Mathematics has always held a central position in the liberal arts, and over time it has also come to play an important role in more and more aspects of our lives. Mathematical analysis and modeling are involved in many areas, far beyond the traditional association of mathematics with the physical sciences and engineering. This fact is reflected in the diversity of the students who study at least some mathematics during their time at Washington University — students who recognize the importance of quantitative skills in a world that becomes more and more technological.
Students major in mathematics for many reasons. Some are planning academic careers in mathematics or statistics that involve teaching or research. Others plan to work as actuaries or at other jobs in industry or government. Some plan careers in secondary education. Many majors do not intend to become "mathematicians" at all, but simply realize that quantitative training is a valuable asset in many kinds of careers; often, work in mathematics or statistics is meant to complement their study in other areas. Still other majors just view mathematics as an interesting concentration in their liberal arts education, even though they plan to enter professional fields such as medicine or law.
The Mathematics program gives majors and minors a broad introduction to the subject. Majors choose among several tracks to complete their study; these tracks add different emphases to their programs, reflecting individual interests or professional goals. Majors are encouraged to complete additional work (perhaps even a minor or second major) in other related areas.
Contact:  Blake Thornton 

Phone:  3149356301 
Email:  bthornton@wustl.edu 
Website:  http://wumath.wustl.edu 
Chair
John E. McCarthy
Spencer T. Olin Professor of Mathematics
PhD, University of California, Berkeley
Analysis; operator theory; one and several complex variables
Directors
Brett Wick
Director of Graduate Studies; Professor of Mathematics
PhD, Brown University
Complex analysis, harmonic analysis, operator theory, and several complex variables
John Shareshian
Director of Undergraduate Studies; Professor of Mathematics
PhD, Rutgers University
Algebraic and topological combinatorics
Endowed Professor
John E. McCarthy
Spencer T. Olin Professor of Mathematics
PhD, University of California, Berkeley
Analysis; operator theory; one and several complex variables
Professors
QuoShin Chi
PhD, Stanford University
Differential geometry
Renato Feres
PhD, California Institute of Technology
Differential geometry; dynamical systems
José FigueroaLópez
PhD, Georgia Institute of Technology
Statistics; probability and stochastic processes; mathematical finance
Steven G. Krantz
PhD, Princeton University
Several complex variables; geometric analysis
Rachel Roberts
PhD, Cornell University
Lowdimensional topology
John Shareshian
PhD, Rutgers University
Algebraic and topological combinatorics
Edward Spitznagel
PhD, University of Chicago
Statistics; statistical computation; application of statistics to medicine
Xiang Tang
PhD, University of California, Berkeley
Symplectic geometry; noncommutative geometry; mathematical physics
Brett Wick
PhD, Brown University
Analysis of several complex variables; harmonic analysis and operatory theory
Mladen Victor Wickerhauser
PhD, Yale University
Harmonic analysis; wavelets; numerical algorithms for data compression
David Wright
PhD, Columbia University
Affine algebraic geometry; polynomial automorphisms
Associate Professors
Roya Beheshti Zavareh
PhD, Massachusetts Institute of Technology
Algebraic geometry
Brian E. Blank
PhD, Cornell University
Representations of Lie groups; harmonic analysis
Jimin Ding
PhD, University of California, Davis
Statistics
Matthew Kerr
PhD, Princeton University
Algebraic geometry; Hodge theory
Gregory Knese
PhD, Washington University
Complex function theory, operators; harmonic analysis
Nan Lin
PhD, University of Illinois at UrbanaChampaign
Statistics
Jack Shapiro
PhD, City University of New York
Algebraic Ktheory
Ari Stern
PhD, California Institute of Technology
Geometric numerical analysis, computational mathematics
Assistant Professors
Steven Frankel
PhD, University of Cambridge
Todd Kuffner
PhD, Imperial College London
Statistics; likelihood and asymptotics, econometrics
Yanli Song
PhD, Pennsylvania State University
Professors Emeriti
William M. Boothby
PhD, University of Michigan
Differential geometry
Lawrence Conlon
PhD, Harvard University
Differential topology
Ron Freiwald
PhD, University of Rochester
General topology
Gary R. Jensen
PhD, University of California, Berkeley
Differential geometry
Robert McDowell
PhD, Purdue University
General topology
Richard Rochberg
PhD, Harvard University
Complex analysis, interpolation theory
Guido L. Weiss
PhD, University of Chicago
Interpolation of operators; harmonic analysis, Lie groups
Edward N. Wilson
PhD, Washington University
Harmonic analysis, differential geometry
William Chauvenet Postdoctoral Lecturers
Yakov BerchenkoKogan
PhD, Massachusetts Institute of Technology
Patricio Gallardo
PhD, Stony Brook University
Associate Director of Undergraduate Studies
Blake Thornton
PhD, University of Utah
Program Coordinator
Lisa M. Kuehne
Program Coordinator, University College & Center for Advanced Learning
AM Mathematics, Washington University
Undergraduate Mathematics Education
The Major in Mathematics
Total units required: 3639
All mathematics majors are required to complete
Code  Title  Units 

Math 131  Calculus I  3 
Math 132  Calculus II  3 
Math 233  Calculus III  3 
Total Units  9 
In addition, each major is required to complete the courses in one of the following five tracks and to participate in a departmental exit interview shortly before graduation.
