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Mathematics

Professors. HuaiDong Cao, Ph.D. (Princeton), A. Everett Pitcher Professor; Donald M. Davis, Ph.D. (Stanford); Vladimir Dobric, Ph.D. (Zagreb, Croatia); Bennett Eisenberg, Ph.D. (M.I.T); WeiMin Huang, Ph.D. (Rochester), chair; Garth Isaak, Ph.D. (Rutgers); Terrence Napier, Ph.D. (Chicago); Eric P. Salathe, Ph.D. (Brown), director of the Institute for Biomedical Engineering and Mathematical Biology; Lee J. Stanley, Ph.D. (Berkeley); Steven H. Weintraub, Ph.D. (Princeton); Joseph E. Yukich, Ph.D. (M.I.T.).

Associate professors. Bruce A. Dodson, Ph.D. (S.U.N.Y. at Stony Brook); David L. Johnson, Ph.D. (M.I.T.); Mark A. Skandera, Ph.D. (M.I.T.); Xiaofeng Sun, Ph.D. (Stanford); Susan Szczepanski, Ph.D. (Rutgers); Ramamirthan Venkataraman, Ph.D. (Brown); Linghai Zhang, Ph.D. (Ohio State).

Assistant professors. Soutir Bandyopadhyay, Ph.D. (Texas A& M); Daniel Conus, Ph.D. (EPFL, Switzerland); Robert Neel, Ph.D. (Harvard); PingShi Wu, Ph.D. (Davis).

Lecturer. Vincent E. Coll, Ph.D. (U. Penn).

Adjunct professor. Howard Fegan, Ph.D. (Oxford).

Emeriti. Theodore Hailperin, Ph.D. (Cornell); Samir A. Khabbaz, Ph.D. (Kansas); Jerry P. King, Ph.D. (Kentucky); Clifford S. Queen, Ph.D. (Ohio State); Murray Schechter, Ph.D. (N.Y.U.); Andrew Snyder, Ph.D. (Lehigh); Albert Wilansky, Ph.D. (Brown).

Mathematics is a subject of great intrinsic power and beauty. It is the universal language of science, and is essential for a clear and complete understanding of virtually all phenomena. Mathematical training prepares a student to express and analyze problems and relationships in a logical manner in a wide variety of disciplines including the physical, engineering, social, biological, and medical sciences, business, and pure mathematics itself. This is a principal reason behind the perpetual need and demand for mathematicians in education, research centers, government, and industry.

The department offers three major programs leading to the degrees of bachelor of arts in mathematics, bachelor of science in mathematics (with a general mathematics and an applied mathematics option), and bachelor of science in statistics. It also offers several minor programs for undergraduates.

At the graduate level, it offers programs leading to the degrees of master of science in mathematics, master of science in applied mathematics, master of science in statistics, doctor of philosophy in mathematics, and doctor of philosophy in applied mathematics.

The Division of Applied Mathematics and Statistics is a part of the Department of Mathematics.

Calculus Sequences

Many degree programs throughout the university include a mathematics requirement consisting of a sequence in calculus. The Department of Mathematics offers three calculus sequences: MATH 21, 22, 23; MATH 31, 32, 33; MATH 51, 52.

The MATH 21, 22, 23 sequence is a systematic development of calculus. Most students of mathematics, science, engineering, and business will take some or all of this sequence.

As an honors sequence, the MATH 31, 32, 33 sequence covers essentially the same material but in greater depth and with more attention to rigor and proof. This sequence should be considered by students who have demonstrated exceptional ability in mathematics. Students who are contemplating a major in mathematics are strongly encouraged to consider this sequence.

The MATH 51, 52 sequence is a survey of calculus. Math 81 is a survey course with business applications. This sequence is not sufficient preparation for most subsequent mathematics courses. Students contemplating further study in mathematics should consider MATH 21, 22 instead.

MATH 75, 76 is a two-semester sequence that substitutes for MATH 21, covering the same material but at a slower pace.

The MATH 31, 32, 33 sequence will be accepted in place of the other two sequences. MATH 21, 22 will be accepted in place of MATH 51, 52. Credit will be awarded for only one course in each of the following groups: 21, 75/76, 31, 51 and 81; 22, 32, and 52; 23 and 33. If two courses in the same group are taken, credit will be awarded for the more advanced course; 3x is the most advanced, while 5x is the least advanced.

Undergraduate Degree Programs

The Department of Mathematics offers degree programs in Mathematics and Statistics. These programs have the flexibility and versatility needed to prepare students for a wide variety of careers in government, industry, research and education.

Students in the degree programs in mathematics must satisfy three types of requirements beyond those required by the college: Core Mathematics Requirements, Major Requirements and General Electives. The Core Mathematics Requirement ensures a common core of knowledge appropriate for students in each program. The Major Program Electives consist of courses with specific mathematical or statistical content chosen by the student in consultation with the major advisor to complement the student's interest and career aspirations. With these further breadth and greater depth of knowledge are achieved. The General Electives consist of additional courses chosen from among those offered by the university faculty. Students can use these electives to pursue interests beyond the major, or may use these to expand upon the basic requirements of the degree program. Students are strongly encouraged to use some of these electives to earn a minor in another discipline.

