College Bulletin  Course Catalog
Mathematics and Statistics


Courses
Faculty
AIMEE S.A. JOHNSON, Professor and Chair
VICTOR BARRANCA, Associate Professor^{3}
LINDA CHEN, Professor
PHIL EVERSON, Professor
JOSHUA GOLDWYN, Assistant Professor
RALPH R. GOMEZ, Associate Professor^{3}
CHERYL P. GROOD, Professor
CATHERINE M. HSU, Assistant Professor^{2}
THOMAS J. HUNTER, Professor^{1}
AMANDA LUBY, Assistant Professor^{3}
NSOKI MAMIE MAVINGA, Associate Professor
ALLISON MILLER, Assistant Professor
LYNNE STEUERLE SCHOFIELD, Associate Professor
JANET C. TALVACCHIA, Professor
SUSANNE THORNTON, Assistant Professor
STEVE C. WANG, Professor
IAN WHITEHEAD, Assistant Professor^{3}
THOMAS CRAWFORD, Visiting Assistant Professor
PAT DEVLIN, Visiting Assistant Professor
EDUARD EINSTEIN, Visiting Assistant Professor
CAROLYN REINHART, Visiting Assistant Professor
LUCAS VAN METER, Visiting Assistant Professor
ROBERT VIATOR, Visiting Assistant Professor
LAURA DANDRIDGE, Academic Support Coordinator
STEPHANIE J. SPECHT, Administrative Coordinator
^{1 }Absent on leave, Fall 2022.
^{2 }Administrative leave, Spring 2023.
^{3 }Absent on leave, 20222023.

Overview of Curriculum
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, majors and minors in the department receive a strong foundation in pure mathematics and the opportunity to apply it to a variety of disciplines, including statistics, physical science, biological science, computer science, social science, operations research, education, and finance.
By studying mathematics and statistics, majors and minors grow in:
 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;
 Comprehension skills: absorbing mathematical ideas and information presented on paper, orally, and electronically;
 Computation skills: mental, by hand, and by machine, as appropriate.
Through core courses, students learn fundamental concepts, results, and methods. Through elective courses, they pursue special interests. In the process, students develop a further appreciation for the scope and beauty of our discipline.
Graduates of the department follow many career paths. These paths lead to graduate school in mathematics, statistics, and other fields; to professional schools; and to the workplace.
Introductory Courses
Many firstyear 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 later. 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, Fall semester), MATH 003 (Introduction to Mathematical Thinking, Spring semester), and STAT 011 (Statistical Methods I, both semesters). MATH 003 is a writing course. MATH 039 (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, such as linear algebra and severalvariable calculus.
Placement and Credit on Entrance to Swarthmore
Placement Procedure
To gain entrance to mathematics or statistics courses at any time during one’s Swarthmore years, students are expected to take at least one of the following exams: the Advanced Placement (AP) or higherlevel International Baccalaureate (IB) exams, Swarthmore’s Calculus Placement Exam, or Swarthmore’s Math/Stat Readiness Exam. Students who take AP or IB exams may be required to take the departmental exams as well, or parts thereof. In particular, students intending to take MATH 015 must take Swarthmore’s Calculus Readiness Exam and those intending to take MATH 28 must take Swarthmore’s Calculus Placement Exam. Students who receive placement or credit for AP Calculus exams taken in the spring of 2020 will be required to take the department’s Calculus Placement Exam for more thorough placement information. Versions of the Calculus Placement Exam and the Readiness Exam are available to entering firstyear students over the summer, along with detailed information about the rules for placement and credit.
Advanced Placement/International Baccalaureate Credit
Placement and credit mean different things. Placement allows students to skip material they have already learned well 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. The credit awarded on the basis of the AP and IB exams was updated during the 20182019 year and resulted in the following rules:
 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 or 5 on the AB Calculus AP Test of the College Board (or for an AB subscore of 4 or 5 on the BC Test) or for a score of 6 or 7 on the Higher Level Mathematics Test of the IB.
 2 credits (for MATH 015 and 025) for a main score of 4 or 5 on the BC Calculus AP Test.
Students who receive placement but not credit for a course occasionally make use of section 8.1 of the course catalog to arrange to take a course without regular attendance. See section 8.1 for details. Students who are eligible on entrance for credit for a course, but who take the course anyway, will lose the entrance credit.
Firstyear 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 courses into which they may be placed and additional materials they may need to present for this placement. The department will not normally award credit for work above the firstyear calculus level completed before entering Swarthmore.
Major and Minor Application Process
Students apply for a major in the middle of the second semester of the sophomore year. Upon indication of intent to major (or minor) in the department, students will be assigned a departmental sophomore plan advisor who will help them decide on a reasoned plan of study for their last two years. This plan is then submitted to the department via their electronic Sophomore Plan. After the Sophomore Plan process is over, students may apply to add or change a major (or minor) at any time, but applications will normally be held until the next time that sophomore applications are considered (around March 1).
