# Mathematics and Statistics

PHILIP J. EVERSON, Professor and Chair

CHARLES M. GRINSTEAD, Professor

AIMEE S.A. JOHNSON, Professor

^{2}

STEPHEN B. MAURER, Professor (part time)

DON H. SHIMAMOTO, Professor

JANET C. TALVACCHIA, Professor

LINDA CHEN, Associate Professor

CHERYL P. GROOD, Associate Professor

THOMAS J. HUNTER, Associate Professor

^{3}

STEVE C. WANG, Associate Professor

^{1}

RALPH R. GOMEZ, Assistant Professor

^{3}

NSOKI MAVINGA, Assistant Professor

LYNNE STEUERLE SCHOFIELD, Assistant Professor

MICHAEL BIRO, Visiting Assistant Professor

SCOTT COOK, Visiting Assistant Professor

RACHEL EPSTEIN, Visiting Assistant Professor

KELLY MCCONVILLE, Visiting Assistant Professor

KAITLYN E. LITWINETZ, Academic Support Coordinator

STEPHANIE J. SPECHT, Administrative Assistant

^{1}Absent on leave, fall 2014.

^{2}Absent on leave, spring 2015.

^{3}Absent on leave, 2014–2015.

### 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 statistics, physical science, biological science, computer science, social science, operations research, education, and finance—the list grows.

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 careers paths, leading them to graduate school, in mathematics, statistics, or other fields, to professional schools, or to the workplace.

#### Introductory Courses

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 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, both semesters), 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 several-variable 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 either MATH 15 or MATH 28 must take Swarthmore’s Calculus Placement Exam. Versions of the Calculus Placement Exam and the Readiness Exam are sent to entering first-year 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. 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 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 first-year calculus level completed before entering Swarthmore.

### The Academic Program

#### 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).

### Course Major

#### 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 any flavor of 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 upper-level courses in mathematics are typically more demanding and more theoretical than the first- and second-year 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 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. 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. *Note that MATH 097 is given in the fall only.*

Note that placement counts for satisfying the requirements but not for the 10-credit 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.

The two *required core courses*, Introduction to Real Analysis (MATH 063) and Introduction to Modern Algebra (MATH 067), are offered every fall semester, and we try to create enough sections to keep them relatively small and seminar-like. We hope, but cannot promise, to offer one or the other of 063 and 067 each spring as well.

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 computers. Students bound for graduate work should obtain a reading knowledge of French, German, or Russian.

#### Special Emphases

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); (5) the Senior Conference (MATH 097); and (6) another mathematics course numbered over 040. 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 040.

Students interested in mathematics and computer science should consider a mathematics major with a minor 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 061 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 (ENGR 057); (4) the three required core courses (MATH 063, MATH 067 and MATH 097); 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 three required core courses (MATH 063, MATH 067 and MATH 097); (2) Combinatorics (MATH 069) and Topics in Discrete Mathematics (MATH 059 or 079); (3) Mathematical Statistics (STAT 061); and (4) at least one of Number Theory (MATH 058), Modeling (MATH 056), or Probability (MATH 105).

Students interested in quantitative economics, mathematical finance, or similar fields should consider a double major in mathematics and economics, or a major in mathematics with a minor in economics. Students thinking of graduate work in quantitative economics or mathematical finance should consider a math major with a program including at least MATH 43, MATH 54, MATH 63 and STAT 61 together with appropriate additional coursework to round out a mathematics major or a mathematics major with emphasis in statistics.

### Course Minor

#### 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 to complete the mathematics course minor (for Class of ’15 and later)

By graduation, a mathematics course minor must have 6 credits in mathematics or statistics, at least 3 of which must be for courses numbered 045 or higher. Also, 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 6 credits in mathematics or statistics. Every statistics course minor must obtain credit for, or place out of, CPSC 21, STAT 031, and STAT 061. At least one of STAT 031 and STAT 061 must be taken at Swarthmore. Students are advised to take CPSC 21 as early as possible, as it can be difficult to add the course in junior and senior years.

