Handbook for Mathematics Majors and Minors
Information for Mathematics Majors and Minors
Handbook for Majors and Minors: TOCHandbook for Mathematics Majors and MinorsPublisher's Note
The Duke University Handbook for Mathematics Majors and Minors is published annually by the Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA.
Copies of this handbook are available from Cynthia Wilkerson (121C Physics Building, (919) 660-2801, cbw@math.duke.edu).
The 1995-1996 version of this handbook will be installed on the Department of Mathematics World Wide Web site (http://www.math.duke.edu).
Corrections to this handbook, proposed additions or revisions, and questions not addressed herein should be directed to Harold Layton (217B Physics Building, (919) 660-2809, dus@math.duke.edu); electronic mail is preferred.
Acknowledgments
The 1995-96 edition of this handbook depends heavily on earlier editions prepared by Richard Hodel, David Kraines, Gregory Lawler, and Richard Scoville.
The assistance of Lewis Blake, Jack Bookman, Robert Bryant, John Davies, Dick Hain, Joan McLaughlin, William Pardon, Richard Scoville, Carolyn Sessoms, and Cynthia Wilkerson is gratefully acknowledged.
Harold Layton
Director of Undergraduate Studies
Chairman
John Harer
132A Physics Building
(919) 660-2813, harer@math.duke.edu
Associate Chairman
William Pardon
124A Physics Building
(919) 660-2838, wlp@math.duke.edu
Director of Graduate Studies
Richard Hain
135C Physics Building
(919) 660-2819, hain@math.duke.edu
Director of Undergraduate Studies
Harold Layton
217B Physics Building
(919) 660-2809
dus@math.duke.edu
Supervisor of First-year Instruction
Lewis Blake
117 Physics Building
(919) 660-2800
sfi@math.duke.edu
Department Address
Department of Mathematics
Duke University
Box 90320
Durham, NC 27708-0320
Department Phone Number
(919) 660-2800
Facsimile
(919) 660-2821
Electronic Mail
dept@math.duke.edu
World Wide Web Home Page URL
http://www.math.duke.edu
September
4 Monday--Labor Day, classes in session
8 Friday--Drop/Add ends
29-30 Friday-Saturday--Parents' Weekend
October
1 Sunday--Parents' Weekend continues
6-8 Friday-Sunday--Homecoming
13 Friday--Last day for reporting mid-semester grades
13 Friday, 7:00 P.M.--Fall break begins
18 Wednesday, 8:00 A.M.--Classes resume
25 Wednesday--Registration begins for spring semester, 1996
November
14 Tuesday--Registration ends for spring semester,
15 Wednesday--Drop/Add begins
22 Wednesday, 12:40 P.M.--Thanksgiving recess begins
27 Monday, 8:00 A.M.--Classes resume
December
7 Thursday, 7:00 P.M.--Fall semester classes end
8-10 Friday-Sunday--Reading period
10 Sunday--Founders' Day
11 Monday, 9:00 A.M.--Final examinations begin
16 Saturday, 10:00 P.M.--Final examinations end
August
23 Wednesday--Orientation begins; assemblies for all new undergraduate students
28 Monday, 8:00 A.M.--Fall semester classes begin
February
23 Friday--Last day for reporting mid-semester grades
March
8 Friday, 7:00 P.M.--Spring recess beings
18 Monday, 8:00 A.M.--Classes resume
27 Wednesday--Registration begins for fall semester, 1996, and summer, 1996
April
11 Thursday--Registration ends for fall semester, 1996; summer registration continues
12 Friday--Drop/Add begins
24 Wednesday, 7:00 P.M.--Spring semester classes end
25-28 Thursday-Sunday--Reading period
29 Monday, 9:00 A.M.--Final examinations begin
May
4 Saturday, 10:00 P.M.--Final examinations end
10 Friday--Commencement begins
12 Sunday--Graduation exercises. Conferring of degrees
January
10 Wednesday--Registration and matriculation of new undergraduate students
11 Thursday, 8:00 A.M.--Spring semester classes begin
24 Wednesday--Drop/Add ends
This handbook is directed primarily to mathematics majors and minors; its purpose is to provide useful advice and information so that students can get the most out of their studies in mathematics. This handbook should also be a useful resource for potential majors and minors and for university personnel who advise students. The information and policies set forth here are intended to supplement material contained in the Bulletin of Duke University 1995-1996: Undergraduate Instruction.
The most important information in this handbook is organized in three main sections. The first section, Course Selection, is intended to assist students in developing programs of study that meet university requirements and that serve their educational and professional objectives.
The second section is intended to enrich the undergraduate experience by describing Resources and Opportunities available to students of mathematics.
The third section, After Graduation: Educational and Professional Opportunities, is intended to give a brief introduction to the careers and programs of study for which mathematics provides a good foundation.
* * * * *
A popular modern dictionary¹ defines mathematics as
mathematics: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.However (in the view of the current Director of Undergraduate Studies), a more complete and appropriately general definition of mathematics² is given by
mathematics: the science of abstract structure.
Indeed the inestimable importance of mathematics arises directly from the identification of mathematics as the study of the essential structure that remains in a problem or situation after all nonessential elements have been stripped away. Consequently, mathematics is a science of extraordinary intrinsic beauty, highly deserving of study for the sake of that beauty, standing alone. But owing to its generality and breadth, mathematics is an indispensable component of rational discourse, sound public policy, scientific understanding, and technological advancement. On pages 4 and 5, in a section entitled The Nature of Mathematics, some excerpts are reproduced from an essay that seeks to characterize mathematics and to describe its emerging role in today's world.
Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty.
In addition to theorems and theories, mathematics offers distinctive modes of
thought which are both versatile and powerful, including modeling, abstraction,
optimization, logical analysis, inference from data, and use of symbols.
Experience with mathematical modes of thought builds mathematical power--a
capacity of mind of increasing value in this technological age that enables
one to read critically, to identify fallacies, to detect bias, to assess risk,
and to suggest alternatives. Mathematics empowers us to understand better the
information-laden world in which we live.
