equations. Undetermined coefficients and variation of parameters. Power series solutions. Linear systems and eigenvector methods. Linearization and solution sketching for autonomous systems.

### MATH 360. Abstract Algebra I (3)

*Prerequisites: MATH 262, MATH 320.* Survey course in abstract algebra. Introduction to groups, rings, fields and vector spaces.

### MATH 366. Combinatorics (3)

*Corequisite: MATH 320 or MATH 326.* This is a one-semester introduction to combinatorics. Topics include enumerative combinatorics (inclusion-exclusion, generating functions, Polya’s Theorem, etc.) and combinatorial structures (graphs, designs, etc.).

### MATH 370. Foundations of Geometry (3)

*Prerequisite or Corequisite: MATH 320.* One of the goals of this course is to help students write rigorous proofs of results of plane Euclidean geometry. It is also expected that students visualize and develop geometric intuition through the use of dynamic geometry software. The content includes history, axiomatic structure and theorems of plane Euclidean geometry, geometric transformations of the plane, rigid motions, similarities, and inversion, coordinate geometry and an introduction to non-Euclidean geometries.

### MATH 382/L. Introduction to Scientific Computing and Lab (2/1)

*Corequisite: MATH 262.* This course gives students an introduction to basic numerical techniques and to programming using some of the common software packages used in mathematics. Students apply these techniques in projects from different branches of mathematics. (This course does not replace a rigorous course in numerical analysis.) 2 hours lecture, 2 hours lab.

### MATH 391. Field Experience in the Mathematics of the Public Schools (2)

*Prerequisites: Multiple Subject Candidates—MATH 210 and MATH 310 or corequisite with MATH 310. Single Subject Candidates—MATH 150A, MATH 150B; Junior standing. *Field experience course designed to give the prospective teacher an appreciation of a quality mathematics program in public schools. Requirements include 45 hours of participation in an assigned school and regular group meetings to discuss the classroom experience. (Credit/No Credit only)

### MATH 440A. Mathematical Statistics I (3)

*Prerequisites: MATH 262, MATH 340.* Point estimation, bias and mean squared error, optimality theory for estimates, maximum likelihood estimation, confidence intervals, test of hypotheses, power, and optimality theory for tests. Available for graduate credit.

### MATH 440B. Mathematical Statistics II (3)

*Prerequisite: MATH 440A.* Chi-square goodness of fit tests, simple and multiple linear regression, 1- and 2-way analysis of variance, and statistical analysis using the computer. Available for graduate credit.

### MATH 442A-Z. Topics in Mathematical Statistics (3)

*Prerequisite: MATH 340 or MATH 440A.* Topics selected from statistics and/or probability, such as nonparametric statistics, multivariate statistics, experimental design, decision theory and advanced probability theory.

### MATH 450A. Advanced Calculus I (3)

*Prerequisite: MATH 320.* Topics include the real number system, sequences and series of numbers, limits and continuity of functions, differentiation and Riemann integration of functions of one real variable.

### MATH 450B. Advanced Calculus II (3)

*Prerequisite: MATH 450A.* Topics include sequences and series of functions, the Heine-Borel Theorem, continuous functions, differentiation of multi-variable functions, and theInverse and Implicit Function Theorems. Available for graduate credit.

### MATH 455. Complex Variables (3)

*Prerequisite: MATH 450A.* Complex numbers, analytic functions, complex integration, Cauchy’s Theorem, power series, calculus of residues and conformal mappings. Available for graduate credit.

### MATH 460. Abstract Algebra II (3)

*Prerequisite: MATH 360.* Second course in abstract algebra. Group theory, rings and modules, and field extensions. Available for graduate credit.

### MATH 462. Advanced Linear Algebra (3)

*Prerequisites: MATH 262, MATH 320. Recommended Corequisite or Preparatory MATH 360.* This course covers vector spaces and linear transformations from a more theoretical perspective than that covered in MATH 262. The course begins with a review of abstract vector spaces, including the invariance of dimension of a finite dimensional vector space, the correspondence between linear transformations and matrices, similarity, as well as the definition and basic properties of determinants. The concepts of eigenvalues, eigenvectors, and diagonalization will be reviewed. The course will also cover real and complex inner product spaces, Hermitian operators, the spectral theorem, minimal and characteristic polynomials, the Cayley Hamilton theorem, and the Jordan canonical form.