Traditional
Code  Title  Units 

Math 3200  Elementary to Intermediate Statistics and Data Analysis  3 
or Math 493  Probability  
Math 310  Foundations for Higher Mathematics  3 
Math 4111 & Math 4121  Introduction to Analysis and Introduction to Lebesgue Integration  6 
Math 429 & Math 430  Linear Algebra and Modern Algebra  6 
Three additional upperlevel mathematics electives. Two of these could be Math 309 and Math 318 (or Math 308). At least one elective must be at the 400 level.  9 
Total 27 upperlevel mathematics units
Probability/Statistics
Code  Title  Units 

One course in computer science chosen from CSE 131, CSE 132, CSE 200, CSE 247  3  
Math 309  Matrix Algebra  3 
Math 3200  Elementary to Intermediate Statistics and Data Analysis  3 
Math 310  Foundations for Higher Mathematics  3 
Math 318  Introduction to Calculus of Several Variables  3 
or Math 308  Mathematics for the Physical Sciences  
Math 493 & Math 494  Probability and Mathematical Statistics  6 
Two additional upperlevel mathematics electives in the areas of probability or statistics  6 
Total 24 upperlevel mathematics units
Applied
Code  Title  Units 

One course in computer science chosen from CSE 131, CSE 132, CSE 200, CSE 247  3  
Math 217  Differential Equations  3 
Math 309  Matrix Algebra  3 
Math 3200  Elementary to Intermediate Statistics and Data Analysis  3 
or Math 493  Probability  
Math 310  Foundations for Higher Mathematics  3 
Math 318  Introduction to Calculus of Several Variables  3 
or Math 308  Mathematics for the Physical Sciences  
Math 449 & Math 450  Numerical Applied Mathematics and Topics in Applied Mathematics  6 
Two additional upperlevel mathematics electives. Major track must include at least one of Math 410, Math 415, Math 416, or Math 4111.  6 
Total 24 upperlevel mathematics units
Mathematics for Secondary Education
(This track also requires a major in secondary education)
Code  Title  Units 

One course in computer science chosen from CSE 131, CSE 132, CSE 200, CSE 247  3  
Math 309  Matrix Algebra  3 
Math 3200  Elementary to Intermediate Statistics and Data Analysis  3 
or Math 493  Probability  
Math 310  Foundations for Higher Mathematics  3 
Math 318  Introduction to Calculus of Several Variables  3 
or Math 308  Mathematics for the Physical Sciences  
Math 302  Elementary Geometry From an Advanced Point of View  3 
Math 331  Algebraic Systems  3 
Two additional upperlevel mathematics electives  6 
Total 24 upperlevel mathematics units
Mathematics (Economics Emphasis)
Three economics courses:
Code  Title  Units 

Econ 1011 and Econ 1021  6  
One Economics course chosen from among Econ 4011, Econ 4151, or Econ 467  3 
and
Code  Title  Units 

Math 309  Matrix Algebra  3 
Math 3200  Elementary to Intermediate Statistics and Data Analysis  3 
or Math 493  Probability  
Math 310  Foundations for Higher Mathematics  3 
Math 4111 & Math 4121  Introduction to Analysis and Introduction to Lebesgue Integration  6 
Three additional upperlevel courses from among Math 318, Math 429, L24 Math 456, or any 400level probability or statistics course. The major must include at least one of Math 493 or Math 429.  9 
Total 24 upperlevel mathematics units + 3 upperlevel economics units
Notes applying to all major tracks
 Students who entered Washington University before fall 2015 should visit the Prior Bulletins section of this Bulletin to view the major requirements that were in place for the appropriate year of matriculation. These requirements can also be found from the department's undergraduate webpage.
 Upperlevel mathematics courses are those whose course number begins with a "3" or higher; e.g., Math 3***. Lowerlevel courses do not count toward upperlevel mathematics requirements even if crosslisted as an upperlevel course in another department or program. For example, if Math 2200 were crosslisted by another department as 3***, then registering for that 3*** course would not satisfy an upperlevel mathematics requirement.
 All required courses (lower and upperlevel) must be completed with a grade of C or better.
 Math 318 Introduction to Calculus of Several Variables and Math 308 Mathematics for the Physical Sciences cannot both be used to fulfill major requirements.
 Courses transferred from other accredited colleges and universities with department approval can be counted. However,
 courses transferred from a twoyear college (such as a community college) cannot be used to satisfy upperlevel requirements;
 at least half the upperlevel units required in a major must be earned at Washington University or in a Washington Universityapproved Overseas Study program;
 courses from University College cannot be used to fulfill major requirements.
 At most 3 units for independent study or research work can count toward the major requirements.
 No upperlevel course used to satisfy a major requirement can be counted toward the requirements of any other major or minor (no "doublecounting").
 Certain approved substitutions are found on the department's webpage. However, in all cases, at most one substitution can be used that involves a course not homebased in the Department of Mathematics.
Additional Information
Independent Study: Majors with a plan for independent work or a research project may apply for independent study with a faculty member. Majors are encouraged (but not required) to consider a senior project or an honors project.
