### Steps to declare the Math Major (Declaration Periods: Fall 9/15- 12/9 & Spring 2/1 - 4/17)

- Complete the calculus requirement (Math 1400 and Math 1410+2400 or Math 1610+2600) and one proof-based major course.
- Go to Path @ Penn to formally request to add or remove your major to the Math Office (otherwise it won't show on your transcripts).
- Use this link to complete the top portion of our course plan worksheet and email it to the major advisor we assign to you after we process your Path @ Penn request.
- After consulting with your advisor, he/she will review your worksheet for processing.
- Allow 15 business days for your major to appear on your transcript. If its not visible by day 15th email me at rtoney@math.upenn.edu
- Contact your math advisor once a semester.
*Email your advisor to answer questions or make changes to your course plan.* **Once you enter your semester of graduation, you must email your math major advisor to certify your math major worksheet for completion**.

### On This Page

- Admission to the Major Program Entrance Requirements for the Major.
- Background: The Major and its goals.
- What can I do with a math major?
- Advanced Placement Potential Credit for incoming freshmen.
- The Major Program Requirements
*A minimum of 13 credits are needed.* - Planning your Mathematics Major Our suggested plan for completing your major.
- Other Useful Experiences for a Math Major
- Mathematics Major: Biological Concentration
- The Mathematics Minor
- Further Recommendations Advice for students planning graduate study in mathematics or related fields.
- The Honors Program Requirements for a degree with honors.
- The Master's Program Submatriculation into graduate study at Penn.
- Advice General advice regarding the major.
- External links.
- Current Course Descriptions (for reference in course planning)
- Courses offered next semester

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### Admission to the Major Program

Permission to major in mathematics is normally obtained by the end of the sophomore year, but planning for it should begin as early as possible. It is important that majors entering their junior year commence satisfying the algebra and analysis requirements.

To be admitted to the major, a student must have completed successfully (i.e., with grades of C or better) the calculus requirement as well as one proof-based math class (such as Math 2020, 2030, 1610, 2600, 3140) in the freshman and sophomore years. A higher-level proof-based class may be substituted at the discretion of a math major advisor.

Students who plan to have math as their second major should have a cumulative GPA of at least 3.0, an average of at least 3.0 in their math courses, and no math grades lower than B-.

The Major is open to SEAS undergraduates (as a second major) as well as to students in the College.

### The Math Major and its Goals

Mathematical training allows one to take a problem, abstract its essential features, and investigate them further. This ability can assist greatly in such diverse fields as economics, law, medicine, engineering, and computer science -- as well as in the more traditional activities of research and teaching.

The goals of the major program are to assist students in acquiring both an understanding of mathematics and an ability to use it. We wish to inspire the discovery of new mathematics as well as the application of mathematics to other fields.

The mathematics major provides a solid foundation for graduate study in mathematics as well as background for study in economics, the biological sciences, the physical sciences and engineering, as well as many non-traditional areas. This flexibility is available through an appropriate choice of electives within the major. A variety of electives are offered. They are designed to serve the needs of mathematics majors and others who want more advanced training in mathematics and its applications. Most of these courses presume our basic two year calculus sequence.

The mathematics major is also excellent training for students interested in elementary and secondary education. For information on the elementary education undergraduate major or the secondary education submatriculation program which leads to a Master's degree, students should consult the Undergraduate Chair as well as the Director of Teacher Education in the Graduate School of Education.

Highly qualified and motivated students should note the possibility of obtaining both the B.A. and M.A. degrees in four years. This is discussed below.

Given the widening role of mathematics, students with special interests and needs may wish to consider the possibility of an individualized program of study, perhaps in conjunction with a major in another field. The Major Coordinator should be consulted about this.

**How to Plan a Mathematics Major** Prospective majors should first check the information listed under Advanced Placement. We strongly encourage students to master the basic material as early as possible, and AP credit is equal to credit for a course taken at Penn. Students are urged to read the Major Program Requirements carefully, and use it as a guideline to plot the plan. You should also read Other Useful Experience and Further Recommendations for a complete overview.

