DEBORAH J. BERGSTRAND, Professor (part time)
CHARLES M. GRINSTEAD, Professor
EUGENE A. KLOTZ, Professor
STEPHEN B. MAURER, Professor and Chair
HELENE SHAPIRO, Professor
DON H. SHIMAMOTO, Professor
JANET C. TALVACCHIA, Professor
GARIKAI CAMPBELL, Associate Professor1
PHILIP J. EVERSON, Associate Professor
CHERYL P. GROOD, Associate Professor
THOMAS J. HUNTER, Associate Professor 2
AIMEE S.A. JOHNSON, Associate Professor2
DAVID J. RUSIN, Visiting Associate Professor3
WALTER R. STROMQUIST, Visiting Associate Professor
STEVE C. WANG, Assistant Professor
STEVEN AMGOTT, Computer Laboratory Coordinator
STEPHANIE J. SPECHT, Administrative Assistant
1 Absent on leave, 2006–2007.
2 Absent on leave, spring 2007.
3 Spring 2007.
Mathematics and Statistics are among the great achievements of human intellect and at the same time powerful tools. As Galileo said, the book of the universe "is written in the language of mathematics." The goal of the department is to enable students to appreciate these achievements and use their power. To that end students in the department receive a firm foundation in pure mathematics and the opportunity to apply it – to statistics, physical science, biological science, computer science, social science, operations research, education, finance – the list grows. All courses in the department also have as a general goal the continuing development of various mathematical skills, among them:
reasoning skills: logical argument and abstraction;
formulation skills: developing mathematical models;
communication skills: expressing mathematical ideas and information clearly and precisely on paper, orally, and electronically;
computation skills: mental, by hand, and by machine, as appropriate.
Graduates of the department follow many careers paths, leading them after graduation to graduate school, in mathematics, statistics, or other fields, or to professional schools or the workplace.
Note: With this catalogue the department completes a 2-year process to revise its program and course numbering. For mathematics and statistics, earlier issues of the course catalog should not be consulted for 206–2007.
Most first-year students entering Swarthmore have had calculus while in high school and place out of at least one semester of Swarthmore’s calculus courses, whether they continue with calculus or decide, as is often best, to try other sorts of mathematics. See the discussion of placement in the following section. However, some entering students have not had the opportunity to take calculus or need to begin again. Therefore, Swarthmore offers a beginning calculus course (MATH 015) and several courses that do not require calculus or other sophisticated mathematics experiences. These courses are STAT 001 (Statistical Thinking, both semesters), MATH 003 (Introduction to Mathematical Thinking, spring semester), and STAT 011 (Statistical Methods, both semesters). MATH 003 is a writing course. Students who would like to begin calculus (MATH 015) but are not sure they are prepared must take the departmental Calculus Readiness Exam when they arrive on campus. MATH 029 (Discrete Mathematics, both semesters) also does not require any calculus but is a more sophisticated course; thus, some calculus is a useful background for it in an indirect way. Once one has had or placed out of two semesters of calculus, many other courses are available, especially in linear algebra and several-variable calculus.
To gain entrance to any mathematics course (but unnecessary to gain entrance to statistics courses), students must take at least one of the following exams: the Advanced Placement (AP) or International Baccalaureate (IB) exams, Swarthmore’s Calculus Placement Exam, or Swarthmore’s Calculus Readiness Exam. Students who do take AP or IB exams may be required to take the departmental exams as well. The Calculus Placement Exam is sent to entering first-year students over the summer, along with detailed information about the rules for placement and credit. The Calculus Readiness Exam is given on campus only, during first-year orientation.
Placement and credit mean different things. Placement allows students to skip material they have learned well already by starting at Swarthmore in more advanced courses. Credit confers placement as well but also is recorded on the student’s Swarthmore transcript and counts toward the 32 credits needed for graduation.