### Honors Major

All current sophomores who wish to apply for Honors should indicate this in their Sophomore Plan, should work out a tentative Honors Program with their departmental adviser, and should submit the College’s Honors Program Application along with their Sophomore Plan. (All Sophomore Plan forms and Honors forms are available from the registrar or the registrar’s website.) Honors applications are also accepted at the end of the sophomore year or during the junior year. Students, in consultation with their advisers, often change their Honors Programs anyway as time goes on.

#### Basic requirements

To be accepted as an Honors major in mathematics, a student should have a grade point average in mathematics and statistics courses to date of at least B+.

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).

#### Preparations

The department offers preparations in the fields listed below. Each preparation is subject to External Examination, including a 3-hour written examination and a 45-minute 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)

Discrete Mathematics (069 and either 059 or 079)

Geometry (either 055 or 075, and 106)

Probability (061 and 105)

Statistics (061 and 111)

Topology (104, a 2-credit seminar)

Since no course is allowed to count in two honors preparations, it is not possible for a student to offer both Real Analysis and Complex Analysis as fields. Similarly, one may take only one of Probability and Statistics as fields.

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 so—that 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

For the honors portion of their program, minors must complete one preparation chosen from those in the previous section.

### Transfer Credit

Courses taken elsewhere may count for the major. However, the number of upper-level transfer credits for the major is limited. Normally, *at least 3 of the 5 upper-level 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.

### Off-Campus 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 upper-level 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.

### 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 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.*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.

##### STAT 001. Statistical Thinking

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 interpret and evaluate statistical claims critically 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 2014. Everson. Spring 2015. Schofield.

##### MATH 003. Introduction to Mathematical Thinking

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. It is not open to students who already have received credit on their Swarthmore transcripts for mathematics, Advanced Placement credit included, or who concurrently are taking another mathematics course, or who have placed out of any Swarthmore mathematics course. (See “Placement Procedure” earlier.) 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 2015. Grood.

##### MATH 007. Elementary Topics in Mathematics in Applied Contexts

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. In fall 2010 this was a course in biomathematics.

1 credit.

Not offered 2014–2015.

##### STAT 011. Statistical Methods

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. 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.

Eligible for COGS credit.

Each semester.

Fall 2014. Cook, McConville. Spring 2015. McConville, Schofield.

##### MATH 015. Elementary Single-Variable Calculus

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.

Prerequisite: Four years of traditional high school mathematics (precalculus) and placement into this course through Swarthmore’s Math/Stat Readiness Examination. Students with prior calculus experience must also take Swarthmore’s Calculus Placement Examination (see “Placement Procedure” section earlier).

1 credit.

Fall 2014. Grood, Mavinga.

##### MATH 015SP. Calculus STEM Scholars Program

MATH 015SP will provide an enriched experience designed to support MATH 015 students who plan to take at least four other STEM courses during their time at Swarthmore. During class, students work in small groups on challenging problems designed to promote deep understanding and mastery of the material.

Prerequisite: Students must apply for admission to this attachment. Admission will be determined by a commitment to both hard work and excellence, rather than by high school GPA, math SAT scores, or past performance in math classes.

0.5 credit.

Graded credit/no credit.

Fall 2014. Grood.

##### STAT 021. Elementary Topics in Statistics: Quantitative Paleontology

This course will explore current areas of research in paleobiology and macroevolution. For instance, does evolutionary change generally occur gradually or in short bursts? How reliably does the fossil record preserve information about ecosystems? What factors make species more likely to go extinct? To answer these and other questions, paleobiologists use a range of statistical and mathematical techniques. We will emphasize conceptual understanding and applications of such quantitative methods, rather than their underlying theory or proofs. Class meetings will include a combination of lectures, discussion of journal articles, and conversations with leading paleontologists via Skype.

Prerequisite: BIOL 002, or STAT 011 or equivalent.

1 credit.

Not offered 2014–2015.