* * * * *
During the first half of the twentieth century, mathematical growth was stimulated primarily by the power of abstraction and deduction, climaxing more than two centuries of effort to extract full benefit from the mathematical principles of physical science formulated by Isaac Newton. Now, as the century closes, the historic alliances of mathematics with science are expanding rapidly; the highly developed legacy of classical mathematical theory is being put to broad and often stunning use in a vast mathematical landscape.
Several particular events triggered periods of explosive growth. The Second World War forced development of many new and powerful methods of applied mathematics. Postwar government investment in mathematics, fueled by Sputnik, accelerated growth in both education and research. Then the development of electronic computing moved mathematics toward an algorithmic perspective even as it provided mathematicians with a powerful tool for exploring patterns and testing conjectures.
At the end of the nineteenth century, the axiomatization of mathematics on a foundation of logic and sets made possible grand theories of algebra, analysis, and topology whose synthesis dominated mathematics research and teaching for the first two thirds of the twentieth century. These traditional areas have now been supplemented by major developments in other mathematical sciences--in number theory, logic, statistics, operations research, probability, computation, geometry, and combinatorics.
In each of these subdisciplines, applications parallel theory. Even the most esoteric and abstract parts of mathematics--number theory and logic, for example--are now used routinely in applications (for example, in computer science and cryptography). Fifty years ago, the leading British mathematician G.H. Hardy could boast that number theory was the most pure and least useful part of mathematics. Today, Hardy's mathematics is studied as an essential prerequisite to many applications, including control of automated systems, data transmission from remote satellites, protection of financial records, and efficient algorithms for computation.
In 1960, at a time when theoretical physics was the central jewel in the crown of applied mathematics, Eugene Wigner wrote about the ``unreasonable effectiveness'' of mathematics in the natural sciences: ``The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.'' Theoretical physics has continued to adopt (and occasionally invent) increasingly abstract mathematical models as the foundation for current theories. For example, Lie groups and gauge theories--exotic expressions of symmetry--are fundamental tools in the physicist's search for a unified theory of force.
During this same period, however, striking applications of mathematics have emerged across the entire landscape of natural, behavioral, and social sciences. All advances in design, control, and efficiency of modern airliners depend on sophisticated mathematical models that simulate performance before prototypes are built. From medical technology (CAT scanners) to economic planning (input/output models of economic behavior), from genetics (decoding of DNA) to geology (locating oil reserves), mathematics has made an indelible imprint on every part of modern science, even as science itself has stimulated the growth of many branches of mathematics.
Applications of one part of mathematics to another--of geometry to analysis, of probability to number theory--provide renewed evidence of the fundamental unity of mathematics. Despite frequent connections among problems in science and mathematics, the constant discovery of new alliances retains a surprising degree of unpredictability and serendipity. Whether planned or unplanned, the cross-fertilization between science and mathematics in problems, theories, and concepts has rarely been greater than it is now, in this last quarter of the twentieth century.
111. Applied Mathematical Analysis I114. Applied Mathematical Analysis II
120S. Introduction to Theoretical Mathematics
121. Introduction to Abstract Algebra
123S. Geometry
124. Combinatorics
126. Introduction to Linear Programming and Game Theory
128. Number Theory
131. Elementary Differential Equations
132S. Qualitative Theory of Ordinary Differential Equations
135. Probability (C-L. STA 104)
136. Statistics (C-L. STA 114)
139. Advanced Calculus I
149S. Problem Solving Seminar
160. Mathematical Numerical Analysis
171S. Elementary Topology
181. Complex Analysis
187. Introduction to Mathematical Logic
196S. Seminar in Mathematical Model Building
197S. Seminar in Mathematics
200. Introduction to Algebraic Structures I
201. Introduction to Algebraic Structures II
203 . Basic Analysis I
204. Basic Analysis II
205. Topology
206. Differential Geometry
221. Numerical Analysis (C-L. CPS 250)
233. Asymptotic and Perturbation Methods
238, 239. Topics in Applied Mathematics
240. Applied Stochastic Processes (C-L. STA 253)
241. Introduction to Linear Models (C-L. STA 244)
242. Introduction to Multivariate Statistics (C-L. STA 245)
The numbering scheme of upper level courses in the
Department of Mathematics (which differs somewhat
from that of other departments) is given below.
Numbers
< or = 199 Undergraduate courses.
200-206 Primarily undergraduate courses.
These courses are recommended for students
planning graduate work in mathematics.
221-245 Primarily graduate courses for students in
departments other than Mathematics. These
courses may also be appropriate for advanced
undergraduates with interest in the applications
of mathematics.
> or = 250 Primarily graduate courses for students in
mathematics. However, sufficiently
prepared undergraduates may be
permitted to enroll.
Every semester: 111, 114, 121, 131, 135, 139
Every fall: 200, 203, 205, 241
Every spring: 120S, 136, 201, 204, 206, 242
Fall or spring: 124, 126, 132S, 187
The Department of Mathematics offers both the A.B. degree and the B.S. degree. Students who plan to attend graduate school in mathematics or the sciences should consider working toward the B.S. degree, which requires at least eight courses in mathematics numbered above Mathematics 111. The A.B. degree requires at least six and one-half courses numbered above Mathematics 111. The specific requirements for each degree are listed below.
Prerequisites: Mathematics 31 or 31L or an equivalent course (Advanced Placement course credit allowed); Mathematics 32 or 32L or 41 or an equivalent course (Advanced Placement course credit allowed); Mathematics 103 and Mathematics 104 or equivalent courses. (Many upper level mathematics courses assume programming experience at the level of Computer Science 4. Students without computer experience are encouraged to take Computer Science 6.)Major Requirements: Six and one-half courses in mathematics numbered above 111 including Mathematics 121 or 200 and Mathematics 139 or 203.