### What can I do with a math major?

See Careers in Mathematics . It is maintained by the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.

See Math in the Media for pointers to current general articles involving mathematics and its applications.

### Advanced Placement

Note: The AP policy and more details on this subject (and on "transfer credit") can be found on our web page AP/Transfer Credit Information.

We strongly encourage students to master the basic material as early as possible. It is our policy to waive prerequisite course requirements for those students who can pass an examination that demonstrates that they know the material. These remarks apply especially to the first-year calculus courses. For these, a student may receive credit towards the degree (in addition to the waiving of prerequisites) by either of the following methods:

- Passing the external Advanced Placement BC Exam administered by the College Entrance Examination Board with a score of 5 gives credit for Math 1400. Lower scores on the BC Exam receive no course credit. No credit is given for the AB Exam. Students taking first semester calculus, math 1400, are expected to have had an AB calculus course in high school.
- Passing the internal Advanced Placement Examination administered in the first week of the fall and spring semester by the mathematics department. A student may take the examination regardless of whether he or she took the external exam described under (1) above.

Those receiving advanced placement and planning to enroll in more advanced courses should see the Major Coordinator, who will help them plan a program of study.

### The Major Program Requirements (Minimum 13 c.u.)

The Mathematics Major comprises 13 courses organized into eight basic requirements. Each of the 13 courses must be taken for a grade (i.e., not pass/fail), and must be completed with a grade of C (2.0) or better. (A student who receives a grade lower than C in a requirement consisting of more than one course may still count that course toward the major by achieving a grade of C or above in a more advanced course within the same requirement.) The math department also expects the completion of at least one proof-based math class in the freshman or sophomore year in order to be admitted into the major (usually Math 2020, 2030, 1600, 2600, or 3140), or permission of a math major advisor. **Courses taken on a pass/fail basis will not count toward fulfilling the following requirements.**

**Three Semester Calculus Requirement.**This is satisfied by any of the two sequences 1400-1410-2400 or 1400-1610-2600. The 1400 requirement can be satisfied by AP credit for the Calculus BC exam with a score of 5. The courses 1610-2600 are proof-based and provide the best preparation for higher mathematics, and in particular for Math 3600 and 3610. Math 1300 does*not*count toward the Math major.**Advanced Calculus Requirement**Math majors must take either a fourth semester of calculus, Math 2410, or partial differential equations Math 4250.**Complex Analysis Requirement**All math majors must take Complex Analysis Math 4100.**Seminar Requirement**This is satisfied by taking either Math 2020 (intro to analysis) or Math 2030 (intro to algebra). These courses carry one-unit of credit and are intended to be taken concurrently with calculus. For students taking honors calculus (Math 1610-2400) the seminar requirement is replaced by a higher math elective course.

Students who begin with Math 1400 in their freshman year usually postpone this requirement until their second year. Students who have already taken one of Math 2410, Math 4100 or Math 4250 can substitute a higher math elective course for the Seminar Requirement. Under exceptional circumstances, other students may also make such a substitution with the permission of the Undergraduate Chair. In general, though, we recommend that prospective math majors take a freshman seminar to gain an overview of the subject.**Linear algebra requirement.**

Math Majors must take Advanced Linear Algebra Math 3140. Math 3140 is a prerequisite for Math 3700 and Math 5020.**Algebra Requirement**This is satisfied by taking the sequence Math 3700-3710 or the more theoretical Math 5020-5030. However, you can't get credits for both Math 3700 and Math 5020, or both Math 3710 and Math 5030.