Acceptance into the Major
The Department offers three different pathways to a course major: a course major without special emphasis, a course major with an emphasis in statistics, and a course major with an emphasis in applied mathematics. The normal preparation for all majors in mathematics is to have obtained credit for, or placement out of, at least four of the following five course groups by the end of the sophomore year: Calculus I (MATH 015), Calculus II (MATH 025 or 026), Discrete Mathematics (MATH 039  previously called Math 029), Linear Algebra (MATH 027 or 028), and Several Variable Calculus (MATH 033, 034, or 035). In any event, all majors must complete the Linear Algebra and Several Variable Calculus requirement by the end of the first semester of the junior year.
To be accepted as a major or a minor, a candidate normally should have a grade point average of at least C+ in courses taken in the department to date, including courses in the fall term of the first year, for which we have shadow grades. (We do not have shadow grades from Spring 2020, so those courses will not be included in the grade point average.) A candidate should have at least one grade at the B level. Students should be aware that upperlevel courses in mathematics are typically more demanding and more theoretical than the firstand secondyear courses. This is an important factor in considering borderline cases. In some cases, applicants may be deferred pending successful work in courses to be designated by the department.
Requirements for the Course Major
Graduation Requirements for all Course Majors (see additional requirements below):
 At least 10 credits in mathematics and statistics courses. (Certain courses  mostly numbered under 10  do not count toward the major. These are indicated under the individual course listings.)
 Credit for, or placement out of, the following courses: MATH 015; MATH 025 or 026; MATH 027 or 028; and MATH 033, 034, or 035.
 Credit for, or placement out of, Introduction to Real Analysis (MATH 63).
 Credit for Senior Conference (MATH 97), the department’s comprehensive requirement, which has a new format as of the 202122 academic year. Students must now take this zerocredit course in both the fall and spring of their senior year. This comprehensive will have students engage with mathematicians and statisticians who present lectures on various topics, and will have students participate in a range of departmental activities. Any student who will not be on campus during one of those semesters must replace that semester’s MATH 97 course with a departmentapproved substitution. The substitution should be approved during the sophomore planning stage, or, if that is not possible, by speaking with their department advisor as soon as possible.
In the multivariate group (MATH 33, 34, or 35), it is strongly preferred that candidates for the major take 34 or 35, which are the versions that assume a background in linear algebra.
Note that placement out of a course does not add to a student’s credit total; students need to earn 10 credits of coursework in the department. If you believe you are eligible for credit for courses taken before Swarthmore (because of AP or IB scores) but these credits are not showing on your transcript, please address this matter immediately with the registrar. Your application to our department may be held up otherwise.
Each of the three pathways to the course major (without special emphasis, emphasis in Statistics, emphasis in Applied Mathematics) has its own additional requirements, which are detailed below.
Course Majors without special emphasis must also:
 Earn credit for, or placement out of, Introduction to Modern Algebra (MATH 67)
 Earn at least 5 credits in mathematics and statistics courses for courses numbered over 38. At most one of these courses may be taken CR/NC. No course numbered over 100 may be taken CR/NC.
 Take at least one of MATH 63 or MATH 67 at Swarthmore.
The departmental schedule offers the two required core courses, Introduction to Real Analysis (MATH 63) and Introduction to Modern Algebra (MATH 67), in alternate semesters (MATH 63 in the fall and MATH 67 in the spring). Because MATH 63 is only guaranteed to be offered in the fall, students should plan to take it before the spring semester of senior year.
Course Majors with an emphasis in Statistics must also:
 Earn credit for, or placement out of:
 Statistical Methods II (STAT 21)
 Probability (STAT 51)
 Mathematical Statistics I (STAT 61)
 Mathematical Statistics II (STAT 111)
 Computer Science 21. Students are advised to take CPSC 21 as early as possible, as it can be difficult to add this course in the junior or senior year.
 Earn at least 5 credits in mathematics and statistics courses numbered over 38 OR earn credit (not placement) for STAT 21 and at least 4 credits in math/stat courses numbered over 38. At most one of these five credits may be taken CR/NC. No course numbered over 100 may be taken CR/NC.
 STAT 111 and at least one of STAT 51 or STAT 61 must be taken at Swarthmore.
Course majors with an emphasis in Applied Mathematics must also:
 Earn credit for, or placement out of:
 Stochastic and Numerical Methods (MATH 66)
 Differential Equations (MATH 43 or MATH 44)
 At least one of
 Partial Differential Equations (MATH 54)
 Modeling (MATH 56)
 At least one additional course from the following:
 Partial Differential Equations (MATH 54)
 Modeling (MATH 56)
 Probability (STAT 51)
 Complex Analysis (MATH 103)
 Computer Science 21. Students are advised to take CPSC 21 as early as possible, as it can be difficult to add this course in the junior or senior year.