The Swarthmore Calculus Placement Exam is used for placement only, not credit. Credit is awarded on the basis of the AP and the IB exams, as follows:
• 1 credit (for STAT 011) for a score of 4 or 5 on the Statistics AP Test of the College Board.
• 1 credit (for MATH 015) for a score of 4 on the AB or BC Calculus AP Test of the College Board (or for an AB subscore of 4 on the BC Test) or for a score of 5 on the Higher Level Mathematics Test of the IB.
• 1.5 credits (for MATH 015 and the first half of Math 025) for a score of 5 on the AB Calculus AP Test (or for an AB subscore of 5 on the BC Test) or a score of 6 or 7 on the higher-level IB. Students who receive this credit and want to continue calculus take MATH 026.
• 2 credits (for MATH 015 and 025) for a main score of 5 on the BC Calculus AP Test.
Alternatively, any entering student who places out of MATH 015 or 025 may receive credit for those courses by passing the final exams in these courses with a grade of straight C or better. These exams must normally be taken during the student’s first semester at Swarthmore, at the time when the final exam is given for the course. Students who wish to take these exams must arrange to do so with the departmental placement coordinator and should do so during their first semester at Swarthmore.
Students who are eligible on entrance for credit for a course, but who take the course anyway, will lose the entrance credit.
First-year students seeking advanced placement and/or credit for calculus taken at another college or university must normally validate their work by taking the appropriate external or Swarthmore placement examination, as described earlier. The department does not grant credit directly for college courses taken while a student is in high school. For work beyond calculus completed before entering Swarthmore, students should consult the departmental placement coordinator to determine the Swarthmore course into which they should be placed. The department will not normally award credit for work above the first-year calculus level completed before entering Swarthmore.
Students who do not know calculus can take STAT 001 or 011. STAT 001 shows how statistics is used to gain an understanding of the world around us and to prepare students to critically interpret and evaluate statistical claims. STAT 011 is a practical course for students who expect to analyze data in their own work. Any students who think they might ever need to do statistical analyses (not just critically interpret statistical claims in the media) should take STAT 011, not STAT 001. STAT 011 leads to STAT 031 on data analysis and visualization. Students with a strong background in mathematics can begin with the theoretical course STAT 061 and continue with the 1-credit seminar STAT 111.
Students apply for a major in the middle of the second semester of the sophomore year. By the end of the sophomore year, an applicant should have received credit for, or placement out of, at least four of the following five course groups: Elementary Single-Variable Calculus (MATH 015); Further Single-Variable Calculus (MATH 025, 026, or 026S); Linear Algebra (MATH 027, 028, or 028S); Discrete Mathematics (MATH 029); and Several-Variable Calculus (MATH 033, 034, or 035). All majors must complete Linear Algebra and Several-Variable Calculus by the end of the first semester of the junior year.
In addition, a candidate should have a grade-point average in mathematics and statistics courses of at least C+. This should include at least one grade at the B level. In some cases, applicants may be deferred, pending successful work in courses to be designated by the department.
By graduation, a mathematics major must have at least 10 credits in mathematics and statistics courses. At least 5 of the credits counted in the 10 must be for courses numbered over 40. (Courses numbered under 10 do not count toward the major in any event.) Furthermore, every major is required to obtain credit for, or place out of, each of the following course groups: MATH 015; MATH 025, 025S, or 026; MATH 027, 028, or 028S; MATH 033, 034, or 035; MATH 063; and MATH 067. The two upper-level core courses, MATH 063 (Introduction to Real Analysis) and MATH 067 (Introduction to Modern Algebra), will be offered at least every fall semester. At least one of these two should be taken no later than the fall semester of the junior year, and both must be taken before the spring semester of the senior year. Finally, course majors must satisfy the departmental comprehensive requirement by passing MATH 097, Senior Conference. Normally, at least 3 of the 5 credits for courses numbered over 40 must be taken at Swarthmore, including MATH 097 and at least one of the core courses MATH063 and MATH067.