These courses all overlap considerably.**Analysis Requirement**This is satisfied by taking the sequence Math 3600-3610 or the more theoretical Math 5080-509- However, you can't get credits for both Math 3600 and Math 5080, or both Math 3610 and Math 5090.*Note: Majors who begin their mathematics studies with Math 1410-2400 plus a seminar should fulfill at least one of the linear algebra, algebra, and analysis requirements in their sophomore year.***Mathematics Electives**The total number of approved math course units required for a math major is 13. Students should determine how many course units they still need for a math major after completing requirements 1 through 6 above. This will depend on which options have been chosen in completing the requirements. The remaining courses may then be made up from Math 2100 and mathematics courses numbered 3200 or above. One mathematics elective course unit may be taken from the list of approved Cognate Courses given outside the math department. Students who are double majors may take two Cognate courses units.

Students may, for example, take Statistics 4300 (or Systems Engineering 3010 or Econ 103 or ENM 5030), and count such a course as being within the Mathematics Department. Thus by taking one of these courses, one does not lower the number of cognate courses one can take outside the math department, as explained on the page of Cognate Courses.

*Example 1:*A student is double majoring in math and engineering, did not take a freshman seminar, and completed the Advanced Calculus requirement by taking math 2410. This student thus takes 4 courses related to the Calculus requirements, 4 courses to complete the algebra and analysis requirements, and Math 3140 and Math 4100 for a total of 10 courses. They must take 3 electives to bring their course total up to 13. Because the student is double majoring in math and engineering, two of these electives can be Cognate Courses in other departments. Notice that on the above list of cognate courses, some courses given in other departments are listed as being counted as within the math department as far as the math major and minor are concerned. For example, the student could take Stat 4300 (which is counted as within the math department), use Physics 0150 and Physics 0151 as their cognate courses not counted as within the math department, and then choose two more electives from within the math department to complete their math major requirements.*Example 2:*A student is majoring only in math, took a freshman seminar, and completed the Advanced Calculus requirement by taking math 4250. This student thus takes the freshman seminar, 3 calculus courses, math 4100 and 4250 and four algebra and analysis courses in the course of completing the above requirements, for a total of 10 courses. They must take three further electives for a math major. Only one of these can be a Cognate Course, because the student is not a double major.

### Planning your Mathematics Major

Students who do not plan graduate study in mathematics or in a highly mathematics-related subject should, as a means of acquiring more background, consider Math 4100, 4200, 4250, and 4300. For glimpses of several beautiful mathematical subjects beyond the basic core, students should consider Math 3500, 5420, 5480, 5490, 5800, 5000, 5300, 4800.

Students who are interested in the physical sciences should consider Physics 0150-0151 or 0170-0171 and the courses beyond. Those interested in the social or biological sciences should consider Math 4300 or Statistics 4300-4310. Those interested in computer science should consider CSE 110, 120-121 and the courses beyond as well as Math 4500, 5700 (previously 473 and 670). For computer programming and numerical methods, students should learn a programming language such as Pascal or C and learn to use symbolic manipulation software such as Mathematica or Maple. They also should consider Math 3200-3210. For discrete methods, in addition to Math 3400 and 3410, 450, 5700 (previously 473), and 5800 (previously 440), students should consider Math 5240-5250 (previously 470) and 5810 (previously 441).

For students who plan to do research in mathematics, or in a highly mathematical subject such as statistics, the considerations which are listed just above still apply. However, since a great deal of further theoretical training is necessary, such students are directed to the basic graduate courses in mathematics: 6000, 6010, 6020, 6030, 6080, and 6090.

All this material must eventually be mastered. It needs to be understood clearly that what is required is a comprehensive grasp of theoretical mathematics. Thus, the student's attention is directed to Method B for obtaining honors in mathematics, and to the joint B.A./M.A. program, pursuing a master's degree at the same time as their undergraduate degree.

### Other Useful Experiences for a Math Major

The first order of business is to satisfy the first four requirements discussed above. When this has been done, the student usually has sufficient experience and direction to complete the program in consultation with the Major Advisor . It needs to be emphasized strongly, however, that apart from the strict requirements, there are certain other things which all mathematics majors should do. These are:

- Learn to program a computer and learn how to use mathematical symbolic manipulation packages. The latter skill is taught in our Calculus courses.
- Learn statistics. This may be done by taking Math 4300 or Stat 4300 followed by Stat 4310.
- Learn how mathematics is actually used. This can be done by learning something of an applied but highly mathematical field. Operations research, engineering and physics provide examples, but there are many others. (See below.)
- Obtain some job experience. This should be done, if possible, in the summer following the junior year. It should involve some interface between mathematics and the real world.