 Earn at least 5 credits in math/stat courses numbered over 38. At most one of these 5 credits may be taken CR/NC. No course numbered over 100 may be taken CR/NC.
 Take Math 66 at Swarthmore.
All majors are encouraged to study in some depth an additional discipline that makes use of mathematics or statistics. We also recommend that they acquire some facility with computer programming.
Credit/No Credit Policy
At most one upper level course counted towards the major can be taken credit/no credit. This does not include courses which are only offered credit/no credit, but does include courses for which the grade is uncovered after completion of the course. In any case, no seminars can be taken credit/no credit.
Acceptance into the minors
The department offers two types of course minors: a course minor in mathematics and a course minor in statistics. There are two pathways to the course minor in mathematics: a minor without special emphasis and a minor with an emphasis in applied mathematics. Students may not have more than one minor in the department. The requirements for acceptance into any course minor, such as prerequisite courses and grade average, are the same as for acceptance into the major.
Requirements for the Course Minor
Graduation Requirements for all Course Minors (see additional requirements below)
 At least 6 credits in mathematics and statistics courses. Those courses offered by the Department that do not count towards the course major also do not count towards the course minor.
 Credit for, or placement out of, the following courses: MATH 015; MATH 025 or 026; MATH 027 or 028; and MATH 033, 034, or 035.
Course Minors in Mathematics without special emphasis must also:
 Earn at least 3 credits in mathematics and statistics courses for courses numbered over 43. (Note the difference from the course major requirement, which is 5 courses over 38.)
 At least 2 of these 3 credits must be taken at Swarthmore.
 One of these 3 credits must be either Introduction to Real Analysis (MATH 63) or Introduction to Modern Algebra (MATH 67)
 At most one of these 3 credits may be taken CR/NC. No course numbered over 100 may be taken CR/NC.
Course Minors in Mathematics with an emphasis in Applied Mathematics must also:
 Earn credit for, or placement out of, each of the following: CPSC 21, MATH 43 or 44, MATH 66, and at least one additional course from MATH 54, MATH 56, or STAT 51.
 Take MATH 66 at Swarthmore.
 Earn at least 3 credits in mathematics and statistics courses numbered over 40 (Note the difference from the course major requirement, which is 5 courses over 38).
 At most one of these 3 credits may be taken CR/NC. No course numbered over 100 may be taken CR/NC.
Course Minors in Statistics must also:
 Earn credit for, or placement out of, each of the following courses: CPSC 21, Stat 21, Stat 51, Stat 61
 Take at least one of Stat 51 or Stat 61 at Swarthmore College.
 Earn at least 3 credits in mathematics and statistics courses numbered over 40 OR earn credit (not placement) for Stat 21 and at least 2 credits in math/stat courses numbered over 40 (Note the difference from the course major requirement, which is 5 courses over 38).
 At most one of these 3 credits may be taken CR/NC. No course numbered over 100 may be taken CR/NC.
Credit/No Credit Policy
For the math minor, at most one of the required upper level courses counted towards the minor can be taken credit/no credit. For the statistics minor, only one of Stat 21, Stat 51, and Stat 61 may be taken credit/no credit. For the applied math minor, at most one of the 4 required courses listed previously may be taken credit/no credit. This does not include courses which are only offered credit/no credit, but does include courses for which the grade is uncovered after completion of the course. In any case, no seminar can be taken credit/no credit.
Honors Major
All current sophomores who wish to apply for Honors should indicate this in their Sophomore Plan and should work out a tentative Honors Program with their departmental adviser.
Acceptance into the Honors Major
The requirements to be accepted as an honors major are the same as those to be accepted as a course major except that such students should have a grade point average in mathematics and statistics courses to date of at least B+..
Requirements for the Honors Major
 At least 10 credits in mathematics and statistics courses
 Credit for, or placement out of, the following courses: MATH 015; MATH 025 or 026; MATH 027 or 028; and MATH 033, 034, or 035
 Three preparations of two credits each, for a total of six distinct credits, in the following areas:
 Real Analysis (MATH 63 and 101) or Complex Analysis (MATH 63 and 103)
 Modern Algebra (MATH 67 and 102)
 One of:
 Geometry (MATH 65 and MATH 105)
 Statistics (STAT 61 and 111)
 Topology (MATH 104, a 2credit seminar)
 At most one of the courses in the three preparations may be taken CR/NC. No course numbered over 100 may be taken CR/NC.
Students who are doing an Honors math major with the Statistics preparation are required to take a datadriven statistics course as well (e.g. Stat 11 or 21).