##### MATH 024. Numerical Methods–Engineering Applications

(See ENGR 019)

1 credit.

Not offered 2014–2015.

##### MATH 025. Further Topics in Single-Variable Calculus

The continuation of MATH 015, this 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” section).

1 credit.

Each semester.

Fall 2014. Epstein. Spring 2015. Biro.

##### MATH 026. Advanced Topics in Single-Variable Calculus

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.

Prerequisite: Placement by examination (see “Advanced Placement and Credit Policy” section).

1 credit.

Fall 2014. Biro, Maurer.

##### MATH 027. Linear Algebra

This course covers systems of linear equations, matrices, vector spaces, linear transformations, determinants, and eigenvalues. Applications to other disciplines are presented. This course is a step up from calculus: It includes more abstract reasoning and structures. Formal proofs are discussed in class and are part of the homework. 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 025 or higher or placement by examination (see “Advanced Placement and Credit Policy” section).

1 credit.

Each semester.

Fall 2014. Biro. Spring 2015. Bergstrand, Chen.

##### MATH 028. Linear Algebra Honors Course

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. 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, including both placement out of calculus and placement into this course via Part IV of Swarthmore’s Calculus Placement Exam (see “Placement Procedure” section).

1 credit.

Fall 2014. Johnson. Spring 2015. Cook.

##### MATH 028S. First-Year Seminar: Linear Algebra Honors Seminar

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, including both placement out of calculus and placement into this course via Part IV of Swarthmore’s Calculus Placement Exam (see “Placement Procedure” section).

1 credit.

Fall 2014. Maurer.

##### MATH 029. Discrete Mathematics

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 mathematical induction and other methods of proof, recurrence relations, counting, and graph theory. Additional topics may include algorithms, and probability. There is a strong emphasis on good mathematical writing, especially proofs. While it does not use any calculus, MATH 029 is a more sophisticated course than MATH 15 or MATH 25; thus success in a calculus course demonstrates the mathematical maturity needed for MATH 29.

Prerequisite: Strong knowledge of at least precalculus, as evidenced by taking another mathematics course numbered 15 or above, or through our placement examinations (see “Placement Procedure” section). Familiarity with some computer language is helpful but not necessary.

Writing course.

Eligible for COGS credit.

1 credit.

Fall 2014. Shimamoto. Spring 2015. Epstein.

##### STAT 031. Data Analysis and Visualization

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.

Statistics Prerequisites: Credit for AP Statistics, STAT 011, STAT 061, or ECON 031; or STAT 001 and permission of the instructor.

Writing course.

Eligible for COGS credit.

1 credit.

Fall 2014. McConville. Spring 2015. Wang.

##### MATH 033. Basic Several-Variable Calculus

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 is offered every semester and 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, or 026 or placement by examination (see “Advanced Placement and Credit Policy” section). Students who have taken linear algebra at Swarthmore or elsewhere may not take MATH 033 without the instructor’s permission.

1 credit.

Fall 2014. Johnson.

##### MATH 034. Several-Variable Calculus

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, or 026; and MATH 027, 028, or 028S.

Eligible for COGS credit.

1 credit.

Each semester.

Fall 2014. Scott. Spring 2015. Cook.

##### MATH 035. Several-Variable Calculus Honors Course

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. 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, or in the fall for entering students who have placed out of linear algebra, permission of the departmental placement coordinator.

1 credit.

Fall 2014. Grinstead. Spring 2015. Grinstead.

##### STAT 032. Topics in Statistics: Data Analysis Projects in Public and Social Policy

This course is offered occasionally, when it was last offered in spring 2011 it was a Community-Based Learning project course in data analysis. Students worked in teams on a semester-long data analysis problem. Projects were drawn from data from local organizations in order to attempt to answer questions of direct importance to them. A key objective of the course is to expose students to the variety of challenges faced by the data analyst. Topics may include multiple regression, analysis of variance, analysis of covariance, and other related methods. Students research the scientific background of their problem and consult with the local organizations from which their data came.