Mathematics majors are urged to study in some depth a discipline that makes use of mathematics and to acquire some facility with computers and software. Students bound for graduate work in mathematics should obtain a reading knowledge of French, German, or Russian.
The preceding requirements allow room to choose an optional special emphasis within the mathematics major. For instance:
A student may major in mathematics with an emphasis on statistics by taking the following courses at the advanced level: (1) the core analysis course (MATH 063); (2) Mathematical Statistics I (STAT 061) (3) Probability (MATH 105) or Mathematical Statistics II (STAT 111) (4) Data Analysis and Visualization (STAT 031); and (5) another mathematics course numbered over 40. Students are encouraged but not required to select the core algebra course (MATH 067) if they choose this emphasis. When a student does an emphasis in statistics, STAT 031 counts as if it were numbered over 40.
Students interested in mathematics and computer science should consider a mathematics major with a concentration in computer science or an honors program with a mathematics major and a computer science minor. Details on these options are in the catalog under computer science.
Students thinking of graduate work in social or management science, or a master’s in business administration, should consider the following options. Basic courses: single-variable calculus (two semesters), one or more practical statistics courses (STAT 011 and 031), linear algebra, discrete math, several-variable calculus, and introductory computer science; advanced courses: (1) Modeling (MATH 056); (2) at least one of Probability (MATH 105), Mathematical Statistics I (STAT 061), and possibly Mathematical Statistics II (STAT 111); (3) at least one of Combinatorics (MATH 069) or Operations Research (ECON 032); (4) the two required core courses (MATH 063 and MATH 067); and (5) Differential Equations (MATH 043 or 044). Because this program is heavy (one who hopes to use mathematics in another field must have a good grasp both of the relevant mathematics and of the intended applications), one of the core course requirements may be waived with permission of the department.
Students thinking of graduate work in operations research should consider the following options. Basic courses: same as previous paragraph. Advanced courses: (1) the two required core courses (MATH 063 and MATH 067); (2) Combinatorics (MATH 069) and Topics in Combinatorics (MATH 072); (3) Mathematical Statistics (STAT 061); and (4) at least one of Number Theory (MATH 058), Modeling (MATH 056), or Probability (MATH 105).
Swarthmore offers teacher certification in mathematics through a program approved by the state of Pennsylvania and administered by the College’s Educational Studies Department. In additional to meeting the general certification requirements, students seeking certification in mathematics have two choices. Either they complete a mathematics major and must include among their electives:
• One semester of computer science (CPSC 021)
• One semester of discrete mathematics (MATH 029, 059, 069, or 079)
• One semester of geometry (MATH 055 or 075)
• One semester of statistics or probability (STAT 011, 031, 061, 111 or MATH 105)
or they do a special major in mathematics and education. Such a major must include the general certification requirements, seven credits in mathematics, including MATH 063 or 067, one other course numbered over 044, and a mathematical education thesis. See the Educational Studies Department for more details.
Either way, students seeking certification are strongly advised to take further mathematics or statistics courses emphasizing modeling and applications and/or to take at least one course in the natural or social sciences in which mathematics or statistics is significantly used. They are also highly encouraged to work as a tutor in the math clinic or to do individual tutoring for a semester. To receive certification, a student must receive a grade of C or better in all mathematics courses.
By graduation, a mathematics course minor must have 6 credits in mathematics or statistics. Furthermore, every mathematics course minor is required to obtain credit for, or place out of, each of the following subjects: single-variable calculus (two semesters), linear algebra, and several-variable calculus. In addition, every mathematics course minor must obtain at least 2 credits in mathematics or statistics courses whose numbers are greater than 044. At least 1 of these 2 credits must be for MATH 063 or 067. Also, at least one of these two credits must be taken at Swarthmore.