Prerequisites: Mathematics 31 or 31L or an equivalent course (Advanced Placement course credit allowed); Mathematics 32 or 32L or 41 or an equivalent course (Advanced Placement course credit allowed); Mathematics 103 and 104 or the equivalent. (Many upper level mathematics courses assume programming experience at the level of Computer Science 4. Students without computer experience are encouraged to take Computer Science 6.)Major Requirements: Eight courses in mathematics numbered above 111 including Mathematics 121 or 200; Mathematics 139 or 203; and one of Mathematics 136, 181, 204, 205. Also, Physics 41L-42L, 51L-52L, or 53L-54L.
Prerequisites: Mathematics 103 or the equivalent. (Many upper-level courses assume programming experience at the level of Computer Science 4. Students without programming experience are encouraged to take Computer Science 6.)
Minor requirements: Five courses as follows: either Mathematics 104 or Mathematics 111, but not both, and four additional courses in mathematics numbered above 111, to include at least one course (or its equivalent) selected from the following: Mathematics 121, 132S, 135, 139, 160, 181, 187, and courses in Mathematics numbered at the 200 level.
Advising. A student majoring in mathematics should develop a program based on personal interests and needs in consultation with a faculty advisor in the Department of Mathematics. Faculty advisors for first majors are assigned by the Director of Undergraduate Studies; first majors are required to meet with their advisors each term during the registration interval. Second majors and students considering a major, but who have yet to declare, can see the Director of Undergraduate Studies for advice.
A student minoring in mathematics will usually receive formal advising in the department of his or her major. However, a mathematics minor should seek supplementary advice from the mathematics faculty or consult the Director of Undergraduate Studies for advice or for referral to an appropriate member of the mathematics faculty.
Choosing courses. Every mathematics major must take one course in algebra (Mathematics 121 or Mathematics 200) and one course in advanced calculus (Mathematics 139 or Mathematics 203). An essential part of these courses is proving mathematical theorems. Students with little exposure to proofs should probably take the 100-level version of these courses, possibly prefacing them with Mathematics 120S (see paragraph below). Students who are comfortable with abstract ideas, and especially those students who are contemplating graduate work in mathematics, should consider taking the 200-level courses. The remaining courses may be chosen from both pure and applied areas of mathematics.
Mathematics 120S is a half-course recommended for prospective mathematics majors who feel the need to improve skills in logical reasoning and theorem-proving before taking Mathematics 121 and 139. Ideally, Mathematics 120S should be taken before the junior year and concurrently with Mathematics 103 or Mathematics 104. Students working toward the A.B. degree who do not take Mathematics 120S will usually fulfill their major requirements by taking at least seven full courses above Mathematics 111.
Probability and statistics courses. The standard sequence in probability and statistics is Mathematics 135-136. Mathematics 135 covers the basics of probability and Mathematics 136 covers statistics, building on the material in Mathematics 135. Those desiring a further course in probability should select Mathematics 240; a further course in statistics is Mathematics 241.
The Institute of Statistics and Decision Sciences (ISDS) offers a number of courses in statistics at various levels for students of varied mathematics backgrounds. Usually, such courses cannot be counted for mathematics major or minor credit unless they are cross-listed in the Department of Mathematics. The Director of Undergraduate Studies may approve certain statistics courses numbered above 200 for credit, but usually only for courses that have a prerequisite of Mathematics 136 or its equivalent.
For university policy on transfer credit for courses taken elsewhere, see pages 39-40 in the Bulletin of Duke University 1995-96: Undergraduate Instruction. Note specifically the sentence on page 40 that reads, ``Students wishing to transfer credit for study at another accredited college while on leave or during the summer must present a catalog of that college to the appropriate dean and director of undergraduate studies and obtain their approval prior to taking the courses.''
Thus, before enrolling at another school in a course for which transfer credit is wanted, a student must (1) obtain departmental approval for the course, and (2) obtain approval from the student's academic dean.
To obtain departmental approval a student must meet with the Director of Undergraduate Studies for courses numbered above Mathematics 103 and with the Supervisor of First-year Instruction for courses numbered 103 and below. (Additional considerations, not cited below, may apply to courses numbered 103 and below.)
Although the decision to approve or disapprove a particular course will be made by the Director of Undergraduate Studies or the Supervisor of First-year Instruction, a student can often make a preliminary determination by following the procedure below.
The university does not include a grade earned at another school as part of a student's official transcript.
A student who has obtained transfer credit may still enroll in the corresponding Duke course, but transfer credit will then be lost.
General questions about university policy on transfer credit should be addressed to John Rider, to whom the required approval forms and transcripts are sent (103A Allen Building, 684-5353, facsimile: 684-4500, JohnR@PlanOfc.duke.edu).
This section provides recommended course sequences appropriate to areas where a mathematics background is helpful, recommended, or required. For additional information on such areas, see the subsequent section, After Graduation: Educational and Professional Opportunities (page 24).
Students with an interest in the applications of mathematics should take Mathematics 131, 135, 136, and 160 (or 221). Other electives depend on particular interests; recommendations are given below.
Engineering and Natural Science MTH 114, 132S, 181, 196S, 238, 240
Business and Economics MTH 126, 132S, 240
Computer Science MTH 124, 126, 187, 200, 201
To help decide if one is suited to an actuarial career, a summer internship with an insurance company or consulting firm may be helpful. Summer openings are limited and are often filled by January or February; one's chances of being accepted are greatly improved by having passed the first examination.
Some of the topics of the earlier examinations along with recommended supporting Duke courses are:
Calculus and linear algebra MTH 31, 32, 103, 104
Probability and statistics MTH 135, 136
Applied statistical methods MTH 241
Operations research MTH 126, 240
Numerical methods MTH 160, 221
Courses in accounting, finance, economics, and computer science are also helpful preparation for a career in actuarial science.