Graduation with Awards for Distinction or Latin Honors: The department offers the awards Distinction in Mathematics, High Distinction in Mathematics, and Highest Distinction in Mathematics. Recommendations for students to receive Latin Honors (which are awarded by the College of Arts & Sciences, not the department) are keyed to the requirements for the Distinction Awards, as indicated below. In these requirements, "regularly scheduled mathematics course" refers to a course that is homebased in the mathematics department (L24) and is not an independent study or research course.
 Core Course Work for All Distinction Awards (Core Course Work alone earns no award)
 at least 3.65 GPA in upperlevel mathematics (L24) courses
 completion with grades of B or better (not B) of
 one of the course sequences Math 41114121, 429430, 493494, 449450, and
 three additional regularlyscheduled 400500 level mathematics courses
 Distinction in Mathematics. Awarded for Core Course Work plus
 completion of one additional regularlyscheduled mathematics course at the 400500 level with grade of B or better (not B), or
 passing the first actuarial exam (Exam P) from Society of Actuaries
 High Distinction in Mathematics. Awarded for Core Course Work plus satisfactory completion of an honors thesis. Details about honors theses are available on the department's webpage.
 Highest Distinction in Mathematics. Awarded for Core Course Work plus satisfactory completion of an honors thesis plus one of the following:
 course work includes completion of at least one of the graduate sequences Math 50215022, 50315032, 50415042(3), 50515052, 50615062 and passing the graduate qualifying exam for that course sequence, or
 course work includes all the requirements for the department's Honors Program in Statistics
 Latin Honors. For majors in the College of Arts & Sciences, the department will recommend that the AB degree be awarded with Latin Honors if the student has an overall GPA of 3.65, as required by the college, and has earned the department award of High or Highest Distinction. These majors must apply to the department for admission to Candidacy for Latin Honors by the end of their junior year. The level of Latin Honors (cum laude, magna cum laude or summa cum laude) is determined by the college as described in the Arts & Sciences Academic Honors & Awards section of this Bulletin.
More details are available on the department's webpage.
Study Abroad: Students interested in a semester or year abroad studying mathematics intensively should consider the Budapest Semesters in Mathematics Program.
The Major in Mathematics and Computer Science
The School of Engineering & Applied Science (SEAS) and the College of Arts & Sciences (A&S) developed a new major that efficiently captures the intersection of the complementary studies of computer science and math.
SEAS students who declare this major must fulfill the distribution and all other requirements for the "Applied Science" degree in the School of Engineering & Applied Science. A&S students who declare this major must fulfill the distribution and all other requirements for an AB degree in addition to the specific requirements listed below.
Core Course Requirements
Code  Title  Units 

Math 131  Calculus I (AP credit may satisfy this requirement)  3 
Math 132  Calculus II (AP credit may satisfy this requirement)  3 
Math 233  Calculus III  3 
CSE 131  Introduction to Computer Science  3 
Math 310  Foundations for Higher Mathematics  34 
or Math 310W  Foundations for Higher Mathematics with Writing  
or CSE 240  Logic and Discrete Mathematics  
CSE 247  Data Structures and Algorithms  3 
Math 309  Matrix Algebra  3 
Math 3200  Elementary to Intermediate Statistics and Data Analysis  3 
or ESE 326  Probability and Statistics for Engineering  
CSE 347  Analysis of Algorithms  3 
Electives
Eight upperlevel approved courses from Math or CSE

No fewer than three courses from each department

Up to two preapproved courses from outside both departments
The Minor in Mathematics
Units required: 24
Required courses:
Code  Title  Units 

One course in computer science chosen from CSE 131, CSE 132, CSE 200, CSE 247  3  
Math 131  Calculus I  3 
Math 132  Calculus II  3 
Math 233  Calculus III  3 
Math 309  Matrix Algebra  3 
Math 318  Introduction to Calculus of Several Variables  3 
or Math 308  Mathematics for the Physical Sciences  
Math 3200  Elementary to Intermediate Statistics and Data Analysis  3 
or Math 493  Probability  
One additional upperlevel (300 or 400level) elective  3  
Total Units  24 
Additional Information
 Students who entered Washington University before fall 2015 should visit the Prior Bulletins section of this Bulletin to view the minor requirements that were in place for the appropriate year of matriculation. These requirements can also be found from the department's undergraduate webpage.
 All required courses (lower and upperlevel) must be completed with a letter grade of C or better.
 Math 318 and Math 308 cannot both be used to fulfill minor requirements.
 Courses transferred from other accredited colleges and universities with department approval can be counted. However,
 courses transferred from a twoyear college (such as a community college) cannot be used to satisfy upperlevel requirements;
 at least 6 of the upperlevel units required in a minor must be earned at Washington University or in a Washington Universityapproved Overseas Study Program;
 courses from University College cannot be used to fulfill major requirements.
 No upperlevel course used to satisfy a minor requirement can be counted toward the requirements of any other major or minor (no "doublecounting").
 At least three of the four upperlevel courses required in the minor must be courses "homebased" in the math department. One approved course from another department may be substituted. Approved substitutions can be found on the department webpage.
Visit online course listings to view semester offerings for L24 Math.
L24 Math 100 Foundations for Calculus
A limited enrollment class for students planning to take calculus but who need additional precalculus preparation. The course aims to build both the technical skills and the conceptual understanding needed to succeed in calculus. Course emphasizes links between the graphical, numeric and algebraic viewpoints. A variety of approaches are used to present the material. Prerequisites: two years of high school algebra and a course in geometry (or the equivalent).