Students in the degree program in statistics must satisfy four types of requirements beyond those required by the college: Required Major Courses, Major Electives, Professional Electives and Free Electives.

Each student is provided a faculty advisor to guide an individual program and supervise the selection of electives.

B.A. with a major in Mathematics

The B.A. program in mathematics emphasizes fundamental principles as well as the mastery of techniques required for the effective use of mathematics. The program provides a solid foundation for those who want to pursue a mathematically oriented career or advanced study in any mathematically oriented field.

Requirements:

College Distribution Requirements excluding mathematics (31-34 credits)

Core Mathematics Requirements (32-35 credits)

Chosen in consultation with faculty advisor.

This program requires a total of 120 credit hours.

A student must achieve an average of 2.0 or higher in major courses.

B.S. in Mathematics

The BS in Mathematics program provides a more extensive and intensive study of mathematics and its applications. Students can pursue the General Mathematics Option or the Applied Mathematics Option. These programs are especially recommended for students intending to pursue advanced study in mathematics or applied mathematics. The General Mathematics Option is recommended for students who wish to pursue mathematics either by itself or in combination with a related field (e.g., physics, computer science or economics). The Applied Mathematics Option provides a broad background in the major areas of applicable mathematics.

General Mathematics Option

Requirements:

College Distribution Requirements excluding mathematics (31-34 credits)

Core Mathematics Requirements (32-34 credits)

This program requires a total of 120 credit hours. A student must achieve an average of 2.0 or higher in major courses.

Applied Mathematics Option

Requirements:

College Distribution Requirements excluding mathematics (31-34 credits)

Core Mathematics Requirements (32-34 credits)

In consultation with the major advisor, a student must establish a concentration in a particular area of applied mathematics. The courses chosen must have specific mathematical or statistical content and together constitute a coherent program. At most two courses may be taken outside the Department of Mathematics. Students, in consultation with the major advisor, can design a concentration which reflects a particular area of interest or choose to pursue one of the following:

Concentration in Applied Analysis:

Electives must include MATH 230, MATH 322 and MATH 341

Concentration in Discrete Mathematics and Theoretical Computer Science:

Electives must include at least three courses selected from MATH 305, MATH 311, MATH 329, MATH 340

Concentration in Probability and Statistics:

Electives must include at least three courses selected from MATH 309, MATH 310, MATH 312, MATH 334, MATH 338

This program requires a total of 120 credit hours.

A student must achieve an average of 2.0 or higher in major courses.

B.S. in Statistics

Statistics provides a body of principles for designing the process of data collection, for summarizing and interpreting data, and for drawing valid conclusions from data. It thus forms a fundamental tool in the natural and social sciences as well as business, medicine, and other areas of research. Mathematical principles, especially probability theory, underlie all statistical analyses.

College and university requirements excluding Mathematics (3134 credit hours)

Required Major courses (45-47 credit hours)

Two approved computing science courses or one approved computer science course and Engineering 1 (6) or (7).

Major Electives (12 credit hours)

At least three courses with specific mathematical or statistical content chosen with the approval of the faculty advisor.

Professional Electives (21 credit hours)

Courses selected from two or three fields of application of statistics and probability.

Free Electives (6-11 credits)

This program requires a total of 120 credit hours.

A student must achieve an average of 2.0 or higher in major courses.

Departmental Honors

Students may earn departmental honors by writing a thesis during their senior year. Students are accepted into the program during their junior year by the department chairperson. This acceptance is based upon the student's grades and a thesis proposal, which the student must prepare in conjunction with a thesis advisor selected by the student. An oral presentation as well as a written thesis are required for completion of the program.

Minor Programs

The department offers minor programs in different branches of the mathematical sciences. The requirement consists of MATH 23 or 33 and four additional courses shown below for each of the programs. At most one of these five courses in the minor program may also be required in the major program. For substitutions, the student should consult the chairperson.

Minor in Pure Mathematics

MATH 242, 243, 301

MATH 302 or 303 or 307 or 316 or 342

Minor in Applied Mathematics

Three of MATH 205, 208, 230, 231, 242, 320, 322, 323

MATH 341

Minor in Probability and Statistics

MATH 12 or 231

MATH 309

Two of MATH 310, 312, 334, 338

Minor in Actuarial Science

MATH 309, 310 and one of Math 202, 203

ECON 029

ACCT 108 or 151

For information on examinations of actuarial societies, students may consult their minor advisor.

Undergraduate Courses

MATH 0. Preparation for Calculus (2) summer-fall

Intensive review of fundamental concepts in mathematics utilized in calculus, including functions and graphs, exponentials and logarithms, and trigonometry. This course is for students who need to take MATH 51 or 21, but who require remediation in precalculus. In particular, students who fail the MATH 51 Readiness Exam must pass MATH 0 before being admitted to MATH 51. The credits for this course do not count toward graduation, but do count toward GPA and current credit count. Prerequisite: department permission.