The curriculum in Mathematical Sciences at the University of North Carolina at Chapel Hill includes an Actuarial Science option through which students may take specialized courses in actuarial mathematics during the spring semester. Under a reciprocal agreement between the two universities, students at Duke may enroll concurrently in these courses offered by UNC-Chapel Hill (see page 76 of the Bulletin of Duke University, 1995-96: Undergraduate Instruction). Note, however, that prior approval from the Director of Undergraduate Studies must be sought for such courses to count toward mathematics major or minor credit.
Inquiries about the courses at UNC or about actuarial science in general may be made to Charles W. Dunn, a former Duke graduate and Fellow of the Society of Actuaries. His office is in the First Federal Building at Five Points in downtown Durham (688-8913).
Required Recommended
Analysis (139 or 203) Number theory (128)
Algebra (121 or 200) Topology (171S or 205)
Geometry (123) Numerical Analysis (160, or CPS 150, 250)
Probability/Statistics (135, 136) Logic (187)
Computer Science (CPS 4 or 6) Combinatorics (124)
Differential Equations (131)
Algebra (201)
Physics (PHY 41L-42L, 51L-52L, or 53L-54L)
For further information, see David Malone, Program in Education (213 West Duke Building, East Campus, 660-3074, dmalone@acpub.duke.edu), or Jack Bookman (211 Physics Building, 660-2831, bookman@math.duke.edu).
Students who do not intend to pursue graduate work should elect Mathematics 135, 136, 241, CPS 6 or 100 as well as some of the following courses: Mathematics 242, 240, 160 (or 221), STA 242, 203S, CPS 108.
Statistics students at all levels are encouraged to take computer programming courses.
At present, job prospects are good at all degree levels for those who have a strong background in statistics and some computer programming experience. For further information, see Valen Johnson, Director of Undergraduate Studies in the Institute of Statistics and Decision Sciences, in 219A Old Chemistry (684-8753, valen@isds.duke.edu).
Below are catalog descriptions of the mathematics courses above Mathematics 104 that are most often taken by undergraduates.
111. Applied Mathematical Analysis I. First and second order differential equations with applications; matrices, eigenvalues, and eigenvectors; linear systems of differential equations; Fourier series and applications to partial differential equations. Intended primarily for engineering and science students with emphasis on problem solving. Not open to students who have had Mathematics 131. Prerequisite: Mathematics 103.
(Note: Mathematics 111, which is not intended for mathematics majors, overlaps material in Mathematics 104; mathematics majors should take Mathematics 131, rather than Mathematics 111, for a first course in ordinary differential equations.)
114. Applied Mathematical Analysis II. Boundary value problems, complex variables, Cauchy's theorem, residues, Fourier transform, applications to partial differential equations. Not open to students who have had Mathematics 181 or 230. Prerequisites: Mathematics 111 or 131, or 103 and consent of instructor.
120S. Introduction to Theoretical Mathematics. Topics from set theory, number theory, algebra and analysis. Recommended for prospective mathematics majors who feel the need to improve skills in logical reasoning and theorem-proving before taking Mathematics 121 and 139. Not open to students who have had Mathematics 121, Mathematics 139, or equivalents. Prerequisite: Mathematics 103; corequisite: Mathematics 104. Half course.
121. Introduction to Abstract Algebra. Groups, rings, and fields. Students intending to take a year of abstract algebra should take Mathematics 200-201. Not open to students who have had Mathematics 200. Prerequisites: Mathematics 104 or 111.
123S. Geometry. Euclidean geometry, inversive and projective geometries, topology (Möbius strips, Klein bottle, projective space), and non-Euclidean geometries in two and three dimensions. Prerequisite: Mathematics 32 or 41 or consent of the instructor.
124. Combinatorics. Permutations and combinations, generating functions, recurrence relations; topics in enumeration theory, including the Principle of Inclusion-Exclusion and Polya Theory; topics in graph theory, including trees, circuits, and matrix representations; applications. Prerequisites: Mathematics 104 or consent of the instructor.
126. Introduction to Linear Programming and Game Theory. Fundamental properties of linear programs; linear inequalities and convex sets; primal simplex method, duality; integer programming; two-person and matrix games. Prerequisite: Mathematics 104.
128. Number Theory. Divisibility properties of integers, prime numbers, congruences, quadratic reciprocity, number-theoretic functions, simple continued fractions, rational approximations. Prerequisite: Mathematics 32 or 41, or consent of the instructor.
131. Elementary Differential Equations. Solution of differential equations of elementary types; formation and integration of equations arising in applications. Not open to students who have had Mathematics 111. Prerequisite: Mathematics 103; corequisite: Mathematics 104.
132S. Qualitative Theory of Ordinary Differential Equations. Qualitative behavior of general systems of ordinary differential equations, with application to biological and ecological systems, oscillations in biochemistry, electrical networks, and the theory of deterministic epidemics. Prerequisite: Mathematics 131 or 111 or consent of the instructor.
135. Probability. Probability models, random variables with discrete and continuous distributions. Independence, joint distributions, conditional distributions. Expectations, functions of random variables, central limit theorem. Prerequisite: Mathematics 103. C-L. Statistics 104.
136. Statistics. Sampling distributions, point and interval estimation, maximum likelihood estimators. Tests of hypotheses, the Neyman-Pearson theorem. Bayesian methods. Not open to students who have had Statistics 112 or 213. Prerequisites: Mathematics 104 and 135. C-L. Statistics 114.
139. Advanced Calculus I. Algebraic and topological structure of the real number system; rigorous development of one-variable calculus including continuous, differentiable, and Riemann integrable functions and the Fundamental Theorem of Calculus; uniform convergence of a sequence of functions. Not open to students who have had Mathematics 203. Prerequisite: Mathematics 103.
149S. Problem Solving Seminar. Techniques for attacking and solving challenging mathematical problems and writing mathematical proofs. Course may be repeated. Prerequisite: consent of the instructor.