Statistics Course Minor
By graduation, a statistics course minor must have 6 credits in mathematics or statistics. Furthermore, every statistics course minor is required to obtain credit for, or place out of, each of the following subjects: single-variable calculus (two semesters), linear algebra, and several-variable calculus. In addition, every statistics course minor must obtain credit for, or place out of, STAT 031 and STAT 061. At least one of STAT 031 and STAT 061 must be taken at Swarthmore.
Requirements for acceptance as a mathematics major in the Honors Program are more stringent than those for the course major and include a grade-point average in mathematics and statistics courses of B+ or better. Potential honors majors may want to consider including in the sophomore year a course that emphasizes theory and provides an opportunity for writing proofs. Department faculty can give advice on appropriate courses.
The program for an honors major in mathematics consists of preparations for external examination in three fields of 2 credits each. For each field chosen, the courses or seminars are specified by the department. For the honors major, one preparation shall be in algebra (MATH 067 and 102) and one in analysis (MATH 063 and either 101 or 103). Each student may select the third preparation from discrete mathematics, geometry, probability, statistics, and topology.
Students who wish to complete an honors minor in mathematics must have credit for, or place out of, single-variable calculus (two semesters), linear algebra, and several-variable calculus. For the honors portion of their program, minors must complete one preparation chosen from among any of the fields described earlier.
Note: In the department’s new numbering scheme for courses numbered under 100, the ones digit indicates the subject matter, and the other digits indicate the level. In most cases, a ones digit of 1 means statistics, 2 to 6 means continuous mathematics, and 7 to 9 means noncontinuous mathematics (algebra, number theory, and discrete math). Courses below 10 do not count for the major, from 10 to 39 are first- and second-year courses, from 40 to 59 are intermediate, in the 60s are core upper-level courses; from 70 to 89 are courses that have one or more core courses as prerequisites, and in the 90s are independent reading courses.
Statistics provides methods for collecting and analyzing data and generalizing from their results. Statistics is used in a wide variety of fields, and this course provides an understanding of the role of statistics in these fields and in everyday life. It is intended for students who want an appreciation of statistics, including the ability to critically interpret and evaluate statistical claims, but who do not imagine they will ever need to carry out statistical analyses themselves. (Those who may need to carry out statistical analyses should take STAT 011.) This course cannot be counted toward a major in mathematics, is not a prerequisite for any other course, and cannot be taken for credit after or simultaneously with any other statistics course, including AP Statistics and ECON 031.
Prerequisite: Four years of traditional high school mathematics (precalculus).
1 credit.
Each semester.
Fall 2006. Everson. Spring 2007. Everson.
Students will explore the world of mathematical ideas by sampling logic, number theory, geometry, infinity, topology, probability, and fractals, while we emphasize the thinking and problem-solving skills these ideas stimulate. Class meetings will involve presentation of new material; group work on problems and puzzles; and lively, maybe even passionate discussions about mathematics. This course is intended for students with little background in mathematics or those who may have struggled with math in the past. Students planning to go on to calculus should consult with the instructor. This course does not count toward a major in mathematics.
Writing course.
1 credit.
Spring 2007. Bergstrand.
This course is offered occasionally and is interdisciplinary in nature. It provides an introduction to some area of mathematics in the context of its use in another discipline. A recent version of this course was taught in the Linguistics Program. This course does not count toward a major in mathematics.
1 credit.
Not offered 2006–2007.
(Cross-listed as SOAN 010E)
STAT 011 prepares students to carry out basic statistical analyses with the aid of computer software. Topics include basic summary statistics and graphics, design of surveys and experiments, one and two-sample t-tests and tests of proportions, chi-square tests, and an introduction to linear regression and analysis of variance. The course is intended for students who want a practical introduction to statistical methods and who intend to do, or think they may eventually do, statistical analysis, especially in the biological and social sciences. Students who receive credit on entrance for the Statistics AP Exam should not take this course; they have placed out of it and will lose their AP credit if they take it. Students who have earned credit for the former STAT 002 or STAT 002C will not receive credit for STAT 011. Note that STAT 011 overlaps considerably with ECON 031; both courses cover similar topics, although ECON 031 focuses more on economic applications while STAT 011 draws examples from a variety of disciplines.