MATH 5. Introduction to Mathematical Thought (3) spring

Meaning, content, and methods of mathematical thought illustrated by topics that may be chosen from number theory, abstract algebra, combinatorics, finite or nonEuclidean geometries, game theory, mathematical logic, set theory, topology. (MA)

MATH 9. Introduction to Finite Mathematics (4)

Systems of linear equations, matrices, introduction to linear programming. Sets, counting methods, probability, random variables, introduction to Markov chains. (MA)

MATH 12. Basic Statistics (4) fall/spring

A first course in the basic concepts and methods of statistics with illustrations from the social, behavioral, and biological sciences. Descriptive statistics; frequency distributions, mean and standard deviation, twoway tables, correlation and regression; random sampling, rules of probability, probability distributions and parameters, parameter estimation, confidence intervals, hypothesis testing, statistical significance. (MA) Note: Students may not receive credit for both MATH 12 & ECO 045.

MATH 21. Calculus I (4) fall/spring

Functions and graphs; limits and continuity; derivative, differential, and applications; indefinite and definite integrals; trigonometric, logarithmic, exponential, and hyperbolic functions. (MA)

MATH 22. Calculus II (4) fall/spring

Applications of integration; techniques of integration; separable differential equations; infinite sequences and series; Taylor's Theorem and other approximations; curves and vectors in the plane. Prerequisite: MATH 21 or MATH 31. (MA)

MATH 23. Calculus III (4) fall/spring

Vectors in space; partial derivatives; Lagrange multipliers; multiple integrals; vector analysis; line integrals; Green's Theorem, Gauss's Theorem. Prerequisite: MATH 22 or MATH 32. (MA)

MATH 31. Honors Calculus I (4) fall

Same topics as in MATH 21, but taught from a more thorough and rigorous point of view. (MA)

MATH 32. Honors Calculus II (4) fall/spring

Same topics as in MATH 22, but taught from a more thorough and rigorous point of view. Prerequisite: MATH 31. (MA)

MATH 33. Honors Calculus III (4) fall/spring

Same topics as in MATH 23, but taught from a more thorough and rigorous point of view. Prerequisite: MATH 32. (MA)

MATH 43. Survey of Linear Algebra (3) fall

Matrices, vectors, vector spaces and mathematical systems, special kinds of matrices, elementary matrix transformations, systems of linear equations, convex sets, introduction to linear programming. (MA).

MATH 51. Survey of Calculus I (4) fall

Limits. The derivative and applications to extrema, approximation, and related rates. Exponential and logarithm functions, growth and decay. Integration. Trigonometric functions and related derivatives and integrals. (MA)

MATH 52. Survey of Calculus II (3) spring

Techniques of integration. Differential equations. Probability and calculus. Partial derivatives and extrema. Multiple integrals and applications. Prerequisite: MATH 21 or 31 or 51 or 81. (MA)

MATH 75. Calculus I, Part A (2) fall

Covers the same material as the first half of MATH 21. Meets three hours per week, allowing more class time for each topic than does MATH 21. (MA)

MATH 76. Calculus I, Part B (2) spring

Continuation of MATH 75, covering the second half of MATH 21. Meets three hours per week. Final exam for this course is similar to the MATH 21 final. Prerequisite: MATH 75. (MA)

MATH 81. Calculus with Business Applications (4) fall-spring

Limits and continuity; exponential, logarithmic and trigonometric functions; derivatives; extrema; approximations; indefinite and definite integrals. Applications with emphasis on business and economics. (MA)

MATH 130. Biostatistics (4)

Elements of statistics and probability with emphasis on biological applications. Statistical analysis of experimental and observational data. Prerequisite: MATH 22 or MATH 52. (ND)

MATH 163. Introductory Seminar (3) spring

An introduction to the discipline of mathematics designed for students considering a major in mathematics. The course will provide an introduction to rigorous mathematical reasoning and will survey some area of mathematics. Topics covered will vary. (MA)

MATH 171. Readings (1-3) fall/spring

Study of a topic in mathematics under individual supervision. Intended for students with specific interests in areas not covered in the listed courses. Prerequisite: permission of the department chair. (MA)

For Advanced Undergraduates and Graduate Students

Courses listed as (3-4) are 3 credits for graduate students and 4 credits for undergraduates. The extra credit will frequently involve some extra workshops or projects.

MATH 201. Problem Solving (1) fall

Practice in solving problems from mathematical contests using a variety of elementary techniques. (MA)

MATH 202. Actuarial Exam I (1) spring

Preparation for the first actuarial exam – probability. Problems in calculus and probability with insurance applications. Prerequisites: MATH 23 and 231. (MA)

MATH 203. Actuarial Exam II – Financial Mathematics (1) spring

Preparation for the second actuarial exam - financial mathematics. Mathematics of interest and investments, interest rate measurement, present value, annuities, loan repayment schemes, bond valuation. Practice in solving problems from past exams. Prerequisite: Math 22. (MA)

MATH 205. Linear Methods (3) fall/spring

Linear differential equations and applications; matrices and systems of linear equations; vector spaces; eigenvalues and application to linear systems of differential equations. Prerequisite: MATH 22 or 32. (MA)