160. Mathematical Numerical Analysis. Zeros of functions; polynomial interpolation and splines; numerical integration and differentiation; applications to ordinary differential equations; numerical linear algebra; error analysis; extrapolation and acceleration. Not open to students who have had Computer Science 121, 150, 221, or 250. Mathematics 160 or 221, but not both, may count toward the major requirements. Prerequisites: Mathematics 103 and 104 and knowledge of an algorithmic programming language, or consent of the instructor.
171S. Elementary Topology. Metric spaces and topological spaces; basic topological properties including compactness and connectedness; Brouwer fixed point theorem for n = 2, classification of compact, connected, 2-manifolds. Prerequisites: Mathematics 103 and 104.
181. Complex Analysis. Complex numbers, analytic functions, complex integration, Taylor and Laurent series, theory of residues, argument maximum principles, conformal mapping. Not open to students who have had Mathematics 114 or 231. Prerequisite: Mathematics 139 or 203.
187. Introduction to Mathematical Logic. Propositional calculus; predicate calculus. Gödel completeness theorem, applications to formal number theory, incompleteness theorem, additional topics in proof theory or computability. Prerequisites: Mathematics 103 and 104 or Philosophy 103.
196S. Seminar in Mathematical Model Building. Real models, mathematical models, axiom systems as used in model building, deterministic and stochastic models, linear optimization, competition, graphs and networks, growth processes, evaluation of models. Term project: model of a non-mathematical problem. Prerequisites: Mathematics 103 and 104.
(Note: In recent years, Mathematics 196S has been offered in the spring semester of even-numbered years and has emphasized applications of mathematical modeling to physiology and medicine.)
197S. Seminar in Mathematics. Intended primarily for juniors and seniors majoring in mathematics. Topics vary. Prerequisites: Mathematics 103 and 104.
200. Introduction to Algebraic Structures I. Laws of composition, groups, rings; isomorphism theorems; axiomatic treatment of natural numbers; polynomial rings; division and Euclidean algorithms. Not open to students who have had Mathematics 121. Prerequisite: Mathematics 104 or equivalent.
201. Introduction to Algebraic Structures II. Vector spaces, matrices and linear transformations, fields, extensions of fields, construction of real numbers. Prerequisite: Mathematics 200, or Mathematics 121 and consent of the instructor.
203. Basic Analysis I. Topology of n-dimensional real space, continuous functions, uniform convergence, compactness, infinite series, theory of differentiation, and integration. Not open to students who have had Mathematics 139. Prerequisite: Mathematics 104.
204. Basic Analysis II. Inverse and implicit function theorems, differential forms, integrals on surfaces, Stokes' theorem. Not open to students who have had Mathematics 140. Prerequisite: Mathematics 203, or 139 and consent of the instructor.
205. Topology. Elementary topology, surfaces, covering spaces, Euler characteristic, fundamental group, homology theory, exact sequences. Prerequisite: Mathematics 104.
206. Differential Geometry. Geometry of curves and surfaces, the Serret-Frenet frame of a space curve, the Gauss curvature, Cadazzi-Mainardi equations, the Gauss-Bonnet formula. Prerequisite: Mathematics 104.
221. Numerical Analysis. Error analysis, interpolation and spline approximation, numerical differentiation and integration, solutions of linear systems, non-linear equations, and ordinary differential equations. Prerequisites: knowledge of an algorithmic programming language, intermediate calculus including some differential equations, and Mathematics 104. C-L. Computer Science 250.
233. Asymptotic and Perturbation Methods. Asymptotic solution of linear and nonlinear ordinary and partial differential equations. Asymptotic evaluation of integrals. Singular perturbation. Boundary layer theory. Multiple scale analysis. Prerequisite: Mathematics 114 or equivalent.
238, 239. Topics in Applied Mathematics. Conceptual basis of applied mathematics, combinatorics, graph theory, game theory, mathematical programming, or numerical solution of ordinary and partial differential equations. Prerequisites: Mathematics 103 and 104 or equivalents.
240. Applied Stochastic Processes. An introduction to stochastic processes without measure theory. Topics selected from: Markov chains in discrete and continuous time, queuing theory, branching processes, martingales, Brownian motion, stochastic calculus. Prerequisite: Mathematics 135 or equivalent.
241. Introduction to Linear Models. Multiple linear regression. Estimation and prediction. Likelihood, Bayesian, and geometric methods. Analysis of variance and covariance. Residual analysis and diagnostics. Model building, selection, and validation. Prerequisites: Mathematics 104 and Statistics 113 or 210. C.-L. Statistics 244.
242. Introduction to Multivariate Statistics. Multinormal distributions, multivariate general linear models, Hotelling's T² statistic, Roy union-intersection principle, principal components, canonical analysis, factor analysis. Prerequisite: Mathematics 241 or equivalent. C.-L. Statistics 245.
All mathematics majors and minors are encouraged to develop computer skills and to make use of electronic mail. Some sections of Linear Algebra, Probability, Statistics and Differential Equations may require students to use computers. In some cases, university-maintained personal computer clusters may suffice; in other cases students may be required to use the Sun Cluster, described below.
General information. The department maintains a cluster of 10 Sun Workstations and a printer. The cluster is located in room 250A-B of the Physics Building and is open 24 hours a day. The cluster is used for undergraduate and graduate instruction and other appropriate purposes. Students doing mathematics work have priority for use of the workstations.
The Sun Workstations, which utilize the UNIX operating system, provide access to electronic mail and the World Wide Web; moreover, original or previously written programs in FORTRAN and C may be run on these machines, and the symbolic manipulation program MAPLE is available to all users.
Opening an account. Mathematics majors may obtain accounts on the Sun Cluster by application to the system administrators. Non-majors who are enrolled in mathematics courses that use the Sun Cluster may be granted temporary accounts that expire automatically at the end of the academic term. The system administrators are Yunliang Yu (024A Physics Building, 660-2803, yu@math.duke.edu) and John Davies (217A Physics Building, 660-2802, jdavies@math.duke.edu).