Prerequisite: Four years of traditional high school mathematics (precalculus).
1 credit.
Each semester.
Fall 2006. Wang. Spring 2007. Everson.
A first-semester calculus course with emphasis on an intuitive understanding of the concepts, methods, and applications. Graphical and symbolic methods will be used. The course will mostly cover differential calculus, with an introduction to integral calculus at the end. Applications to biological science and social science will receive special attention.
Prerequisite: Four years of traditional high school mathematics (precalculus) and placement into this course through Swarthmore’s Calculus Readiness Examination or Calculus Placement Examination (see “Placement Procedure” earlier).
1 credit.
Fall 2006. Grinstead.
Survey of key topics in single- and several-variable calculus for students who do not plan to take any more calculus. In single-variable calculus, topics may include antiderivatives, the fundamental theorem, probability, geometric series, and modeling with differential equations. Topics in several variables may include contour plots, partial derivatives, and Lagrange multipliers. Emphasis on applications in biological and social sciences. Cannot be substituted for either MATH 025 or 033 as courses required for the major.
Prerequisites: MATH 015 or placement by examination (see “Advanced Placement and Credit Policy” earlier).
1 credit.
Each semester.
Fall 2006. Maurer. Spring 2007. Klotz.
The continuation of MATH 015 for students who wish to major in mathematics, physics, chemistry, or engineering, or who want the option of continuing to several-variable calculus. The course covers the fundamental theorem, integration, geometric series, Taylor polynomials and series, and an introduction to differential equations.
Prerequisites: MATH 015 or placement by examination (see “Advanced Placement and Credit Policy” earlier).
1 credit.
Each semester.
Fall 2006. Talvacchia. Spring 2007. Bergstrand.
MATH 025S covers the same material as the lecture-based MATH 025 but uses a seminar format (maximum 12 students) with additional meetings and lots of hands-on activities (e.g., writing, oral presentations, group work, and computer work). Intended for students who think they could benefit from the collaborative seminar format and who wish to be challenged to excel in calculus so that they gain more confidence to continue with mathematics and science.
Prerequisite: Placement by examination (see “Advanced Placement and Credit Policy” earlier).
First-year seminar. 1 credit.
Fall 2006. Bergstrand.
For students who place out of the first half of MATH 025. This course goes into more depth on sequences, series, and differential equations than does MATH 025 and includes power series and convergence tests. This course, or MATH 025, is required of all students majoring in mathematics, physics, chemistry, or engineering. Students may not take MATH 026 for credit after MATH 025 without special permission.
Prerequisites: Placement by examination (see “Advanced Placement and Credit Policy” earlier).
1 credit.
Fall 2006. Hunter.
This course covers systems of linear equations, matrices, vector spaces, linear transformations, determinants and eigenvalues. Many applications to other disciplines are presented. Students may take only one of MATH 027, MATH 028, and MATH 028S for credit.
Prerequisite: A grade of C or better in some math course numbered 023 or higher or placement by examination (see “Advanced Placement and Credit Policy” earlier).
1 credit.
Each semester.
Fall 2006. Klotz, Shapiro. Spring 2007. Rusin.
More theoretical, abstract, and rigorous than MATH 027. The subject matter will be equally as valuable in applied situations, but applications will be emphasized less and students will do many proofs. MATH 028 is intended for students with exceptionally strong mathematical skills, especially if they are thinking of a mathematics major. Students may take only one of MATH 027, MATH 028, and MATH 028S for credit.
Prerequisite: A grade of B or better in some math course numbered 025 or higher or placement by examination (see “Advanced Placement and Credit Policy” earlier).
1 credit.