Credit 3 units. A&S IQ: NSM
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L24 Math 1011 Introduction to Statistics
Basic concepts of statistics. Data collection (sampling, designing experiments), data organization (tables, graphs, frequency distributions, numerical summarization of data), statistical inference (elementary probability and hypothesis testing). Prerequisite: two years of high school algebra.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 109 Mathematics and Music
An elementary introduction to the connections between mathematics and musical sound. Review of integers, ratios, prime numbers, functions, rationality, exponents, logarithms, trigonometry. Review of scales, clefs, key signatures, intervals, time signatures. Frequency and pitch. The connection between intervals and logarithms. Tuning and temperament, just intonation. Scales and modular arithmetic. The mathematics of harmony; the sound of the low prime numbers and their roles in harmony. Harmonics, partials and overtones. Numerical integration and basic Fourier analysis. The nature of complex tones. Analysis of instrument sounds. Human vowels and formants. Prerequisites: two years of high school algebra, and trigonometry.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 131 Calculus I
Derivatives of algebraic, trigonometric and transcendental functions, techniques of differentiation, Mean Value Theorem, applications of the derivative. The definite integral and Fundamental Theorem of Calculus. Areas. Simpler integration techniques. Prerequisites: highschool algebra and precalculus, including trigonometry.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 132 Calculus II
Continuation of Math 131. A brief review of the definite integral and Fundamental Theorem of Calculus. Techniques of integration, applications of the integral, sequences and series, Taylor polynomials and series, and some material on differential equations. Prerequisite: Math 131 or a B or better in a oneyear high school calculus course, or permission of the department.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 139A Real Mathematical Applications: Solving Problems with Calculus I
This is a 1credit course that can only be taken concurrently with Math 131, Calculus I. The purpose of the course is to show how mathematics can solve realworld problems, and how calculus dramatically expands the range of problems that can be tackled. Each class will be devoted to the analysis of some problems, which may include: dimensional analysis, the mathematics of convoys, Fibonacci numbers, fractals, linear regression, Euclid's algorithm, Stein's algorithm, network capacities, Braess' paradox, Galton's approach to surnames, how genes spread through populations, SIR model of infectious diseases. The first few classes will not use differentiation. Must be taken concurrently with Math 131.
Credit 1 unit. A&S: FYO Arch: NSM Art: NSM
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L24 Math 203 Honors Mathematics I
This is the first half of a oneyear calculus sequence for first year students with a strong interest in mathematics with an emphasis on rigor and proofs. The course begins at the beginning but assumes the students have already studied the material from a more "mechanical" view. Students who complete both semesters will have completed the material Calc III and other topics that may let them move through the upperlevel math curriculum more quickly. Sets, functions, real numbers, and methods of proof. The RiemannDarboux integral, limits and continuity, differentiation, and the fundamental theorems of calculus. Sequences and series of real numbers and of functions. Vector spaces and linear maps. Prerequisite: Score of 5 on the AP Calculus Exam, BC version, or the equivalent.
Credit 4 units. A&S IQ: NSM, AN
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L24 Math 204 Honors Mathematics II
Matrices, linear systems, and determinants. Eigenvalues and eigenvectors, diagonalization, and the spectral theorem. Scalar and vector fields, differential and integral calculus of several variables, and the fundamental theorems of Green, Gauss, and Stokes. Restricted to first year students who have completed Math 203 in the fall semester. Math 204 can replace Math 233 in major/minor requirements.
Credit 4 units. A&S IQ: NSM, AN Art: NSM
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L24 Math 217 Differential Equations
Introduction to ordinary differential equations: firstorder equations, linear equations, systems of equations, series solutions, Laplace transform methods, numerical solutions. Prerequisite: Math 233 (or Math 233 concurrently).
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 220 Finite Mathematics
Topics selected from number theory, combinatorics and graph theory. Methods of proof and practical applications: for example, calendars, scheduling, communications, encryption. Prerequisite: high school algebra.
Credit 3 units. A&S IQ: NSM, AN Art: NSM
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L24 Math 2200 Elementary Probability and Statistics
An elementary introduction to statistical concepts, reasoning and data analysis. Topics include statistical summaries and graphical presentations of data, discrete and continuous random variables, the logic of statistical inference, design of research studies, point and interval estimation, hypothesis testing, and linear regression. Students will learn a critical approach to reading statistical analyses reported in the media, and how to correctly interpret the outputs of common statistical routines for fitting models to data and testing hypotheses. A major objective of the course is to gain familiarity with basic R commands to implement common data analysis procedures. Students intending to pursue a major or minor in mathematics or wishing to take 400level or above statistics courses should instead take Math 3200. Prerequisite: Math 131.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 233 Calculus III
Multivariable calculus. Topics include differential and integral calculus of functions of two or three variables: vectors and curves in space, partial derivatives, multiple integrals, line integrals, vector calculus at least through Green's Theorem. Prerequisite: Math 132 or a score of 45 on the Advanced Placement Calculus Exam (BC version).