MATH 208. Complex Variables (3) fall/spring

Functions of a complex variable; calculus of residues; contour integration; applications to conformal mapping and Laplace transforms. Prerequisite: MATH 23 or MATH 33. (MA)

MATH 214 (PHIL 214). Topics in Philosophical Logic (4)

The course materials are drawn from the many topics and figures in philosophical logic, widely construed, that are not covered by the other logic courses. Examples of such topics are the many systems of nonclassical logic, truth theory, the impact of incompleteness and undecidability results on philosophy, and the foundational projects of many philosophers/mathematicians. The topic may also concern the work of a certain important figure in the history of philosophical logic. Prerequisite: Permission of the instructor. (MA)

MATH 229. Geometry (3-4)

Discussion of geometry as an axiomatic system. Euclid's postulates. History of and equivalent versions of Euclid's fifth postulate. Finite projective geometries. NonEuclidean geometries based upon negation of the fifth postulate: Geometry on the sphere; Hyperbolic and elliptic geometries. Examination of the concepts of “straight”, angle, parallel, symmetry and duality in each of these geometries. Applications of the different geometries will be considered. Prerequisite: Math 205 or Math 242 or permission of instructor. (MA)

MATH 230. Numerical Methods (3)

Representation of numbers and rounding error; numerical solution of equations; quadrature; polynomial and spline interpolation; numerical solution of initial and boundary value problems. Prerequisites: MATH 205 (previously or concurrently) and knowledge of either FORTRAN or PASCAL. (MA)

MATH 231. Probability and Statistics (3) fall/spring

Probability and distribution of random variables; populations and random sampling; chisquare and t distributions; estimation and tests of hypotheses; correlation and regression theory of two variables. Prerequisite: MATH 22 or MATH 32 or MATH 52. (MA)

MATH 234. Fractal Geometry (3-4)

Metric spaces and iterated function systems; various types of fractal dimension; Julia and Mandelbrot sets. Other topics such as chaos may be included. Small amount of computer use. Prerequisite: MATH 23 or MATH 33. (MA)

MATH 242. Linear Algebra (3-4) fall

Solution of systems of linear equations, matrices, vector spaces, bases, linear transformations, eigenvalues, eigenvectors, additional topics as time permits. Prerequisite: MATH 22 or MATH 32 or permission of instructor. (MA)

MATH 243. Algebra (3-4) spring

Introduction to basic concepts of modern algebra: groups, rings, and fields. (MA)

MATH 261. (CSE 261) Discrete Structures (3) fall-spring

Topics in discrete mathematical structures chosen for their applicability to computer science and engineering. Sets, propositions, induction, recursion; combinatorics; binary relations and functions; ordering, lattices and Boolean algebra; graphs and trees; groups and homomorphisms. Prerequisite: MATH 21. (MA)

MATH 271. Readings (13) fall/spring

Study of a topic in mathematics under individual supervision. Intended for students with specific interests in areas not covered in the listed courses. Prerequisite: permission of the department chair. May be repeated for credit. (MA)

MATH 291. Undergraduate Research (1-4) fall/spring

Research in mathematics or statistics under the direction of a faculty member. Prerequisite: permission of the department chair. May be repeated for credit. (ND)

MATH 301. Principles of Analysis I (3-4) fall

Existence of limits, continuity and uniform continuity; HeineBorel Theorem; existence of extreme values; mean value theorem and applications; conditions for the existence of the Riemann integral; absolute and uniform convergence; emphasis on theoretical material from the calculus of one variable. Prerequisite: MATH 23 or MATH 33. (MA)

MATH 302. Principles of Analysis II (3-4) spring

Continuation of MATH 301. Functions of several variables; the implicit function theorem, and further topics with applications to analysis and geometry. Prerequisite: MATH 301. (MA)

MATH 303. (Phil 303) Mathematical Logic (3-4) fall

Detailed proofs are given for the basic mathematical results relating the syntax and semantics of firstorder logic (predicate logic): the Soundness and Completeness (and Compactness) Theorems, followed by a brief exposition of the celebrated limitative results of Gödel, Turing, and Church on incompleteness and undecidability. The material is conceptually rigorous and mathematically mature; the necessary background is a certain degree of mathematical sophistication or a basic knowledge of symbolic logic. Prerequisite: Permission of instructor. (MA)

MATH 304 (PHIL 304). Axiomatic Set Theory (3-4) fall

A development of set theory from axioms; relations and functions; ordinal and cardinal arithmetic; recursion theorem; axiom of choice; independence questions. Prerequisite: permission of instructor. (MA)

MATH 305. Enumerative Combinatorics (3)

An introduction to basic theoretical results and techniques of enumerative combinatorics such as combinatorial identities, generating functions, inclusionexclusion, recurrence relations, bijective proofs and permutations. Additional topics will be covered as time permits. Prerequisite: MATH 163 or MATH/CSE 261 or MATH 205 or permission of instructor. (MA)

MATH 306. (CHE 306) Introduction to Biomedical Engineering and Mathematical Biology (3)