Each of the three preparations is subject to External Examination, including a 3hour written examination and a 45minute oral examination. The External Examination component of the program is meant to prompt students to learn their core subjects well and to show the examiners that they have done so  that is, show that they deserve Honors. However, no three fields cover everything a student would ideally learn as an undergraduate. Honors majors should consider including in their studies a number of advanced courses and seminars beyond what they present for Honors should their schedules allow it.
No course is allowed to count in two honors preparations, so it is not possible for a student to do honors preparations in both Real Analysis and Complex Analysis.
Especially strong students who take many advanced courses may petition to substitute an advanced preparation for either Algebra or Analysis. For instance, a student who has taken essentially all our seminars might petition to be examined in Algebra, Topology, and Geometry, omitting an analysis examination. However, all honors students must take the algebra sequence and one of the analysis sequences, even if they are given permission to be examined in something else.
Note that to be an Honors math major, a student is required to also have an Honors minor in another subject. Senior Honors Study or Portfolio is not required or offered, and Honors majors do not need to take Senior Conference (MATH 97).
Honors Minor
All current sophomores who wish to apply for Honors should indicate this in their Sophomore Plan and should work out a tentative Honors Program with their departmental adviser.
Acceptance into the Honors Minor
The requirements to be accepted as an honors minor are the same as those to be accepted as a course major except that such students should have a grade point average in mathematics and statistics courses to date of at least B.
Requirements for the Honors Minor
 Credit for, or placement out of, the following courses: MATH 015; MATH 025 or 026; MATH 027 or 028; and MATH 033, 034, or 035
 One preparation consisting of two credits in one of the following areas:
 Real Analysis (MATH 63 and 101)
 Complex Analysis (MATH 63 and 103)
 Modern Algebra (MATH 67 and 102)
 Geometry (MATH 65 and 105)
 Statistics* (STAT 61 and 111)
 Topology (MATH 104, a 2credit seminar)
All prospective minors who are majoring in a subject related to mathematics or statistics are encouraged to consult with a member of the Department to see which preparation is most appropriate to their interests. Honors minors are encouraged to take at least one of MATH 63 and 67 even if it is not part of their Honors preparation. As mentioned before, no seminar may be taken credit/no credit.
* Students who are doing an Honors math minor with the Statistics preparation are required to take a datadriven statistics course as well (e.g., STAT 11 or 21).
Note that to be an Honors math minor, a student is required to also have an Honors major in another subject.
Transfer Credit
Courses taken elsewhere may count for the major. However, the number of upperlevel transfer credits for the major is limited. Normally, at least 3 of the 5 upperlevel courses used to fulfill the major must be taken at Swarthmore, including at least one of the core courses MATH 063 and MATH 067. Exceptions should be proposed and approved during the Sophomore Plan process, not after the fact. Also, the usual College rules for transfer credit apply: students must see the professor in charge of transfer twice: in advance to obtain authorization, and afterwards to get final approval and a determination of credit. In particular, for MATH 063 and 067, students are responsible for the syllabus we use. If a course taken elsewhere turns out not to cover our entire syllabus, the student will not get full credit (even though the transfer course was authorized beforehand) and the student will not complete the major until they have demonstrated knowledge of the missing topics.
Similarly, for honors preparations, students are responsible for the syllabi we use; we will not offer special honors exams based on work done at other institutions.
OffCampus Study
Students planning to study abroad should obtain information well in advance about the courses available at the institution they plan to attend and check with the department about selecting appropriate courses. It may be difficult to find courses abroad equivalent to our core upperlevel courses, or to our honors preparations, since curricula in other countries are often organized differently.
Teacher Certification
Swarthmore offers teacher certification in mathematics through a program approved by the state of Pennsylvania and administered by the College’s Educational Studies Department. For further information about the relevant set of requirements, please refer to the Educational Studies section of the Bulletin. One can obtain certification either through a mathematics major or through a Special Major in Mathematics and Education, in either case if taken with appropriate electives.
Mathematics and Statistics Courses
Note 1: For courses numbered under 100, the last digit indicates the subject matter, and the other digit indicates the level. In most cases, a last digit of 1 or 2 means statistics, 3 to 6 means continuous mathematics (calculus and analysis), and 7 to 9 means noncontinuous mathematics (algebra, number theory, and discrete math). Courses below 10 do not count for the major, 10 to 39 are first and secondyear courses, 40 to 59 are intermediate courses, 60 to 69 are core upperlevel courses, 70 to 89 are courses that have one or more core courses as prerequisites, and 90 to 99 are independent reading courses.
Note 2: There are several sets of courses below from which a student may take only one for credit. For instance, see the descriptions of MATH 033, 034 and 035. In such cases, if a student does take more than one of them, each group is treated for the purpose of college regulations as if they have the same course number. See the Repeated Course Rule in section 8.2.4