### MATH 463. Number Theory (3)

*Prerequisite: MATH 320. Recommended Corequisite or Preparatory: MATH 360.* Euclidean algorithm and the unique factorization theorem, congruences, primitive roots and indices, quadratic residues and the law of quadratic reciprocity, and distribution of primes. Available for graduate credit.

### MATH 480. Partial Differential Equations (3)

*Prerequisite: MATH 351 or MATH 280.* Orthogonal functions, Laplaces equation, Poissons equation, Bessels equation, self-adjoint operators, Sturm-Liouville theory, Fourier series, separation of variables applied to the heat equation and wave equation, nonhomogeneous problems, Greens functions for time-independent problems, and infinite domain problems. Available for graduate credit.

### MATH 481A. Numerical Analysis (3)

*Prerequisites: COMP 106/L or COMP 110/L; MATH 262.* Techniques of applied mathematics, solution of equations, interpolation, numerical integration and numerical solution of differential equations. Available for graduate credit.

### MATH 482. Combinatorial Algorithms (3)

*Prerequisites: MATH 150B, MATH 262; Some computer programming experience.* Computer-oriented study of seminumerical and non-numerical algorithms. Sorting, tree searching, generation of combinatorial structures, algorithm proof techniques, best algorithms and programming complexity. Available for graduate credit.

### MATH 483. Mathematical Modeling (3)

*Prerequisites: MATH 340; MATH 351.* Applications of mathematical techniques to solve selected problems in ecology, biology, economics, finance, social sciences, life sciences, physical sciences and engineering. Models discussed include deterministic, stochastic, optimization, static and dynamic ones. Emphasis is placed on the initial phase of building mathematical models and the final phase of interpreting the solutions in terms of real-life applications.

### MATH 490. Capstone Course (3)

*Prerequisite: Senior standing. *A course where prospective teachers see high-school level mathematics from a more advanced perspective, where there is considerably more emphasis on issues of pedagogy than in other content courses, and where students will see connections between the mathematics they have learned and some of the activities that they will themselves be engaged in as teachers. MATH 490 is required for the Secondary Teaching option, but a student may choose, in consultation with his or her advisor, to take the course a second time as an elective.

### MATH 493. Undergraduate Seminar in Mathematics (3)

*Prerequisite: Junior standing in the major.* Students will study current topics in mathematical and/or statistical literature and will prepare written summaries and give oral presentations to the class. Students will be active participants in all seminars by asking questions and providing written critiques and summaries of the presentations of other students.

### MATH 494. Practical Experience in Mathematics (3)

*Prerequisite: Junior standing in the major.* Students will gain practical experience in the profession by either participating in an internship doing mathematical/statistical work at an outside organization or by doing directed research within the department. All students are expected to work a minimum of 8 hours per week on this assignment and meet with the course instructor on a regular basis. All students are required to produce a written report on their work at the end of the semester. Students will give oral reports to the department and their peers.

### MATH 496A-Z. Experimental Topics in Modern Mathematics (3)

*Prerequisites: Senior standing and instructor consent.*

### MATH 499A-C. Independent Study (1-3)

See Independent Study under courses of study.

### MATH 501. Topology (3)

*Prerequisite: MATH 450A.* Metric spaces, topological spaces, compactness, completeness and connectedness. Introduction to function spaces, with emphasis on the uniform topology.

### MATH 540. Regression Analysis (3)

*Prerequisite: MATH 440A.* General linear model in matrix form, simple and multiple regression analysis, transformations, variable selection, multicollinearity, analysis of variance, robust regression, logistic regression, principal components and factor analysis. Statistical software utilized.

### MATH 542A-D. Probability and Statistics (3-3-3-3)

*Prerequisite: MATH 440A or consent of instructor.* This course will cover topics in probability and statistics not covered elsewhere in the program. Part A is usually devoted to multivariate statistics, Part B to stochastic processes, and Part C to probability theory. Part D is left to a topic chosen by the individual instructor.

### MATH 550. Calculus on Manifolds (3)

*Prerequisite: MATH 450B.* Integration of functions of several variables. Differential forms and differential manifolds, Line integrals, integration on manifolds, Stoke’s Theorem and Poincare’s Lemma.