The importance of the above four recommendations cannot be sufficiently emphasized. Equipped with them, a mathematics major is an attractive candidate for entrance into a great many fields. Without them, job opportunities are limited. These remarks apply to the most theoretical, as well as to the most practical of careers.

### The Honors Program

To be eligible for honors in mathematics, a student must have an average of at least 3.5 in his/her major and major-related courses. If this condition is satisfied, honors may be obtained by either of the following methods.

**Method A.** By preparing, through independent study, a body of material approximately equal in amount to a one-semester course and giving a lecture on it as the Honors Committee shall direct.

The area of study chosen should be one that is not normally covered in the department and should involve reading sources outside normal course material. The selected topic may be picked from one field of mathematics or may involve assimilation of topics from different fields. Before beginning the project, the student should ask two members of the faculty, at least one affiliated with the Mathematics Department, to serve as the Honors Committee. The Honors Committee must approve the selected topic and serve as examiners for the lecture (which should be approximately an hour long, seminar-style talk).

**Method B.** By passing the written Preliminary Examination in undergraduate mathematics. This is required of all incoming Ph.D candidates. Details concerning this examination may be found in the Graduate Admissions Catalogue (also see below).

For further guidance, prospective honors students should consult with their Major Advisor during their junior year. The honors project must be completed by the end of February of the senior year.

### The Master's Program

Undergraduates who wish to take courses beyond the math major program should consider submatriculation and pursuing a master's degree. The minimum requirements are a A- average in 3600-3610 or 5080-5090 and 3700-3710 or 5020-5030, and permission of the Graduate Chair. Students who plan on a master's degree should submatriculate as early as possible because only courses taken subsequently to this may be counted toward the degree. The degree itself requires the successful completion of eight graduate courses and the written examinations for the Ph.D. The requirements can sometimes be completed by the end of the fourth undergraduate year, but often a fifth year is required.

For more information see the SAS web page Submatriculation and the Math Department Submatriculation page.

### Advice

- Gifted high school students from the Philadelphia area are encouraged to take courses (usually Math 2400-2410) in the department while they are still in high school. This is done through the Young Scholars Program which is administered by the College of Liberal and Professional Studies.
- High school seniors who wish to major in mathematics and think that they might like to attend the University are invited to visit the mathematics department to meet the faculty and visit classes. They should email ugrad AT math.upenn.edu or call 898-8178 for an appointment.
- For our Math Majors and Minors there is a list of courses often approved as COGNATES for Mathematics Majors (these are courses from other departments often approved for mathematics majors or minor credit). All cognates require the approval of the Undergraduate Chair and must be part of a well-planned selection of electives within the major. The statistics courses enjoy a special status: since they count as being inside the Mathematics Department as far as the major or minor is concerned. Thus, a student who takes one or two of these may count additional outside courses toward the Mathematics Elective requirement. Additional courses may also be approved as cognates upon application to the Undergraduate Chair.
- Undergraduates who plan to teach in secondary schools should refer to the section on the Bachelor of Arts/Master of Science in Education.
- Penn has an active
**Undergraduate Mathematics Society**which conducts seminars, colloquia and other activities for those who wish to encounter Math outside the classroom. Information about Society membership and schedules of its activities can be obtained in the Math Department office or by clicking on the link above. - Opportunities for summer research exist at many Universities. The Undergraduate Chair is a good source of information about such programs, which are usually announced in October or November.

### Some External Links relevant to math majors

- National Science Foundation Research Experiences for Undergraduates
- Internships and Summer Employment (maintained by Penn Career Service)
- Penn's Graduate program in Education, including information about obtaining a Masters Degree in Education by submatriculation.