Study of human physiology, including the cardiovascular, nervous and respiratory systems, and renal physiology. Mathematical analysis of physiological processes, including transport phenomena. Mathematical models of excitation and propagation in nerve. Biomechanics of the skeletal muscle system. Mathematical models in population dynamics and epidemiology. Independent study projects. Prerequisite: MATH 205. (MA)

MATH 307. General Topology I (3-4) fall

An introductory study of topological spaces, including metric spaces, separation and countability axioms, connectedness, compactness, product spaces, quotient spaces, function spaces. Prerequisite: MATH 301. (MA)

MATH 309. Theory of Probability (3) fall

Probabilities of events on discrete and continuous sample spaces; random variables and probability distributions; expectations; transformations; simplest kind of law of large numbers and central limit theorem. The theory is applied to problems in physical and biological sciences. Prerequisite: MATH 23 or MATH 33 or MATH 52. (MA)

MATH 310. Random Processes and Applications (3-4) spring

Theory and applications of stochastic processes. Limit theorems, introduction to random walks, Markov chains, Poisson processes, birth and death processes, and Brownian motion. Applications to financial mathematics, biology, business and engineering. Prerequisite: MATH 309 or MATH 231. (MA)

MATH 311. Graph Theory (3)

An introduction to basic theoretical results and techniques of graph theory such as trees, connectivity, matchings, coloring, planar graphs and Hamiltonicity. Additional topics will be covered as time permits. Prerequisite: MATH 163 or MATH/CSE 261 or MATH 205 or permission of instructor. (MA)

MATH 312. Statistical Computing and Applications (3-4)

Use of statistical computing packages; exploratory data analysis; Monte Carlo methods; randomization and resampling, application and interpretation of a variety of statistical methods in real world problems. Prerequisite: MATH 12 or 231. (MA)

MATH 316. Complex Analysis (3-4)

Concept of analytic function from the points of view of the CauchyRiemann equations, power series, complex integration, and conformal mapping. Prerequisite: MATH 301. (MA)

MATH 320. Ordinary Differential Equations (3-4)

The analytical and geometric theory of ordinary differential equations, including such topics as linear systems, systems in the complex plane, oscillation theory, stability theory, geometric theory of nonlinear systems, finite difference methods, general dynamical systems. Prerequisite: MATH 205 or Math 242 and one of Math 23 or Math 33. (MA)

MATH 321. Topics in Discrete Mathematics (3)

Selected topics in areas of discrete mathematics. May be repeated for credit. Prerequisite: permission of the department chair. (MA)

MATH 322. Methods of Applied Analysis I (3) fall

Fourier series, eigenfunction expansions, SturmLiouville problems, Fourier integrals and their application to partial differential equations; special functions. Emphasis is on a wide variety of formal applications rather than logical development. Prerequisite: MATH 205 or permission of instructor. (MA)

MATH 323. Methods of Applied Analysis II (3) spring

Green's functions; integral equations; variational methods; asymptotic expansions, method of saddle points; calculus of vector fields, exterior differential calculus. Prerequisite: MATH 322. (MA)

MATH 327. Groups and Rings (3-4) fall

An intensive study of the concepts of group theory including the Sylow theorems, and of ring theory including unique factorization domains and polynomial rings. Prerequisite: MATH 243 or permission of instructor. (MA)

MATH 329. Computability Theory (3-4) spring

Core development of classical computability theory: enumeration, index and recursion theorems, various models of computation and Church's Thesis, uncomputability results, introduction to reducibilities and their degrees (in particular, Turing degrees, or degrees of uncomputability), computable operators and their fixed points. (MA)

MATH 331. Differential Geometry of Curves and Surfaces (3)

Local and global differential geometry of curves and sufaces in Euclidean 3space. Frenet formulas for curves, isoperimetric inequality, 4vertex theorem; regular surfaces, first fundamental form, Gauss map, second fundamental form; curvatures for curves and surfaces and their relations; The GaussBonnet theorem. Prerequisites: MATH 23 or MATH 33 and MATH 205, or permission of instructor. (MA)

MATH 334. Mathematical Statistics (3-4) spring

Populations and random sampling; sampling distributions; theory of statistical estimation; criteria and methods of point and interval estimation; theory of testing statistical hypotheses. Prerequisite: MATH 231 or MATH 309. (MA)

MATH 338 (Stat 438). Linear Models in Statistics with Applications (3-4) spring

Least square principles in multiple regression and their interpretations; estimation, hypotheses testing, confidence and prediction intervals, modeling, regression diagnostic, multicollinearity, model selection, analysis of variance and covariance; logistic regression. Introduction to topics in time series analysis such as ARMA, ARCH, and GARCH models. Applications to natural sciences, finance and economics. Use of computer packages. Prerequisite: MATH 12 or 231. (MA)

MATH 340. (CSE 340) Design and Analysis of Algorithms (3) spring

Algorithms for searching, sorting, manipulating graphs and trees, finding shortest paths and minimum spanning trees, scheduling tasks, etc.: proofs of their correctness and analysis of their asymptotic runtime and memory demands. Designing algorithms: recursion, divide-andconquer, greediness, dynamic programming. Limits on algorithm efficiency using elementary NP-completeness theory. Credit will not be given for both MATH 340 (CSE 340) and MATH 441 (CSE 441). Prerequisites: MATH 22 and CSE 261 (MATH 261).

