College Bulletin  Course Catalog
Mathematics and Statistics


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Faculty
AIMEE S. A. JOHNSON, Professor and Chair
VICTOR BARRANCA, Assistant Professor^{3}
DEB BERGSTRAND, Professor (part time)
LINDA CHEN, Associate Professor
PHIL EVERSON, Professor
NOAH GIANSIRACUSA, Assistant Professor
JOSHUA GOLDWYN, Assistant Professor
RALPH R. GOMEZ, Associate Professor^{3}
CHERYL P. GROOD, Professor^{1}
THOMAS J. HUNTER, Professor^{2}
NSOKI MAMIE MAVINGA, Associate Professor
LYNNE STEUERLE SCHOFIELD, Associate Professor^{6}
JANET C. TALVACCHIA, Professor
STEVE C. WANG, Professor
REBECCA BLACK, Visiting Assistant Professor
LU CHEN, Visiting Assistant Professor (part time)
THOMAS CRAWFORD, Visiting Assistant Professor
DIANA DAVIS, Visiting Assistant Professor
ELIZABETH DRELLICH, Visiting Assistant Professor
DANIELLE A. LEDFORD, Academic Support Coordinator
STEPHANIE J. SPECHT, Administrative Assistant
^{1} Absent on leave, fall 2018.
^{2 }Absent on leave, spring 2019.
^{3 }Absent on leave, 20182019.
^{6} Spring 2019.

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 firm 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.
Students typically enter our department with strong skills, but there is always room for improvement and new knowledge. 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 or Spring semester), MATH 003 (Introduction to Mathematical Thinking, Spring semester), and STAT 011 (Statistical Methods, both semesters). MATH 003 is a writing course. 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 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 International Baccalaureate (IB) exams, Swarthmore's Calculus Placement Exam, or Swarthmore's Math/Stat Readiness Exam. Students who do 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 15 must take Swarthmore's Calculus Readiness Exam and those intending to take MATH 28 must take Swarthmore's Calculus Placement Exam. 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 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. The credit awarded on the basis of the AP and IB exams will be under review during the 20182019 year and may change for students who matriculate in 2019 and later. For students who matriculate in or before 2018, credit is awarded 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 higherlevel 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.
Students who receive placement but not credit for a course occasionally make use of 8.1 of the course catalog to arrange to take a course without regular attendance. See 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. Students should consult the department webpage during the College's Sophomore Plan process for more details on how to apply for the major. 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 normal preparation for a major 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 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. 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.
Basic Requirements
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 040. (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 or 026; MATH 027or 028; MATH 033, 034, or 035; MATH 063; and MATH 067. The two upperlevel 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. Majors are expected to complete both MATH 063 and 067 before the spring semester of the senior year; permission to delay taking either course until the senior spring must be requested in writing as early as possible but in any event no later than the beginning of the fall 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 040 must be taken at Swarthmore, including MATH 097 and at least one of the core courses MATH 063 and 067. MATH 097 is given in the fall only, and meets Tuesdays, 2:403:55.
Note that placement counts for satisfying the requirements but not for the 10credit rule. Those students who are placed out of courses without credit must take other courses to obtain 10 credits. 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 see the registrar.
Mathematics majors are encouraged to study in some depth an additional discipline that makes use of mathematics. We also recommend that they acquire some facility with coding.
Special Emphases
A student may major in mathematics with an emphasis in statistics by taking at least 10 credits in math or statistics, including the core analysis course (Math 63), Probability (Stat 51), Mathematical Statistics I&II (Stat 61 and 111), Statistical Methods II (Stat 21), and Senior Conference (Math 97), along with placement or credit for Introduction to Computer Science (CS 21). Stat 21 counts as a course numbered over 40 for majors with an emphasis in statistics. Students are advised to take CS 21 as early as possible, as it can be difficult to add the course in junior and senior years. At least one of Stat 51 or Stat 61 must be taken at Swarthmore.
Students interested in Applied Math should consider taking Differential Equations (Math 44), Probability (Stat 51), Partial Differential Equations (Math 54), Mathematical Modeling (Math 56), Real Analysis (Math 63), Fundamentals of Applied Math (Math 66) and Complex Analysis (Math 103), along with Introduction to Computer Science (CS 21).
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 requirements for acceptance into either course minor, such as prerequisite courses and grade average, are the same as for acceptance into the major. Students may not minor in both mathematics and statistics.
Basic requirements of the mathematics course minor
By graduation, a mathematics course minor must have 6 credits in mathematics or statistics, at least 3 of which must be for courses numbered 044 or higher. At least 1 of these 3 credits must be for MATH 063 or 067. Also, at least 2 of these 3 credits must be taken at Swarthmore.
Basic requirements of the statistics course minor
By graduation, a statistics course minor must have at least 6 credits in mathematics and statistics courses. Every statistics course minor must receive credit for, or place out of, CS 21, Stat 21, Stat 51 and Stat 61. At least one of Stat 51 or Stat 61 must be taken at Swarthmore. Students are advised to take CS 21 as early as possible, as it can be difficult to add the course in junior and senior years.
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. 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.
Basic requirements
To be accepted as an Honors major in mathematics, a student should have a grade point average of at least B+ in courses taken to date, including courses taken in the fall term of their first year, for which the department has shadow grades.
An Honors math major program consists of three preparations of two credits each, for a total of six distinct credits. One preparation must be in algebra and one in analysis (real or complex). The student must also satisfy all requirements of the mathematics major with the exception of the comprehensive requirement (MATH 097, Senior Conference). Note that to be an Honors math major, a student is required to also have an Honors minor in another subject.
Of the six credits used for a student's honor preparation, at most one may be taken credit/no credit (whether or not the grade is uncovered after the course is completed). In any case, no seminar may be taken credit/no credit.
Preparations
The department offers preparations in the fields listed below. Each preparation is subject to External Examination, including a 3hour written examination and a 45minute oral examination. Each preparation consists of a specified pair of credits. The specified credits are listed after each field.
Algebra (067 and 102)
Real Analysis (063 and 101)
Complex Analysis (063 and 103)
Geometry (either 055 or 075, and 106)
Statistics (061 and 111)
Topology (104, a 2credit seminar)
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.
The external examination component of the program is meant to prompt students to learn their core subjects really well and to show the examiners that they have done sothat is, show that they deserve Honors. However, no three fields cover everything a strong 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.
Senior Honors Study/Portfolio
None is required or offered.
Honors Minor
To be accepted as an Honors minor in mathematics, a student should have a grade point average of at least B in courses taken in the department to date, including courses in the fall term of the first year, for which the department has shadow grades.
An Honors math minor consists of one preparation of two credits, chosen from those in the previous section. As mentioned before, no seminar may be taken credit/no credit. 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 it all, the student will not get full credit (even though the transfer course was authorized beforehand) and the student will not complete the major until he or she has 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 ones digit indicates the subject matter, and the other digit indicates the level. In most cases, a ones digit of 1 or 2 means statistics, 3 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 secondyear courses, from 40 to 59 are intermediate, in the 60s are core upperlevel courses; from 70 to 89 are courses that have one or more core courses as prerequisites, and in the 90s are independent reading courses.
Note 2: There are several sets of courses below where a student may not take more than one of them 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.

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