Electronic mail. Any user who has an account on the system can send and receive e-mail; a user's address is userid@math.duke.edu. The command for sending mail is mail userid@node, where userid is the user login identity of the recipient, and node is the address of the machine one is mailing to. To send and receive mail, use the command mail, elm, or pine; or use mailtool in OpenWindows. A user can get help by typing ? to the mail prompt.
World Wide Web (WWW): Department of Mathematics Home Page. A wide variety of current departmental information, including course information, departmental policies, and pointers to other mathematical web servers, can be found on the WWW home page. The home page can be called up from a suitably connected computer by running the program netscape. From a remote site, use the Uniform Resource Locator (URL), http://www.math.duke.edu.
Current versions of the Local Unix Guide and this handbook are available as links from the department's home page.
Inquiries and help. Routine questions (e.g., ``How do I use this program? Why doesn't this work? How do I set up the defaults?'') should be addressed by electronic mail to problem@math.duke.edu.
The Suns have an on-line manual, called up by the man command. With the use of this command, one can find out how to use many of the built-in SunOS commands. In general, man command or man -k command gives information about a particular command . Enter man man to find out how to use the manual pages.
Another good resource is the AnswerBook in OpenWindows. Typing the command answerbook at the prompt: one will see two windows ``Navigator'' and ``Viewer'' open on the screen. Use the Navigator to select pages for inspection, which will be displayed on the Viewer.
Security. The UNIX operating system is not a completely secure computing environment. Every user is responsible for the security of his or her own account. Departmental policy prohibits the sharing of passwords or accounts and any other activity that undermines the security of the university's computer systems. Users should be sure to log out when they finish using the machines in university clusters. Any suspicious activities related to the computers or accounts should be reported to the system administrators immediately.
User policy. The computer system of the Department of Mathematics is provided to support mathematical instruction and research. To ensure that the system is fully available for these purposes, the Department of Mathematics has established a policy on responsible use of its computer system. This policy can be found in the Local Unix Guide. Violations of the user policy may lead to suspension of the user's account or referral to the appropriate authority for disciplinary action. University policies and regulations, including the Student Honor Code, and state statutes and rules, including the North Carolina Computer Crimes Act, cover many potential abuses of the computer system.
The Math-Physics Library is located in the Physics Building, Room 233 (660-5960, facsimile: 681-7618, mplib@phy.duke.edu). The library has a comprehensive collection of mathematics textbooks, monographs, journals, and reference works. In addition, the library maintains materials on reserve for specific mathematics courses.
For further information, see the Math-Physics Library
World Wide Web home page at
http://www.phy.duke.edu/~mplib.
The Duke University Mathematics Union (DUMU) is a club for math majors. Recent activities have included sponsorship of talks for undergraduates (see below) and hosting a successful mathematics contest for high school students. The contest attracted participants from throughout the Southeast.
This year, DUMU will be run by Michael Rierson (rierson@math.duke.edu) and Tung Tran (trantt@math.duke.edu). Information about meetings and activities will be posted in the Mathematics portion of the Physics Building. For additional information about DUMU, see the department's WWW server.
From time to time a mathematician is invited to give a talk that is specifically for undergraduates. Recent speakers and their topics are listed below.
Thomas Banchoff The Fourth Dimension: Computer Animated
(Brown) Geometry and Modern Art
Carl Pomerance Prime Numbers: How to Talk to Aliens from
(U. Georgia) Space and Other Matters
Peter Hilton Sums of Random Integers
(SUNY Binghamton)
David Morrison Stalking the Shape of the Universe
(Duke)
J. H. Conway Some Tricks with String
(Princeton)
Persi Diaconis The Mathematics of Shuffling Cards
(MIT)
Joseph Gallian Touring the Torus
(U. Minn., Duluth)
Robert Devaney The Mathematics behind the Mandelbrot Set
(Boston)
Donald Knuth Leaper Graphs
(Stanford)
Colin Adams Real Estate in Hyperbolic Space
(Williams)
David Bayer How Many Shuffles Does It Take to Mix Up
(Columbia) a Deck of Cards?
The Department of Mathematics employs undergraduate students as office assistants, graders, help room/session tutors, and laboratory teaching assistants. Working as a laboratory teaching assistant can be valuable preparation for students planning to become teachers of mathematics.
Applicants for the positions of grader, help room/session tutor, and laboratory teaching assistant should have taken the course involved and received a grade no lower than B. However, a student who received a good grade in a higher level course or who has advanced placement may be eligible to grade for a lower level course not taken.
Students wishing to apply for available positions may obtain an application in the Department of Mathematics Offices, Physics Building, Suite 121.
Julia Dale, an Assistant Professor of Mathematics at Duke University, died early in her career on January 13, 1936. Friends and relatives of Professor Dale established the Julia Dale Memorial Fund in the Spring of 1937; the Julia Dale Prize is supported by the income from this fund. Recent first-prize recipients are listed below.
Year Student
1991 Jeanne Nielsen
1992 William Alan Schneeberger
1993 Sang Chin
Jennifer Slimowitz
1994 Jeffrey Vanderkam
1995 Paul A. Dreyer, Jr.
Craig B. Gentry
Karl Menger (1902-1985) was one of the most distinguished mathematicians of the twentieth century. He made major contributions to a number of areas of mathematics, including dimension theory, logic, lattice theory, differential geometry, and graph theory. Menger, who held academic positions in Europe and the United States, was widely published. The Karl Menger Award was established by a gift to Duke University from George and Eva Menger-Hammond, the daughter of Karl Menger. Recent recipients of the Karl Menger Award are listed below.