MATH 341. Mathematical Models and Their Formulation (3) spring

Mathematical modeling of engineering and physical systems with examples drawn from diverse disciplines. Emphasis is on building models of real world problems rather than learning mathematical techniques. Prerequisite: MATH 205. (MA)

MATH 342. Number Theory (3-4)

Basic concepts and results in number theory, including such topics as primes, the Euclidean algorithm, Diophantine equations, congruences, quadratic residues, quadratic reciprocity, primitive roots, number-theoretic functions, distribution of primes, Pell’s equation, Fermat’s theorem, partitions. Prerequisite: permission of instructor. (MA)

MATH 343. Introduction to Cryptography (3-4)

Classical elementary cryptography: Caesar cipher, other substitution ciphers, block ciphers, general linear ciphers. Fast random encryption (DES and AES: Advanced Encryption Standard). Public key systems (RSA and discrete logs). Congruences, modular arithmetic, fast exponentiation, polynomials, matrices. Distinction between polynomial time (primality), Subexponential time (factoring) and fully Exponential computation (elliptic curves). Introduction to sieving and distributed computation. Prerequisite: permission of instructor. (MA)

MATH 350. Special Topics (3) fall/spring

A course covering special topics not sufficiently covered in listed courses. Prerequisite: permission of the department chair. May be repeated for credit. (MA)

MATH 371. Readings (13) fall/spring

The study of a topic in mathematics under appropriate supervision, designed for the individual student who has studied extensively and whose interests lie in areas not covered in the listed courses. Prerequisite: permission of the department chair. May be repeated for credit (MA)

MATH 374. Statistical Project (3)

Supervised field project or independent reading in statistics or probability. Prerequisite: permission of the department chair. (MA)

MATH 391. Senior Honors Thesis (3) fall/spring

Independent research under faculty supervision, culminating in a thesis presented for departmental honors. May be repeated once for credit. Prerequisite: permission of the department chair. (MA)

Graduate Programs in Mathematics

The department offers graduate programs leading to the degrees of master of science in mathematics, applied mathematics, or statistics, and the doctor of philosophy in mathematics or applied mathematics.

The Department does not offer a doctorate in statistics. However, students may choose statistics or mathematical statistics as a concentration in the doctor of philosophy programs in mathematics and applied mathematics. The Department is a part of the interdisciplinary program in Analytical Finance. For details on the Master of Science in Analytical Finance see the Interdisciplinary Graduate Study and Research, Analytical Finance section.

To begin graduate work in mathematics a student must present evidence of adequate undergraduate preparation. The undergraduate program should have included a year of advanced calculus, a semester of linear algebra, and a semester of abstract algebra.

M.S. in Mathematics or Applied Mathematics

The master's program requires 30 credit hours of graduate courses with at least 18 hours at the 400 level. With the permission of the chairperson, up to six hours of these courses can be replaced by a thesis. All students in the master's program must also pass a comprehensive examination. The M.S. degree can serve both as a final degree in mathematics or as an appropriate background for the Ph.D. degree.

M.S. in Statistics

This program requires 30 credit hours of graduate courses with at least 18 hours of 400-level STAT or MATH courses. The choice of courses must be approved by the graduate advisor, and up to six hours of coursework may be replaced with a thesis. All students in the program must also pass a comprehensive examination.

The M.S. program in statistics has two tracks. The statistics track has recommended courses MATH 309, STAT 412, 434, and 462; electives STAT 410, 438, and 461; and other possible electives STAT 408 and 409, EDUC 411, I.E. 332, 409, and 410, ECO 460 and 463, CSE 411, and MECH 445. The stochastic modeling track has recommended courses MATH 309 and 401, and STAT 410 and 463; electives MATH 341 and STAT 434, 438, and 464; and other possible electives STAT 408 and 409, MATH 402, 430, 467, and 468, ECO 463, CSE 411, MECH 445, and I.E. 316, 339, 409, 416, and 439.

Ph.D. in Mathematics

The plan of work toward the doctor of philosophy degree will include a comprehensive examination, a qualifying examination, and an advanced topic examination. A language exam may be required at the discretion of the thesis committee. The qualifying examination tests the student’s command of algebra and real analysis. The content of the advanced topic examination is determined by a department committee. A general examination, the doctoral dissertation and its defense complete the work for the Ph.D. degree.

Each candidate's plan of work must be approved by a special committee of the department. A Ph.D. student is required to have 18 credits of approved graduate level course work beyond the master's level. After completion of 18 credits a student is required to take at least one course per academic year other than Math 409, 410, and 499.

Ph.D. in Applied Mathematics

The plan of work toward the doctor of philosophy degree will include a comprehensive examination, a qualifying examination, and an advanced topic examination. A language examination may be required at the discretion of the thesis committee. The qualifying examination tests the student’s command of two areas subject to approval by the department. The content of the advanced topic examination is determined by a department committee. A general examination, the doctoral dissertation and its defense complete the work for the Ph.D. degree.