Fall 2006. Grood, Shimamoto. Spring 2007. Maurer
MATH 028S covers the same material as the lecture-based MATH 028 but uses a seminar format (maximum 12 students) with additional meetings. Hands-on student participation takes the place of most lectures. Students may take only one of MATH 027, MATH 028, and MATH 028S for credit.
Prerequisite: Placement by examination (see “Advanced Placement and Credit Policy” earlier).
First-year seminar. 1 credit.
Fall 2006. Johnson.
An introduction to noncontinuous mathematics. The key theme is how induction, iteration, and recursion can help one discover, compute, and prove solutions to various problems—often problems of interest in computer science, social science, or management. Topics will include algorithms, graph theory, counting, difference equations, and finite probability with special emphasis on how to write mathematics.
Prerequisite: Placement by examination (see “Placement Procedure” earlier). Familiarity with some computer language is helpful but not necessary.
Writing course.
1 credit.
Each semester.
Fall 2006. Grood. Spring 2007. Shimamoto.
This course will study methods for exploring and modeling relationships in data. We introduce modern techniques for visualizing trends and formulating hypotheses. We will also discuss methods for modeling structure and patterns in data, particularly using multiple regression and related methods. The format of the course emphasizes writing assignments and interactive problem solving using real datasets.
Prerequisites: Credit for AP Statistics, STAT 011, STAT 061, or ECON 031; or STAT 001 and permission of the instructor.
Writing course.
1 credit.
Spring 2007. Wang.
This course considers differentiation and integration of functions of several variables with special emphasis on two and three dimensions. Topics include partial differentiation, extreme value problems, Lagrange multipliers, multiple integrals, line and surface integrals, Green’s, Stokes’, and Gauss’ theorems. The department strongly recommends that students take MATH 034 instead, which provides a richer understanding of this material by requiring linear algebra (MATH 027 or 028) as a prerequisite. Students may take only one of MATH 033, MATH 034, and MATH 035 for credit.
Prerequisite: MATH 025, 025S, or 026 or placement by examination (see “Advanced Placement and Credit Policy” earlier).
1 credit.
Each semester.
Fall 2006. Grood. Spring 2007. Grood.
Same topics as MATH 033 except in more depth using the concepts of linear algebra. The department strongly recommends that students take linear algebra first so that they are eligible for this course. Students may take only one of MATH 033, MATH 034, and MATH 035 for credit.
Prerequisite: MATH 025, 025S, or 026; and MATH 027, 028, or 028S.
1 credit.
Each semester.
Fall 2006 and spring 2007. Shapiro.
This version of MATH 034 will be more theoretical, abstract, and rigorous than its standard counterpart. The subject matter will be equally as valuable in applied situations, but applications will be emphasized less and students will do many proofs. It is intended for students with exceptionally strong mathematical skills and primarily for those who have completed MATH 028 or 028S successfully. Students may take only one of MATH 033, MATH 034, and MATH 035 for credit.
Prerequisite: A grade of C or better in MATH 028 or 028S or permission of the instructor.
1 credit.
Spring 2007. Grinstead, Talvacchia.
The theory of primes, divisibility concepts, and multiplicative number theory will be developed, with students doing many proofs.
Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.
1 credit.
Alternate years.
Fall 2006. Stromquist.
The choice of topics will depend somewhat on the interest and mathematical background of the students but may include a study of issues in multivariate analysis and statistical inference (Bayesian statistics in particular).
Prerequisite: One course in statistics.
1 credit.
Not offered 2006–2007.
This course emphasizes the standard techniques used to solve differential equations. It will cover the basic theory of the field with an eye towards practical applications. Standard topics include first-order equations, linear differential equations, series solutions, first-order systems of equations, Laplace transforms, approximation methods, and some partial differential equations. Compare with MATH044. Students may not take both MATH 043 and 044 for credit. The department prefers majors to take MATH 044.
Prerequisites: Several-variable calculus or permission of the instructor.
1 credit.
Spring 2007. Shapiro.