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 266 Math for Elementary School Teachers
A review of elementary school mathematics at a level beyond its usual presentation in the schools. Applications of all concepts are given in abundance. Restricted to majors in Elementary Education. Prerequisite: two years of high school mathematics.
Credit 3 units. A&S IQ: NSM, AN Art: NSM
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L24 Math 302 Elementary Geometry From an Advanced Point of View
A rigorous modern treatment of Euclidean geometry and an introduction to nonEuclidean geometry. Prerequisite: Math 310 or permission of instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 308 Mathematics for the Physical Sciences
Continuation of Math 233 emphasizing topics of interest in the physical sciences. Topics in multivariable and vector calculus (div, grad, curl); line, surface integrals and connections to electromagnetism; Fourier series and integrals; boundary value problems (diffusion and wave equations); additional topics if time permits. Students may not receive credit toward a math major or minor for both Math 308 and Math 318. Prerequisite: Math 233 and 217, or permission of instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM BU: SCI
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L24 Math 309 Matrix Algebra
An introductory course in linear algebra that focuses on Euclidean nspace, matrices and related computations. Topics include: systems of linear equations, row reduction, matrix operations, determinants, linear independence, dimension, rank, change of basis, diagonalization, eigenvalues, eigenvectors, orthogonality, symmetric matrices, least square approximation, quadratic forms. Introduction to abstract vector spaces. Prerequisite: Math 132.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 310 Foundations for Higher Mathematics
Introduction to the rigorous techniques used in more advanced mathematics. Topics include postpositional logic, use of quantifiers, set theory, methods of proof and disproof (counterexamples), foundations of mathematics. Use of these tools in the construction of number systems and in other areas such as elementary number theory, combinatorial arguments and elementary proofs in analysis. Prerequisite: Math 233.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 310W Foundations for Higher Mathematics with Writing
Refer to the description for Math 310. Math 310W is a writingintensive version of Math 310. Students participate in the regular Math 310 lectures and are responsible for all the exams and assignments associated with Math 310. Students in Math 310W have one additional meeting each week to deal with writing issues. At least three papers (four to five pages in length) are required, each with at least one revision. Prerequisite: Math 233 or permission of instructor.
Credit 4 units. A&S IQ: NSM, WI
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L24 Math 312 Differential Equations and Dynamical Systems
Qualitative theory of ordinary differential equations. Picard's existence and uniqueness theorem, the phase plane, PoincareBendixon theory, stationary points, attractors and repellors, graphical methods. Physical applications, including chaos, are indicated. Prerequisite: Math 217.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 318 Introduction to Calculus of Several Variables
Selected topics for functions of several variables involving some matrix algebra and presented at a level of rigor intermediate between that of Calculus III and higherlevel analysis courses. Students may not receive credit toward a mathematics major or minor for both Math 308 and 318. Prerequisites: Math 233 and Math 309.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 3200 Elementary to Intermediate Statistics and Data Analysis
An introduction to probability and statistics. Major topics include elementary probability, special distributions, experimental design, exploratory data analysis, estimation of mean and proportion, hypothesis testing and confidence, regression, and analysis of variance. Emphasis is placed on development of statistical reasoning, basic analytic skills, and critical thinking in empirical research studies. The use of the statistical software R is integrated into lectures and weekly assignments. Required for students pursuing a major or minor in mathematics or wishing to take 400level or above statistics courses. Prerequisite: Math 132. Though Math 233 is not essential, it is recommended.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 322 Biostatistics
A second course in elementary statistics with applications to life sciences and medicine. Review of basic statistics using biological and medical examples. New topics include incidence and prevalence, medical diagnosis, sensitivity and specificity, Bayes' rule, decision making, maximum likelihood, logistic regression, ROC curves and survival analysis. Prerequisites: Math 3200, or a strong performance in Math 2200 and permission of the instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 331 Algebraic Systems
Polynomials, binomial expansions, factoring, rings (integers and polynomials), unique factorization, and other topics relevant to the high school curriculum. Designed for future secondary school teachers and other students looking for a course in algebra at a less abstract level than Math 430. Prerequisite: Math 310 or permission of instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 3351 Elementary Theory of Numbers
Divisibility properties of integers, congruences, quadratic reciprocity, Diophantine equations. Introduction to continued fractions and a brief discussion of public key cryptography. Prerequisite: Math 310 or permission of instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 350 Topics in Applied Mathematics
Topics change with each offering of the course. Past topics have included "Mathematics and Multimedia," "The Mathematics and Chemistry of ReactionDiffusion Systems," "Mathematical Biology," and "Simulation Analysis of Random Processes" and "Introduction to Monte Carlo Methods." Prerequisites will vary but always include at least Math 233, Math 309 and basic programming skills in some language.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 370 Introduction to Combinatorics
Basics of enumeration (combinations, permutations and enumeration of functions between finite sets), generating functions; the inclusionexclusion principle, partition theory and introductory graph theory. As time permits, additional topics may include Ramsey's Theorem, probabilistic methods in combinatorics and algebraic methods in combinatorics. Prerequisites: Math 132, 309 and 310, or permission of the instructor.