Each candidate's plan of work must be approved by a special committee of the department. A Ph.D. student is required to have 18 credits of approved graduate level course work beyond the master's level. After completion of 18 credits a student is required to take at least one course per academic year other than Math 409, 410, and 499.

Graduate Courses

MATH 401. Real Analysis I (3) fall

Set theory, real numbers; introduction to measures, Lebesgue measure; integration, general convergence theorems; differentiation, functions of bounded variation, absolute continuity; Lp spaces. Prerequisites: MATH 302 or permission of instructor.

MATH 402. Real Analysis II (3) spring

Metric spaces; introduction to Banach and Hilbert space theory; Fourier series and Fejer operators; general measure and integration theory, RadonNikodym and Riesz representation and theorems; LebesgueStieljtes integral. Prerequisites: MATH 307 and MATH 401.

MATH 403. Topics in Real Analysis (3)

Intensive study of topics in analysis with emphasis on recent developments. Prerequisite: permission of the department chair. May be repeated for credit.

MATH 404. Topics in Mathematical Logic (3)

Intensive study of topics in mathematical logic. Prerequisite: permission of instructor. May be repeated for credit.

MATH 405. Partial Differential Equations I (3) fall

Classification of partial differential equations; methods of characteristics for first order equations; methods for representing solutions of the potential, heat, and wave equations, and properties of the solutions of these equations; maximum principles. Prerequisite: MATH 302 or its equivalent.

MATH 406. Partial Differential Equations II (3) spring

Continuation of MATH 405. Emphasis on second order equations with variable coefficients and systems of first order partial differential equations. Prerequisite: MATH 405.

MATH 408. Algebraic Topology I (3)

Polyhedra; fundamental groups; simplicial and singular homology.

MATH 409. (STAT 409) Mathematics Seminar (16) fall

An intensive study of some field of mathematics not offered in another course. Prerequisite: permission of the department chair.

MATH 410. (STAT 408) Mathematics Seminar (16) spring

Continuation of the field of study in MATH 409 or the intensive study of a different field. Prerequisite: permission of the department chair.

MATH 416. Complex Function Theory (3)

Continuation of MATH 316. Prerequisite: MATH 316 or permission of instructor.

MATH 421. Introduction to Wavelets (3)

Continuous and discrete signals; review of Fourier analysis; discrete wavelets; time frequency spaces; Haar and Walsh systems; multiresolution analysis; Hilbert spaces; quadratic mirror filters; fast wavelet transforms; computer code; applications to filtering, compression, and imaging. Prerequisite: ECE 108, MATH 205, or permission of instructor.

MATH 423. Differential Geometry I (3)

Differential manifolds, tangent vectors and differentials, submanifolds and the implicit function theorem. Lie groups and Lie algebras, homogeneous spaces. Tensor and exterior algebras, tensor fields and differential forms, de Rham cohomology, Stokes' theorem, the Hodge theorem. Prerequisite: MATH 301, 302, or MATH 243 or MATH 205 with permission of instructor.

MATH 424. Differential Geometry II (3)

Curves and surfaces in Euclidean space; mean and Gaussian curvatures, covariant differentiation, parallelism, geodesics, GaussBonnet formula. Riemannian metrics, connections, sectional curvature, generalized GaussBonnet theorem. Further topics. Prerequisite: MATH 423.

MATH 428. Fields and Modules (3) spring

Field theory, including an introduction to Galois theory; the theory of modules, including tensor products and classical algebras. Prerequisite: MATH 327.

MATH 430. Numerical Analysis (3)

Multistep methods for ordinary differential equations; finite difference methods for partial differential equations; numerical approximation of functions. Use of computer required. Prerequisite: MATH 230 or permission of instructor.

MATH 431. Calculus of Variations (3)

Existence of a relative minimum for single and multiple integral problems; variational inequalities of elliptic and parabolic types and methods of approximating a solution. Prerequisite: MATH 302 or its equivalent.

MATH 435. Functional Analysis I (3)

Banach spaces and linear operators; separation and extension theorems; open mapping and uniform boundedness principles; weak topologies; local convexity and duality; Banach algebras; spectral theory of operators; and compact operators. Prerequisites: MATH 307 and MATH 401.

MATH 441 (CSE 441). Advanced Algorithms (3) spring

This is a graduate-level version of CSE/MATH 340, Design and Analysis of Algorithms, covering that course’s content, plus matroid theory, linear programming, max-flow, computational geometry, matching patterns in strings, randomized algorithms, and approximation algorithms for NP-complete problems. Credit will not be given for both MATH 340 (CSE 340) and MATH 441 (CSE 441).

MATH 444. Algebraic Topology II (3)

Continuation of MATH 408. Cohomology theory, products, duality. Prerequisite: MATH 408.

MATH 445. Topics in Algebraic Topology (3)

Selected topics reflecting the interests of the professor and the students. Prerequisite: MATH 444.