An introduction to differential equations that has a more theoretical flavor than MATH 043 and is intended for students who enjoy delving into the mathematics behind the techniques. Problems are considered from analytical, qualitative, and numerical points of view, with an emphasis on the formulation of differential equations and the interpretations of their solutions. This course does not place as strong an emphasis on solution techniques as MATH 043 and thus may not be as useful to the more applied student. Students may not take both MATH 043 and 044 for credit. The department prefers majors to take MATH 044.
Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.
1 credit.
Spring 2007. Stromquist.
Course content varies from year to year. In 2006 the emphasis will be on introductory differential geometry. See also MATH 075.
Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.
1 credit.
Fall 2006. Talvacchia.
An advanced version of MATH 055, sometimes given instead, and typically requiring MATH 063, 067, or both.
Prerequisites: See the instructor.
1 credit.
Not offered 2006-2007.
(Cross-listed as CPSC 046)
Please see computer science for description.
This course concentrates on the careful study of the principles underlying the calculus of real valued functions of real variables. Topics will include continuity, compactness, connectedness, uniform convergence, differentiation, and integration.
Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.
Writing course.
1 credit.
Usually offered fall only.
Fall 2006. Johnson. Spring 2007. Grood.
Course content varies from year to year depending on student and faculty interest. Recent offerings have included coding theory, groups and representations, and finite reflection groups. See also MATH 077.
Prerequisites: Linear algebra.
1 credit.
Alternate years.
Not offered 2006-2007.
An advanced version of MATH 057, sometimes given instead, and requiring the core course in algebra.
Prerequisites: Linear algebra and MATH 067.
1 credit.
Not offered 2006-2007.
This course is an introduction to abstract algebra and will survey basic algebraic systems—groups, rings, and fields. Although these concepts will be illustrated by concrete examples, the emphasis will be on abstract theorems, proofs, and rigorous mathematical reasoning.
Prerequisite: Linear algebra or permission of the instructor.
Writing course. 1 credit.
Usually offered fall only.
Fall 2006. Shimamoto.
This course introduces the mathematical theory of probability, including density functions and distribution functions, joint and marginal distributions, conditional probability, and expected value and variance. It then develops the theory of statistics, including parameter estimation and hypothesis testing. The emphasis is on proving results in mathematical statistics rather than on applying statistical methods. Students needing to learn applied statistics and data analysis should consider STAT 011 or 031 in addition to or instead of this course.
Prerequisites: One of MATH 023, 033 or 034, or permission of the instructor.
1 credit.
Fall 2006. Everson.
An introduction to the methods and attitudes of mathematical modeling. Because modeling in physical science and engineering is already taught in courses in those disciplines, applications in this course will be primarily to social and biological sciences. Various standard methods used in modeling will be introduced: differential equations, Markov chains, game theory, graph theory, and computer simulation. The emphasis, however, will be on how to apply these subjects to specific modeling problems, not on their systematic theory. The format of the course will include projects as well as lectures and problem sets.
Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.
1 credit.
Alternate years.
Not offered 2006–2007.
This course continues the study of noncontinuous mathematics begun in MATH 029. The topics covered include three broad areas: counting theory, graph theory, and design theory. The first area includes a study of generating functions and Polya counting. The second area is concerned with relations between certain graphical invariants. Topics such as extremal graph theory and Ramsey theory may be introduced. The third area introduces combinatorial structures such as matroids, codes, and Latin squares.
Prerequisites: MATH 029 and at least one other course in mathematics.
1 credit.
Alternate years.
Fall 2006. Bergstrand.
Topics vary from year to year. Past topics have included combinatorial matrix theory, linear programming, game theory, graph theory, combinatorial algorithms, number theoretic algorithms, and complexity theory. See also MATH 079.
Prerequisites: MATH 029 and at least one higher-numbered mathematics course.
1 credit.
Alternate years.