Credit 3 units. A&S IQ: NSM, AN Arch: NSM Art: NSM
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L24 Math 371 Graph Theory
Introduction to graph theory including the basic definitions and theorems and some more advanced topics that drive much current research in graph theory: Ramsey's Theorem, random graph theory and, if time permits, Szemeredi's regularity lemma. Graphs are studied as abstract objects; however, graph theory is also of interest to applied mathematicians because graphs are natural models for networks (social, electric). Prerequisite: Math 310 or a roughly equivalent course, or permission of instructor. Students should know what a proof is and how to produce one. Some informal understanding of probability is helpful, but students need not have taken a probability course.
Credit 3 units. A&S IQ: NSM
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L24 Math 400 Undergraduate Independent Study
Approval of instructor required.
Credit variable, maximum 3 units.
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L24 Math 403C Mathematical Logic I
A first course in mathematical logic, an introduction to both proof theory and model theory. The structure and properties of firstorder logic are studied in detail, with attention to such notions as axiomatic theory, proof, model, completeness, compactness and decidability. Prerequisite: Phil 301G or equivalent, or a background in mathematics.
Same as L30 Phil 403
Credit 3 units. A&S IQ: HUM Art: HUM BU: HUM
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L24 Math 404C Mathematical Logic II
Godel's Incompleteness Theorem: its proof, its consequences, its reverberations. Prerequisite: Phil 403 or a strong background in mathematics.
Same as L30 Phil 404
Credit 3 units. A&S IQ: HUM Art: HUM
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L24 Math 406 Topics in Analysis: Special Bases in Analysis
The course considers the construction of specific orthonormal bases for the Hilbert spaces L^2(R^n). We look at the Fourier series bases in L^2([0,1]), which are then used for the construction of two types of orthonormal bases in the more general setting of R^n: wavelets and Gabor bases. Some aspects of extensions to other Hilbert spaces are considered. In general, these bases are used to analyze "signals" or functions on R^n having either real or complex values, and they should be "efficient" in some sense. We look at the types of efficiency that are most useful in several types of application in sciences and engineering. Prerequisites: Math 233, 309, and 310, or permission of instructor. In particular, students are expected to understand derivatives and Riemann integration; topics related to the more general Lebesgue integral are discussed as needed.
Credit 3 units. A&S IQ: NSM
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L24 Math 407 An Introduction to Differential Geometry
A study of properties of curves and surfaces in 3dimensional Euclidean space. The course is essentially a modern recounting of a seminal paper of Gauss. Prerequisites: Math 233 and Math 309.
Credit 3 units.
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L24 Math 408 Nonparametric Statistics
Statistical methods that make few or no assumptions about the data distribution. Permutation tests of different types; nonparametric confidence intervals and correlation coefficients; jackknife and bootstrap resampling; nonparametric regressions. If there is time, topics chosen from density estimation and kernel regression. Short computer programs will be written in a language like R or C. Prerequisites: CSE 131 or 200, Math 3200 and Math 493, or permission of instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 410 Introduction to Fourier Series and Integrals
The basic theory of Fourier series and Fourier integrals including different types of convergence. Applications to certain differential equations. Prerequisites: Math 233 and Math 309.
Credit 3 units. A&S IQ: NSM
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L24 Math 4111 Introduction to Analysis
The real number system and the least upperbound property; metric spaces (completeness, compactness and connectedness); continuous functions (in R^n; on compact spaces; on connected spaces); C(X) (pointwise and uniform convergence; Weierstrass approximation theorem); differentiation (mean value theorem; Taylor's theorem); the contraction mapping theorem; the inverse and implicit function theorems. Prerequisite: Math 310 or permission or instructor.
Credit 3 units. A&S IQ: NSM
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L24 Math 4121 Introduction to Lebesgue Integration
Riemann integration; measurable functions; measures; Lebesgue measure; the Lebesgue integral; integrable functions; L^p spaces; modes of convergence; decomposition of measures; product measures. Prerequisite: Math 4111 or permission of the instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 415 Partial Differential Equations
Introduction to the theory of PDE's with applications to selected classical problems in physics and engineering. Linear and quasilinear firstorder equations, derivation of some of the classical PDE's of physics, and standard solution techniques for boundary and initial value problems. Preliminary topics such as orthogonal functions, Fourier series and variational methods introduced as needed. Prerequisites: Math 217 and Math 309, or permission of instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 416 Complex Variables
Analytic functions, elementary functions and their properties, line integrals, the Cauchy integral formula, power series, residues, poles, conformal mapping and applications. Prerequisites: Math 318, Math 308 or ESE 317, or permission of instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 4171 Topology I
An introduction to the most important ideas of topology. Course includes necessary ideas from set theory, topological spaces, subspaces, products and quotients, compactness and connectedness. Some time is also devoted to the particular case of metric spaces (including topics such as separability, completeness, completions, the Baire Caregory Theorem, and equivalents of compactness in metric spaces). Prerequisite: Math 4111 or permission of instructor.
