Mathematics

Director of Undergraduate Studies: Prof. Panagiota Daskalopoulos, 526 Math; 854-4756; pdaskalo@math.columbia.edu

Departmental Adviser: Prof. Mu-Tao Wang, 514 Math; 854-3052; mtwang@math.columbia.edu

Computer Science-Mathematics Adviser: Prof. Patrick X. Gallagher, 411 Math; 854-4346; pxg@math.columbia.edu

 Economics-Mathematics Advisers:
Mathematics: Prof. Julien Dubedat, 601 Mathematics; 854-8806; dubedat@math.columbia.edu
Economics:  Dr. Susan Elmes, 1006 IAB; 854-9124; se5@columbia.edu

Mathematics-Statistics Advisers:
Mathematics: Prof. Ioannis Karatzas, 619 Mathematics; 854-3177; ik@math.columbia.edu
Statistics: Prof. Daniel Rabinowitz, 1244 Amsterdam Ave., Room 1014; 851-2141; dan@stat.columbia.edu

Calculus Director: Prof. Peter Woit, 421 Math; 854-2642; woit@math.columbia.edu (Fall)
Prof. Ovidiu Savin, 409 Mathematics; 854-8233; savin@math.columbia.edu (Spring)

Departmental Office: 410 Mathematics; 854-2432

• http://www.math.columbia.edu/

The major in mathematics is an introduction to some of the highlights of the development of theoretical mathematics over the past four hundred years from a modern perspective. This study is also applied to many problems, both internal to mathematics and arising in other disciplines such as physics, cryptography, and finance.

Majors begin by taking either Honors mathematics or the calculus sequence. Students who do not take Honors mathematics A and B normally take Linear algebra in the second year. Following this, majors begin to learn some aspects of the main branches of modern mathematics: algebra; analysis; and geometry, as well as some of their subdivisions and hybrids (e.g., number theory, differential geometry, and complex analysis). As the courses become more advanced, they also become more theoretical and proof-oriented and less computational. Aside from the courses offered by the Mathematics Department, cognate courses in areas such as astronomy, chemistry, physics, probability, logic, economics, and computer science can be used toward the major. A cognate course must be of a level 2000 or higher and must be approved by the director of undergraduate studies. In general, a course not taught by the Mathematics Department is a cognate course for the mathematics major if either (a) it has at least two semesters of calculus as a stated prerequisite, or (b) the subject matter in the course is mathematics beyond an elementary level, such as the logic and set theory course PHIL V3411 in the Philosophy Department or the graph theory and combinatorics course COMS W3203 in the Computer Science Department.

Another requirement for majors is participation in an undergraduate seminar, usually in the junior or senior year. In these seminars, students gain experience in learning an advanced topic and lecturing on it. In order to be eligible for departmental honors, majors must write a senior thesis.

Courses for First-Year Students

The systematic study of mathematics begins with one of the following two alterna­tive sequences:

1. Calculus I, II, III, IV (MATH V1101, MATH V1102, MATH V1201, MATH V1202)
2. Honors mathematics A, B (MATH V1207, MATH V1208)

Credit is allowed for only one calculus sequence.

The first sequence, Calculus I,II,III,IV, is a standard course in differential and integral calculus. While Calculus II is no longer a prerequisite for Calculus III, students are strongly urged to take it before taking Calculus III. In particular, students thinking of majoring or concentrating in mathematics or one of the joint majors involving mathematics should take Calculus II before taking Calculus III. Note that both Calculus II and Calculus III are prerequisites for Calculus IV. Honors mathematics A- B is for exceptionally well-qualified students who have strong Advanced Placement scores. It covers multivariable calculus (e.g.,V1201-V1202) and linear algebra (MATH V2010), with an emphasis on theory.

MATH W1003 does not count toward the degree. Students who take this course do not receive college credit.

Advanced Placement

The department grants 3 credits for a score of 4 on the AP Calculus AB exam provided the student completes MATH V1102 with a grade of C or better. The department grants 3 credits for a score of 5 on the AP Calculus AB exam provided the student completes MATH V1102 or MATH V1201 with a grade of C or better. The department grants 3 credits for a score of 4 on the AP Calculus BC exam provided the student completes MATH V1102 or MATH V1201 with a grade of C or better. The department grants 6 credits for a score of 5 on the AP Calculus BC exam provided the student completes MATH V1201 or MATH V1207 with a grade of C or better. Students can receive credit for only one calculus sequence.

Placement in the Calculus Sequences

Calculus I

Students who have essentially mastered a precalculus course and those who have a score of 3 or less on an Advanced Placement (AP) exam (either AB or BC) should begin their study of calculus with Calculus I.

Calculus II

Students with a score of 4 or 5 on the AB exam or 4 on the BC exam or those with no AP score but with a grade of A in a full year of high school calculus course should begin with Calculus II. Those with a score of 4 or 5 on the AB exam or 4 on the BC exam may receive 3 points of AP credit upon completion of this course with a grade of C or higher.

Calculus III

Students with a score of 5 on the BC exam may begin with Calculus III. Students with a score of 4 on the BC exam, a 5 on the AB exam, or a grade of A in a full year of high school calculus may begin with Calculus III, but are strongly urged to complete the equivalent of Calculus II first. Upon completion of Calculus III with a grade of C or higher, those students with a score of 5 on the BC exam may receive 6 points of AP credit, and those students with a score of 4 on the BC exam or 5 on the AB exam may receive 3 points of AP credit.

Honors mathematics A

Students who want a proof-oriented theoretical sequence and have a score of 5 on the BC exam may begin with Honors A, which is especially designed for mathematics majors. Upon completion of this course with a grade of C or higher, they may receive 6 points of AP credit.

Transfers inside the Calculus Sequences

Students who wish to transfer from one calculus course to another are allowed to do so beyond the date specified in the Academic Calendar. They are considered to be adjusting their level, not changing their program. However, students must obtain the approval of the new instructor and the class dean prior to reporting to the Office of the Registrar.

Grading

No course with a grade of D or lower can count toward the major, interdepartmental major, or concentration. Students who are doing a double major cannot double count courses for their majors.

Departmental Honors

In order to be eligible for departmental honors, majors must write a senior thesis. Normally no more than 10 percent of the graduating majors in the department each year may receive departmental honors.