Not offered 2006-2007
An advanced version of MATH 059, sometimes offered instead of MATH 059.
Prerequisites: MATH 029 and 069.
1 credit.
Not offered 2006-2007.
The first part of the course consists of an introduction to linear partial differential equations of elliptic, parabolic, and hyperbolic type via the Laplace equation, the heat equation, and the wave equation. The second part of the course is an introduction to the calculus of variations. Additional topics depend on the interests of the students and instructor.
Prerequisites: Linear algebra, several-variable calculus, and either MATH 043, MATH 44, PHYS 050, or permission of the instructor.
1 credit.
Alternate year.
Not offered 2006-2007
Course content varies from year to year depending on student and faculty interest. Recent topics have included financial mathematics, dynamical systems, and Fourier analysis. The topic in 2007 is expected to be financial mathematics.
Prerequisites: Linear algebra and several-variable calculus. In 2007 STAT 61 is also required, or permission of the instructor.
1 credit.
Alternate years.
Spring 2007. Stromquist.
An advanced version of MATH 053, sometimes offered instead, and requiring the core course in analysis.
Prerequisites: Linear algebra, and Math 63.
1 credit.
Not offered 2006-2007.
MATH 093/STAT 093. Directed Reading
MATH 096/STAT 096. Thesis
This course is required of all senior mathematics majors in the course program and must be taken at Swarthmore. It provides an opportunity to delve more deeply into a particular topic agreed on by the student and the instructor. This focus is accomplished through a written paper and an oral presentation.
0.5 credit.
Fall 2006. Stromquist.
This seminar is a continuation of Introduction to Real Analysis (MATH 063). Topics may include the inverse and implicit function theorems, differential forms, calculus on manifolds, and Lebesgue integration.
Prerequisite: MATH 063.
1 credit.
Spring 2007. Maurer.
This seminar is a continuation of Introduction to Modern Algebra (MATH 067). Topics covered usually include field theory, Galois theory (including the insolvability of the quintic), the structure theorem for modules over principal ideal domains, and a theoretical development of linear algebra. Other topics may be studied depending on the interests of students and instructor.
Prerequisite: MATH 067.
1 credit.
Fall 2006. Shapiro. Spring 2007.Shimamoto.
A brief study of the geometry of complex numbers is followed by a detailed treatment of the Cauchy theory of analytic functions of a complex variable: integration and Cauchy’s theorem, power series, residue calculus, conformal mapping, and harmonic functions. Various applications are given, and other topics—such as elliptic functions, analytic continuation, and the theory of Weierstrass—may be discussed.
Prerequisite: MATH 063.
1 credit.
Alternate years.
Not offered 2006–2007.
An introduction to point-set, combinatorial, and algebraic topology: topological spaces, classification of surfaces, the fundamental group, covering spaces, simplicial complexes, and homology (including related algebra).
Prerequisites: MATH 063 and 067.
2 credits.
Alternate years.
Not offered 2006-2007.
An introduction to measure-theoretic probability theory. Topics may include branching processes, renewal theory, random walks, stochastic processes, laws of large numbers, characteristic functions, the Central Limit Theorem, Markov chains, the Poisson process, and percolation.
Prerequisite: STAT 061.
1 credit.
Alternate years.
Spring 2007. Grinstead.
The course content varies from year to year among differential geometry, differential topology, and algebraic geometry. In 2007 the topic is likely to be advanced differential geometry.
Prerequisites: MATH 045 and 063, or permission of the instructor.
1 credit.
Alternate years.
Spring 2007. Talvacchia.
This seminar is a continuation of STAT 061. It deals mainly with statistical models for the relationships between variables. The general linear model, which includes regression, variance, and covariance analysis, is examined in detail. Topics may also include nonparametric statistics, sampling theory, and Bayesian statistical inference.
Prerequisite: Linear algebra and a grade of C+ or better in STAT 061.
1 credit.
Alternate years.
Not offered 2006-2007.