Credit 3 units. A&S IQ: NSM
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L24 Math 418 Introduction to Topology and Modern Analysis II
Continuation of Math 417. May include some algebraic topology (depending on material covered in 417). Prerequisite: Math 417.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 4181 Topology II
A continuation of Math 4171 featuring more advanced topics in topology. The content may vary with each offering. Prerequisite: Math 4171, or permission of instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 420 Experimental Design
A first course in the design and analysis of experiments, from the point of view of regression. Factorial, randomized block, splitplot, Latin square, and similar design. Prerequisite: CSE 131 or 200, Math 3200, or permission of instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 429 Linear Algebra
Introduction to the linear algebra of finitedimensional vector spaces. Includes systems of equations, matrices, determinants, inner product spaces, spectral theory. Prerequisite: Math 310 or permission of instructor. Math 309 is not an explicit prerequisite but students already should be familiar with such basic topics from matrix theory as matrix operations, linear systems, row reduction and Gaussian elimination. Material on these topics in early chapters of the text are covered very quickly.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 430 Modern Algebra
Introduction to groups, rings and fields. Includes permutation groups, group and ring homomorphisms, field extensions, connections with linear algebra. Prerequisite: Math 429 or permission of the instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 434 Survival Analysis
Life table analysis and testing, mortality and failure rates, KaplanMeier or productlimit estimators, hypothesis testing and estimation in the presence of random arrivals and departures, and the Cox proportional hazards model. Techniques of survival analysis are used in medical research, industrial planning and the insurance industry. Prerequisites: CSE 131 or 200, Math 309 and 3200, or permission of the instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 4351 Number Theory and Cryptography
The course covers many of the basics of elementary number theory, providing a base from which to approach modern algebra, algebraic number theory and analytic number theory. It also introduces one of the most important realworld applications of mathematics, namely the use of number theory and algebraic geometry in public key cryptography. Topics from number theory involve divisibility (Euclidean algorithm, primes, Fundamental Theorem of Arithmetic), congruences (modular arithmetic, Chinese Remainder Theorem, primality testing and factorization). Topics from cryptography include RSA encryption, DiffieHellman key exchange and elliptic curve cryptography. Topics about algebraic numbers may be include if time permits. Prerequisites: Math 233, 309 and 310 (or permission of instructor).
Credit 3 units. A&S IQ: NSM
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L24 Math 436 Algebraic Geometry
Introduction to affine and projective algebraic varieties; the Zariski topology; regular and rational mappings; simple and singular points; divisors and differential forms; genus; the RiemannRoch theorem. Prerequisites: Math 318, Math 429 and Math 430, or permission of the instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 439 Linear Statistical Models
Theory and practice of linear regression, analysis of variance (ANOVA) and their extensions, including testing, estimation, confidence interval procedures, modeling, regression diagnostics and plots, polynomial regression, colinearity and confounding, model selection, geometry of least squares, etc. The theory will be approached mainly from the frequentist perspective, and use of the computer (mostly R) to analyze data will be emphasized. Prerequisite: CSE 131 or 200, Math 3200 and a course in linear algebra (such as Math 309 or 429), or permission of instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 4392 Advanced Linear Statistical Models
Review of basic linear models relevant for the course; generalized linear models including logistic and Poisson regression (heterogeneous variance structure, quasilikelihood); linear mixedeffects models (estimation of variance components, maximum likelihood estimation, restricted maximum likelihood, generalized estimating equations), generalized linear mixedeffects models for discrete data, models for longitudinal data, optional multivariate models as time permits. The computer software R will be used for examples and homework problems. Implementation in SAS will be mentioned for several specialized models. Prerequisites: Math 439 and a course in linear algebra (such as Math 309 or 429), or consent of instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 449 Numerical Applied Mathematics
Computer arithmetic, error propagation, condition number and stability; mathematical modeling, approximation and convergence; roots of functions; calculus of finite differences; implicit and explicit methods for initial value and boundary value problems; numerical integration; numerical solution of linear systems, matrix equations and eigensystems; Fourier transforms; optimization. Various software packages may be introduced and used. Prerequisites: CSE 200 or CSE 131 (or other computer background with permission of the instructor); Math 217 and Math 309.
Credit 3 units. A&S IQ: NSM
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L24 Math 450 Topics in Applied Mathematics
Topic may vary with each offering of the course. Prerequisites: CSE 131 (or 200) and Math 449, or permission of the instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 459 Bayesian Statistics
Introduces the Bayesian approach to statistical inference for data analysis in a variety of applications. Topics include: comparison of Bayesian and frequentist methods, Bayesian model specification, choice of priors, computational methods such as rejection sampling, and stochastic simulation (Markov chain Monte Carlo), empirical Bayes method, handson Bayesian data analysis using appropriate software. Prerequisites: Math 493 and either Math 3200 or 494; or permission of the instructor. Some programming experience such as CSE 131 is also helpful (consult with the instructor).
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 460 Multivariate Statistical Analysis
A modern course in multivariate statistics. Elements of classical multivariate analysis as needed, including multivariate normal and Wishart distributions. Clustering; principal component analysis. Model selection and evaluation; prediction error; variable selection; stepwise regression; regularized regression. Crossvalidation. Classification; linear discriminant analysis. Treebased methods. Time permitting, optional topics may include nonparametric density estimation, multivariate regression, support vector machines, and random forests. Prerequisite: multivariable calculus (Math 233), linear or matrix algebra (Math 429 or Math 309), multivariablecalculusbased probability and mathematical statistics (Math 493, Math 494) and linear models (Math 439). Prior knowledge of R at the level introduced in Math 439 is assumed.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 461 Time Series Analysis
Time series data types; autocorrelation; stationarity and nonstationarity; autoregressive moving average models; model selection methods; bootstrap confidence intervals; trend and seasonality; forecasting; nonlinear time series; filtering and smoothing; autoregressive conditional heteroscedasticity models; multivariate time series; vector autoregression; frequency domain; spectral density; statespace models; Kalman filter. Emphasis on realworld applications and data analysis using statistical software. Prerequisites: Math 493 and either Math 3200 or 494; or permission of the instructor. Some programming experience may also be helpful (consult with the instructor).