Prerequisite: STAT 011, or permission of the instructor.

1 credit.

Not offered 2014–2015.

##### MATH 043. Basic Differential Equations

This course emphasizes the standard techniques used to solve differential equations. It will cover the basic theory of the field with an eye toward 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 MATH 044. 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 2015. Shimamoto.

##### MATH 044. Differential Equations

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 viewpoints, 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.

Eligible for COGS credit.

1 credit.

Spring 2015. Mavinga.

##### MATH 046. Theory of Computation

(See CPSC 046)

Eligible for COGS credit.

1 credit.

Spring 2015. Staff.

##### MATH 048. Mathematical Logic

This course is an introduction to mathematical logic, including model theory, set theory, and computability theory. The purpose of the course is to introduce students to the logical foundations of mathematics as well as to the various fields within the realm of logic that students may wish to study further. The primary focus will be on first-order logic and model theory. Some theorems we will prove carefully, and others, such as Godel's Incompleteness Theorem, we will only sketch, focusing on the important implications instead. Topics include first-order languages, soundness and completeness, models of theories, Incompleteness, the Church-Turing Thesis, Turing reducibility, resolving set-theoretic paradoxes, and the axiom of choice. This course may be taken in addition to the logic courses Phil 12 and Phil 31, which cover some similar topics from a different perspective.

Prerequisite: familiarity with mathematical proofs. Contact the instructor if you are uncertain about your preparation.

1 credit.

Spring 2015. Epstein.

##### MATH 053. Topics in Analysis

Course content varies from year to year depending on student and faculty interest. Recent topics have included financial mathematics, dynamical systems, and Fourier analysis. Prerequisites: Linear algebra and several-variable calculus.

1 credit.

Alternate years.

Spring 2015. Cook.

##### MATH 054. Partial Differential Equations

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 044, PHYS 050, or permission of the instructor.

1 credit.

Alternate years.

Not offered 2014–2015.

##### MATH 055. Topics in Geometry

Course content varies from year to year. In recent years, the emphasis has been on introductory differential geometry. See also MATH 075.

Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.

1 credit.

Alternate years.

Fall 2014. Talvacchia.

##### MATH 056. Modeling

An introduction to the methods and attitudes of mathematical modeling. Course content varies from year to year depending on student and faculty interest. 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. These may include differential equations, Markov chains, game theory, graph theory, and computer simulation. The course will balance theory with how to apply these subjects to specific modeling problems coming from a variety of disciplines. The format of the course will include projects as well as lectures and problem sets with the hope that those enrolling will have the opportunity to apply what they have learned to appropriate problems within their own area of interest.

Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.

1 credit.

Alternate years.

Not offered 2014–2015.

##### MATH 057. Topics in Algebra

Course content varies each year, depending on student and faculty interest. Recent offerings have included coding theory, groups and representations, finite reflection groups, and matrix theory. See also MATH 077.

Prerequisites: Linear algebra.

Eligible for COGS credit.

1 credit.

Alternate years.

Not offered 2014–2015.

##### MATH 058. Number Theory

The theory of primes, divisibility concepts, and multiplicative number theory will be developed.

Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.

1 credit.

Alternate years.

Fall 2014. Grinstead.

##### MATH 059. Topics in Discrete Mathematics

Topics vary each year. Past topics have included combinatorial matrix theory, graph theory, combinatorial algorithms, number theoretic algorithms, and representation theory using combinatorial structures and techniques. See also MATH 079.

Prerequisites: MATH 029 and at least one higher-numbered mathematics course.

1 credit.

Alternate years.

Not offered 2014–2015.

##### STAT 061. Probability and Mathematical Statistics I

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: MATH 033 or 034 or permission of the instructor. STAT 011 or the equivalent is strongly recommended.

1 credit.

Fall 2014. Schofield.