Year Awardees
1991 Jeanne Nielsen
William Schneeberger
Jeffrey Vanderkam
1992 David Bigham
David Jones
Jeffrey Vanderkam
1993 Craig Gentry
Alexander Hartemink
Jeffrey Vanderkam
1994 Andrew Dittmer
Craig Gentry
Jeffrey Vanderkam
1995 James Harrington
Robert Schneck
Noam Shazeer
The department offers to mathematics majors the opportunity for graduation with Latin honors by honors project. The requirements are:
A student must apply for graduation with honors in the spring of the junior year. The application must include the name of the faculty advisor and a general plan for independent study in the senior year. In the spring of the senior year, the Director of Undergraduate Studies will name a two-person committee to review the paper. The faculty will be given the opportunity to read the paper and make comments to the committee. The student will be expected to give a presentation of his or her work in a seminar intended for both faculty and students. The committee, in consultation with the Director of Undergraduate Studies, determines whether Latin Honors will be awarded.
David Jones Primality Testing, Factoring and Schoen
(1992) Continued Fractions
Will Schneeberger The Axiom Diamond Shoenfield
(1992)
Linie Chang Mathematical Models in Immunology Reed
(1993)
Sang Chin The 3-fold Cover of the G-hole Torus Hain
(1993)
Jennifer Slimowitz Transitions of the Gaps between Pardon
(1993) the Integers N Satisfying (N theta)< psi
Jeffrey Vanderkam Eigenfunctions of an Acoustic System Beale
(1994)
Paul Dreyer Knot Theory and the Human Pretzel Game Harer
(1995)
Paul Koss The Effects of Noise on the Kraines
(1995) Iterated Prisoner's Dilemma
Awardee Title of Paper Advisor
Jeanne Nielsen Triply Periodic Minimal Surfaces in R³ Bryant
(1991)
Business School Law School Health Professions
Dean Martina Bryant Dean Gerald Wilson Dean Kay Singer
03 Allen Building 116 Allen Building 303 Union West
684-2075 684-2865 684-6221
mbryant@mail01.adm.duke.edu ksinger@asdean.duke.edu
First-year students and sophomores interested in the health professions should see Professor Donna Kostyu (684-6217, ddkk@acpub.duke.edu).
Although successful actuaries have come from diverse college majors, the obvious candidates are those demonstrating skill in mathematics, verbal communication, and leadership. Indeed, the problems an actuary is likely to face may often involve business, social, and political considerations. Thus there may be more than one solution, or there may be no practical solution at all. Insurance companies actively recruit Duke mathematics majors, and each year several students accept positions with such firms.
Judging from the amount of material received from major companies, actuaries are in substantial demand; a number of announcements, booklets, and pamphlets are available in 217 Physics, including application information for actuarial examinations.
About one-half of Ph.D.'s in mathematics find long-term employment at academic institutions, either at research universities such as Duke or at colleges devoted primarily to undergraduate teaching. At research universities, the effort of most faculty members is divided between teaching and conducting research in mathematics. The employment situation for Ph.D.'s in mathematics for academic positions is currently very tight. Most nonfaculty mathematicians are employed by government agencies, the private service sector, or the manufacturing industry. Current budget-cutting initiatives may adversely affect the number of positions available in government and in defense-related industries.
Students considering graduate school in mathematics are urged to consult with the mathematics faculty and with the Director of Graduate Studies. The choice of graduate school and the area of study may make a significant difference in future job prospects. The Director of Undergraduate Studies receives material on graduate programs in mathematics from all over the country; this material is on file in 217 Physics. (Information about Duke's program should soon appear on the department's WWW server.)
Graduate school in statistics, operations research, computer science, and mathematics-related scientific fields. Some information about graduate programs in fields closely related to mathematics is available in 217 Physics. Students are urged, however, to consult with corresponding Duke departments and with prospective graduate programs.
Mathematical Occupations. For an evaluation of professional opportunities in actuarial science, computer science, mathematics, operations research, and statistics, a section titled ``Computer, Mathematical, and Operations Research Occupations'' from the Occupational Outlook Handbook, published by the U.S. Department of Labor, is available in 217 Physics. The complete Occupational Outlook Handbook is available for examination in the Depository of U.S. Documents in Perkins Library.
United States Government. A number of U.S. Government agencies hire mathematics majors. Information is on file in 217 physics on
Financial Services, Industry, Management, etc. There are many occupations that do not use mathematics directly but for which a major in mathematics is excellent preparation. Many employers are looking for individuals who have skills that are indicated by mathematical training: clear, logical thinking; ability to attack a problem and find the best solution; prompt attention to daily work; sureness in handling numerical data; analytical skills. Because many companies provide specific on-the-job training, a broad range of courses may be the best preparation for such occupations.
Some information about opportunities in the finance, industry, and management is on file in 217 Physics.
Career Development Center. The Career Development Center (located in Room 109, Page Building) is an excellent source of information on career opportunities in mathematics. Patricia O'Connor (660-1059, poconnor@acpub) is the career specialist in mathematics and related fields; appointments should be made at 660-1050.
Summary of Information on File. Information on opportunities for mathematics majors and minors after graduation is on file in 217 Physics as follows:
Recent Graduates. The following is list of positions taken by recent Duke alumni with undergraduate degrees in mathematics:
1993
- Consultant, Andersen Consulting
- Investment banking, Goldman, Sachs & Co.
- Nuclear Power School, U.S. Navy
- Pensions actuary, Wyatt Co.
- Software engineer, SRA
1994
- Assistant trader, Swiss Bank
- Actuary, CIGNA
- Financial analyst, First Boston
- Investment management, J. P. Morgan
- High school teacher
1995
- Credit analyst, International Paper
- Financial services, John Hancock Life
- Financial services, Nations Bank
- Peace Corps
- Software design, Wyatt Co.
Faculty Member Research Area
W. K. Allard Calculus of variations
(Villanova, Brown)
J. T. Beale Partial differential equations
(Cal. Tech, Stanford)
A. L. Bertozzi Nonlinear partial differential equations,
(Princeton, Princeton) applied mathematics
R. Bryant Differential geometry
(N. C. State, UNC)
D. S. Burdick Mathematical statistics
(Duke, Princeton)
R. M. Hain Topology and geometry
(U. Sydney, U. Illinois)
J. Harer Geometric topology
(Haverford, Berkeley)
R. E. Hodel Set-theoretic topology
(Davidson, Duke)
J. W. Kitchen Functional analysis
(Harvard, Harvard)
D. P. Kraines Algebraic topology
(Oberlin, Berkeley)
G. F. Lawler Probability
(Virginia, Princeton)
H. Layton Mathematical physiology
(Asbury, Duke)
L. C. Moore Functional analysis, nonstandard analysis
(N. C. State, Cal. Tech.)