### MATH 552. Real Analysis (3)

*Prerequisite: MATH 501.* Introduction to measure theory and Lebesgue integration, and their application to probability theory. Monotone and dominated convergence theorems, Fubini’s theorem, Fourier analysis and Banach spaces.

### MATH 560. Abstract Algebra III (3)

*Prerequisite: MATH 460.* Graduate course in abstract algebra. Group theory, Galois theory and other topics.

### MATH 570. Differential Geometry (3)

*Prerequisite: MATH 450B.* The local theory of regular curves in R3 and Frenet formulas. Regular surfaces in R3, the first and second fundamental forms, Gaussian and mean curvatures, and the Egregium Gauss theorem. Geodesics and the Gauss-Bonnet theorem.

### MATH 581. Numerical Methods for Linear Systems (3)

*Prerequisite: MATH 462.* Methods for solving large linear problems and eigenvalue problems are presented at an advanced level. Direct methods such as LU factorization, Cholesky factorization and the Least Squares method, and Iterative methods, such as the Jacobi, Gauss-Seidel, SOR and conjugate Gradient methods, are discussed in detail. Eigenvalue problems are solved via power iteration, the QR method and the Jacobi method.

### MATH 582A-D. Topics in Numerical Analysis (3-3-3-3)

*Prerequisite: MATH 581 or consent of instructor.* The course will cover topics in numerical analysis which are important in many applications and which are not covered elsewhere in the program. Part A usually covers numerical methods in optimization, Part B covers numerical methods for ordinary differential equations, and Part C covers numerical solution of partial differential equations. Part D covers a subject chosen by the instructor.

### MATH 589. Seminar in Mathematics (1)

*Prerequisite: Senior or graduate standing in the Mathematics Department.* Students will read about advanced topics in the recent literature in Mathematics and report on them in a lecture. This course may be taken up to two times with the consent of the advisor. (Credit/No Credit only)

### MATH 592A-D. Topics in Applied Mathematics (3-3-3-3)

*Prerequisite: MATH 552 or consent of instructor.* This course is devoted to a variety of important topics in applied mathematics that are not covered elsewhere in the program. In particular, Part A will cover the mathematical theory of partial differential equations, Part B covers mathematical optimization and operations research, and Part C covers mathematical biology. The topic of Part D is left to the individual instructor.

### MATH 595A-Z. Experimental Topics (1-3)

*Prerequisite: Consent of instructor.* Specialized topics from a concentrated field of current interest presented at an advanced level.

### MATH 625. Advanced Mathematical Modeling (3)

Selected problems in ecology, biology, economics, finance, social sciences, life sciences, physical sciences and engineering are used to develop advanced techniques of mathematical modeling.

### MATH 651A-C. Advanced Topics in Analysis, Geometry and Topology (3-3-3)

*Prerequisite: Consent of instructor.* Advanced topics not covered in the previous classes on the subject. Part A covers topics in analysis, Part B covers topics in geometry, and Part C covers topics in topology. May be repeated with the consent of the advisor.

### MATH 655. Complex Analysis (3)

*Prerequisites: MATH 501, MATH 455.* Topics covered include the general Cauchy theorem, power series and analytic continuation, series and product expansions, conformal mapping and the Dirichlet problem.

### MATH 661A-C. Advanced Topics in Algebra, Number Theory and Discrete Mathematics (3-3-3)

*Prerequisite: Consent of instructor.* Advanced topics not covered in the previous classes on the subject. Part A covers topics in algebra, Part B covers topics in number theory, and Part C covers other topics in discrete mathematics. May be repeated with the consent of the advisor.

### MATH 680A/B. Applied Functional Analysis (3-3)

*Prerequisites: MATH 501, MATH 552.* This two-semester sequence gives an introduction to Banach and Hilbert spaces and their applications. Fixed Point Theorems and their applications to differential and integral equations and variational principles. Adjoint and self-adjoint operators and spectral theory of linear operators. MATH 680A is a prerequisite for MATH 680B.

### MATH 697A-C. Directed Comprehensive Studies (1-3)

No course description.

### MATH 698A-C. Thesis or Graduate Project (1-3)

No course description.

### MATH 699A-F. Independent Study (1-6)

See Independent Study under courses of study.