Credit 3 units. A&S IQ: NSM
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L24 Math 462 Mathematical Foundations of Big Data
Mathematical foundations of data science. Core topics include: probability in high dimensions; curses and blessings of dimensionality; concentration of measure; matrix concentration inequalities. Essentials of random matrix theory. Randomized numerical linear algebra. Data clustering. Depending on time and interests, additional topics will be chosen from: compressive sensing; efficient acquisition of data; sparsity; lowrank matrix recovery. Divide, conquer and combine methods. Elements of topological data analysis; point cloud; Cech complex; persistent homology. Selected aspects of highdimensional computational geometry and dimension reduction; embeddings; JohnsonLindenstrauss; sketching; random projections. Diffusion maps; manifold learning; intrinsic geometry of massive data sets. Optimization and stochastic gradient descent. Random graphs and complex networks. Combinatorial group testing. Prerequisite: multivariable calculus (Math 233), linear or matrix algebra (Math 429 or 309), and multivariablecalculusbased probability and mathematical statistics (Math 493494). Prior familiarity with analysis, topology, and geometry is strongly recommended. A willingness to learn new mathematics as needed is essential.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 469 Linear Algebra in Combinatorics
Many problems in combinatorics can be solved in surprising ways using linear algebra. The course will consider several types of problems to which linear algebra methods can be applied, reviewing some requisite ideas from linear algebra (and perhaps learning some new ones) along the way. Prerequisites: Math 310 and Math 429, or permission of instructor.
Credit 3 units. A&S IQ: NSM
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L24 Math 470 Analytic Combinatorics
Analytic combinatorics is the study of large structured combinatorial configurations. The course will be broken into two components. First, generating functions will be used to encapsulate counting sequences and their recurrence structures with a formal power series. Second, analytic methods will be used to obtain the precise asymptotic behavior of counting sequences. The informal prerequisites are: familiarity with basic discrete math objects: sets, permutations, combinations, graphs; power series; mathematical maturity (e.g., the ability to write rigorous proofs and to absorb new definitions quickly). Formal Prerequisites: Math 310.
Credit 3 units. A&S IQ: NSM
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L24 Math 475 Statistical Computation
Introduction to modern computational statistics. Pseudorandom number generators; inverse transform and rejection sampling. Monte Carlo approximation. Nonparametric bootstrap procedures for bias and variance estimation; bootstrap confidence intervals. Markov chain Monte Carlo methods; Gibbs and MetropolisHastings sampling; tuning and convergence diagnostics. Crossvalidation. Time permitting, optional topics include numerical analysis in R, density estimation, permutation tests, subsampling, and graphical models. Prior knowledge of R at the level used in Math 494 is required. Prerequisite: Math 233, 309, 493, 494 (not concurrently); acquaintance with fundamentals of programming in R.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 493 Probability
Mathematical theory and application of probability at the advanced undergraduate level; a calculus based introduction to probability theory. Topics include the computational basics of probability theory, combinatorial methods, conditional probability including Bayes' theorem, random variables and distributions, expectations and moments, the classical distributions, and the central limit theorem. Prerequisites: Math 233. Math 318 or Math 308 is recommended.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 494 Mathematical Statistics
Theory of estimation, minimum variance and unbiased estimators, maximum likelihood theory, Bayesian estimation, prior and posterior distributions, confidence intervals for general estimators, standard estimators and distributions such as the Studentt and Fdistribution from a more advanced viewpoint, hypothesis testing, the NeymannPearson Lemma (about best possible tests), linear models, and other topics as time permits. Prerequisites: CSE 131 or 200, Math 3200 and 493, or permission of the instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 495 Stochastic Processes
Content varies with each offering of the course. Past offerings have included such topics as random walks, Markov chains, Gaussian processes, empirical processes, Markov jump processes and a short introduction to martingales, Brownian motion and stochastic integrals. Prerequisites: Math 318 and Math 493, or permission of instructor.
Credit 3 units. A&S IQ: NSM Arch: NSM Art: NSM
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L24 Math 496 Topics in Statistics
Topic varies with each offering.
Credit 3 units. Art: NSM
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L24 Math 496A Topics in Algebra
Topic varies with each offering.
Credit 3 units. A&S IQ: NSM
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L24 Math 497 Linear Representations of Finite Group and Lie Groups
An introduction to the theory of group representations and some of its applications. Topics include: general facts about group and linear algebra; linear representations of finite and compact groups; Lie groups and Lie algebras; applications to partial differential equations and quantum theory. Prerequisites: Math 309 and 318 or permission from the instructor.
Credit 3 units. A&S IQ: NSM Art: NSM
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L24 Math 499 Study for Honors
Prerequisites: junior or senior standing, eligibility for honors work in mathematics and permission of the department's director of undergraduate studies.
Credit 3 units. Art: NSM
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