MATH 449. Topics in Algebra (3)

Intensive study of topics in algebra with emphasis on recent developments. Prerequisite: consent of the department chairman. May be repeated for credit with the permission of the department chair.

MATH 450. Special Topics (3) fall/spring

Intensive study of some field of the mathematical sciences not covered in listed courses. Prerequisite: consent of the department chair. May be repeated for credit with the permission of the department chair.

MATH 455. Topics in Number Theory (3)

Selected topics in algebraic and/or analytic number theory. Prerequisite: permission of instructor. May be repeated for credit with permission of the department chair.

MATH 461. (STAT 461) Topics in Mathematical Statistics (3)

An intensive study of one or more topics such as theory of statistical tests, statistical estimation, regression, analysis of variance, nonparametric methods, stochastic approximation, and decision theory. Prerequisites: MATH 334 and MATH 401. May be repeated for credit with permission of the department chair.

MATH 462. (STAT 462) Modern Nonparametric Methods in Statistics (3)

Classical and modern methods of nonparametric statistics; order and rank statistics; tests based on runs, signs, ranks, and order statistics; distribution free statistical procedures for means, variances, correlations, and trends; relative efficiency; KolmogorovSmirnov statistics; statistical applications of Brownian process; modern techniques such as robust methods, nonparametric smoothing, and bootstrapping; additional topics such as nonparametric regression and dimension reduction. Prerequisites: MATH 334 (STAT 334) and MATH 338 (STAT 338) or equivalent classes or permission of instructor.

MATH 463. (STAT 463) Advanced Probability (3)

Measure theoretic foundations; random variables, integration in a measure space, expectations; convergence of random variables and probability measures; conditional expectations; characteristic functions; sums of random variables, limit theorems. Prerequisites: MATH 309 and MATH 401.

MATH 464. (STAT 464) Advanced Stochastic Processes (3)

Theory of stochastic processes; stopping times; martingales; Markov processes; Brownian motion; stochastic calculus; Brownian bridge, laws of suprema; Gaussian processes. Prerequisites: MATH 309 and MATH 401.

MATH 465. Topics in Probability (3)

Selected topics in probability. Prerequisite: permission of the department chair. May be repeated for credit with permission of the department chair.

MATH 467. Financial Calculus I (3) fall

Basic mathematical concepts behind derivative pricing and portfolio management of derivative securities. Development of hedging and pricing by arbitrage in the discrete time setting of binary trees and BlackScholes model. Introduction to the theory of Stochastic Calculus, Martingale representation theorem, and change of measure. Applications of the developed theory to a variety of actual financial instruments. Prerequisites: Math 231 or Math 309 or permission of instructor.

MATH 468. Financial Calculus II (3) spring

Models and mathematical concepts behind the interest rates markets. HeathJarrowMorton model for random evolution of the term structure of interest rates and short rate models. Applications of the theory to a variety of interest rates contracts including swaps, caps, floors, swapoptions. Development of multidimensional stochastic calculus and applications to multiple stock models, quantos, and foreign currency interestrate models. Prerequisites: Math 467.

MATH 470. Proseminar (3)

Preparation for entering the mathematics profession. Seminar will concentrate on methods of teaching mathematics, and will include other topics such as duties of a professor and searching for a job. Prerequisite: permission of the department chair.

MATH 471. Homological Algebra (3)

Modules, tensor products, categories and functors, homology functors, projective and injective modules. Prerequisite: MATH 428.

MATH 472. Group Representations (3)

Linear representations and character theory with emphasis on the finite and compact cases. Prerequisite: MATH 428 or permission of the department chair.

MATH 475. Topics in Geometry (3)

Selected topics in geometry, such as geometric analysis, algebraic geometry, complex geometry, characteristic classes, geometric flows or geometric measure theory, with emphasis on recent developments. Prerequisite: permission of the department chair. May be repeated for credit with permission of the department chair.

MATH 485. Topics in Financial Mathematics (3)

Selected topics in financial mathematics. Prerequisite: permission of the department chair. May be repeated for credit with permission of the department chair.

MATH 490. Thesis MATH 499. Dissertation

Statistics

STAT 408. (MATH 410) Seminar in Statistics and Probability (16) spring

Intensive study of some field of statistics or probability not offered in another course. Prerequisite: permission of the department chair.

STAT 409 (MATH 409) Seminar in Statistics and Probability (16) fall

Intensive study of some field of statistics or probability not offered in another course. Prerequisite: permission of the department chair.

STAT 410. Random Processes and Applications (3) spring

See MATH 310.

STAT 412. Statistical Computing and Applications (3)

See MATH 312.

STAT 434. Mathematical Statistics (3) spring

See MATH 334.

STAT 438 (MATH 338). Linear Models in Statistics with Applications (3) spring

See MATH 338

STAT 461 (MATH 461). Topics in Mathematical Statistics (3)

See MATH 461.

STAT 462 (MATH 462). Nonparametric Statistics (3)

See MATH 462.

STAT 463 (MATH 463). Advanced Probability (3)

See MATH 463.

STAT 464 (MATH 464). Advanced Stochastic Processes (3)

See MATH 464.

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