D. R. Morrison Algebraic geometry
(Princeton, Harvard)
W. Pardon Topology
(Michigan, Princeton)
M. C. Reed Nonlinear partial differential equations,
(Yale, Stanford) applications of mathematics to physiology
and medicine
D. Rose Numerical analysis, scientific computing
(Berkeley, Harvard )
L. Saper Differential geometry
(Yale, Princeton)
D. G. Schaeffer Partial differential equations,
(Illinois, MIT) applied mathematics
C. Schoen Algebraic geometry
(Haverford, Chicago)
R. A. Scoville Combinatorial analysis
(Yale, Yale)
D. A. Smith Numerical analysis
(Trinity, Yale)
M. Stern Differential geometry
(Texas A & M, Princeton)
J. A. Trangenstein Scientific computing, numerical solution
(U. Chicago, Cornell) of differential equations, optimization
S. Venakides Partial differential equations
(National Tech. Univ. of Athens,
Courant Institute)
Telephone numbers are (919) 660-XXXX, where XXXX is the extension number. Electronic mail addresses are USERID@math.duke.edu, where USERID is given under ``E-mail.'' Within the department's computer network, only the USERID is required.
NAME OFFICE EXTENSION EMAIL
Allard, William K. 024B 2861 wka
Beale, J. Thomas 124B 2839 beale
Bertozzi, Andrea 135D 2820 bertozzi
Blake, Lewis D. 118 2857 blake
Bookman, Jack 211 2831 bookman
Bryant, Robert L. 128A 2805 bryant
Coyle, Lester 135A 2818 coyle
Davies, John M. 127A 2802 jdavies
Dempster, Elizabeth 021 2866 dempster
Dong, Xiaoying 226 2849 dong
Farrell, Bonnie 121D 2804 bef
Hain, Richard E. 135C 2819 hain
Harer, John L. 132A 2813 harer
Hodel, Margaret J. A.107 2865 hodel
Hodel, Richard E. 222 2846 hodel
Hornung, Richard D. 024C 2828 hornung
Kitchen, Joseph 221 2845
Kraines, David P. 135D 2820 dkrain
Langen, Angelika 214 2825 langen
Lawler, Gregory F. 128B 2842 jose
Layton, Harold 217B 2809 layton
McLaughlin, Joan R. 121B 2807 joan
Moore,Lawrence C. 211A 2822 lang
Morris, Samuel R. 021 2866 sam
Morrison, David R. 213 2862 drm
Pardon, William L. 124A 2838 wlp
Reed, David 133 2816 dreed
Reed, Michael C. 215 2808 reed
Saper, Leslie D. 130 2843 saper
Schaeffer, David G. 132B 2814 dgs
Scheick, John 226 2849
Schoen, Chadmark L. 131 2844 schoen
Scoville, Richard A. 225 2848 ras
Sessoms, Carolyn 121E 2806 sessoms
Shearer, Michael 135B 2863 shearer
Smith, David K. 211B 2823 das
Southern, Mary Ann W. 233 5960 mplib
Stern, Mark A. 126 2840 stern
Tomberg, James A. 116 2858 tomberg
Trangenstein, John 024D 2824 johnt
Venakides, Stephanos 132C 2815 ven
Warner, Seth L. 227 2850 slw
Weisfeld, Morris 230 2812 morris
Wilkerson, Cynthia 121C 2801 cbw
Yang, Jun 119 2836 yang
Yu, Yunliang 024A 2803 yu
Zheng, Fangyang 127 2841 zheng
Zhou, Xin 134 2817 zhou
NAME OFFICE EXTENSION E-MAIL
Arthurs, Kayne 022 2830 ksmith
Ashih, Aaron 250F 2855 aaron
Barnes, Andrew 250D 2854 barnesa
Brooks, Beth 022 2830 brooks
Chaudhary, Sharad 020 2829 sharad
Cheng, Po-Jen 025 2832 pojen
Clelland, Jeanne 022 2830 nielsen
Clelland, Richard 024C 2828 rick
Colon, Ivan 250D 2854 colon
Culver, Tim 020 2829 tculver
Fargason, Chad 025 2832 fargason
Filip, Ann Marie 022 2866 filip
Fisher, Mary Beth 250D 2854 fisher
Foisy, Joel 250C 2853 foisy
Foltinek, Kevin 025 2832 foltinek
Georgieva, Anna 025 2832 georgiev
Horja, Richard 250D 2854 horja
Jones, Ben 250C 2853 benjones
Knudson, Kevin 250D 2854 knudson
McKay, Benjamin 022 2830 mckay
Michael, Chris 250F 2855 michael
Mitchell, Dax 250C 2858 dax
Moser, Nicholai 022 2830 moser
Odden, Christopher 025 2832 odden
Ott, Blaine 250C 2853 blaine
Riley, Phillip 020 2829 priley
Rolf, James 250F 2855 rolf
Schretter, Andrew 250C 2853 schrett
Solodovnikov, Alexander 250D 2854 solodovn
Taalman, Laura 020 2829 taal
Travers, Kirsten 025 2832 kirsten
Vlassopoulos, Ioannis 250C 2853 yiannis
Welsh, Ted 250C 2853 eww
Zhornitskaya, Liya 250F 2855 zhornits
ENTITY OFFICE EXTENSION
Duke Math Journal 229 2811
Fax 121 2821
Lounge 138 2826
CALC Lab 032 2834
CALC Lab 027 2833
Sun Cluster 250AB 2856
Student assistant 116 2800
Upstairs copy room 232 2852
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