##### MATH 063. Introduction to Real Analysis

This course concentrates on the careful study of the principles underlying the calculus of real valued functions of real variables. Topics include continuity, compactness, connectedness, uniform convergence, differentiation, and integration. Required additional meetings.

Prerequisites: Linear algebra and several-variable calculus or permission of the instructor.

Writing course.

Eligible for COGS credit.

1 credit.

Fall 2014. Shimamoto.

##### MATH 067. Introduction to Modern Algebra

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. Required additional meetings.

Prerequisite: Linear algebra or permission of the instructor.

Writing course.

1 credit.

Fall 2014. Bergstrand. Spring 2015, Bergstrand.

##### MATH 069. Combinatorics

This course continues the study of material begun in MATH 029. The primary topics are enumeration and graph theory. The first area includes, among other things, a study of generating functions and Polya counting. The second area is concerned with relations between certain graphical invariants. Additional topics may include one or more of the following topics: design theory, extremal graph theory, Ramsey theory, matroids, matchings, codes, and Latin squares.

Prerequisites: Grades of C or better in MATH 029 and at least one other course in mathematics numbered 27 or higher, or permission of the instructor.

1 credit.

Alternate years.

Fall 2014. Chen.

##### MATH 073. Advanced Topics in Analysis

An advanced version of MATH 053, sometimes offered instead, and requiring the core course in analysis.

Prerequisites: Linear algebra and MATH 063.

1 credit.

Not offered 2014–2015.

##### MATH 075. Advanced Topics in Geometry

An advanced version of MATH 055, sometimes given instead, and typically requiring MATH 063, 067, or both. The topic for 2013–2014 is computational geometry and topology. This version of the course may not be used as part of the Honors preparation in Geometry.

Prerequisites: At least one of MATH 055, MATH 063, MATH 067, or MATH 069. MATH 063 recommended especially.

1 credit.

Not offered 2014–2015.

##### MATH 077. Advanced Topics in Algebra

An advanced version of MATH 057, sometimes given instead, and requiring the core course in algebra. (In 2013–2014 MATH 057 will be offered instead.)

Prerequisites: Linear algebra and MATH 067.

1 credit.

Not offered 2014–2015.

##### MATH 079. Advanced Topics in Discrete Mathematics

An advanced version of MATH 059, sometimes offered instead of MATH 059.

Prerequisites: MATH 029 and 069.

1 credit.

Not offered 2014–2015.

##### MATH 093/STAT 093. Directed Reading

##### MATH 096/STAT 096. Thesis

##### MATH 097. Senior Conference

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 either an oral presentation or participation in a poster session.

0.5 credit.

Fall 2014. Chen.

### Seminars

##### MATH 101. Real Analysis II

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.

Eligible for COGS credit.

1 credit.

Fall 2014. Mavinga. Spring 2015. Talvacchia.

##### MATH 102. Modern Algebra II

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.

Eligible for COGS credit.

1 credit.

Spring 2015. Grood.

##### MATH 103. Complex Analysis

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 2014–2015.

##### MATH 104. Topology

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.

Eligible for COGS credit.

2 credits.

Alternate years.

Not Offered 2014–2015.

##### MATH 105. Probability

Advanced topics in probability theory. Topics may include branching processes, card shuffling, the Central Limit Theorem, generating functions, the Laws of Large Numbers, Markov chains, optimal stopping theory, percolation, the Poisson process, renewal theory, and random walks.

Prerequisite: STAT 061.

1 credit.

Alternate years.

Spring 2015. Grinstead.

##### MATH 106. Advanced Topics in Geometry

The course content varies from year to year among differential geometry, differential topology, and algebraic geometry. In 2013, the topic is expected to be advanced differential geometry.

Prerequisites: MATH 055 and 063 or permission of the instructor.

1 credit.

Alternate years.

Spring 2015. Chen.

##### STAT 111. Mathematical Statistics II

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.

Prerequisites: Linear algebra and a grade of C+ or better in STAT 061; CPSC 021.

Eligible for COGS credit.

1 credit.

Spring 2015. Everson.