Programs · Columbia Engineering · 2026-2027

1. Calculus

   All of these

   All of these: MATH UN1101, MATH UN1102, APMA E2000.

   All three. APMA E2000 carries a required 0-point recitation, APMA E2001, which is not matched here. Students with advanced standing may start the calculus sequence higher up on Advanced Placement credit, which leaves nothing on your record to match.

2. Differential Equations

   Choose N

   Choose 1 of: MATH UN2030, APMA E2101.

   Ordinarily MATH UN2030. APMA E2101 counts only if you took it before declaring the major and your faculty adviser permits it — a condition this audit cannot see, so it is counted here without checking it.

3. Linear Algebra

   Choose N

   Choose 1 of: APMA E3101, MATH UN2010, COMS W3561.

   Applied Math I, or one of the two substitutes the Bulletin names. COMS W3561 is printed by the Bulletin but does not exist in the Bulletin's own course database or in our catalog, so it will not match.

4. Partial Differential Equations

   Choose N

   Choose 1 of: APMA E3102, MATH UN3028, APMA E4200.

   Applied Math II, or either of the two PDE courses the Bulletin accepts in its place.

5. Complex Variables

   Choose N

   Choose 1 of: APMA E4204, MATH UN3007.

   Functions of a Complex Variable, or the Mathematics Department's Complex Variables in its place.

6. Modern Analysis

   Choose N

   Choose 1 of: MATH GU4061, MATH UN2500.

   Introduction to Modern Analysis I, or Analysis and Optimization in its place.

7. Applied Mathematics Core

   All of these

   All of these: APMA E4300, APMA E4101.

   Numerical Methods and Dynamical Systems. These are the two core courses the Bulletin offers no substitute for.

8. Applied Mathematics Seminar

   All of these

   All of these: APMA E4901, APMA E4903.

   Both seminars — the junior-year seminar for 0 points and the senior-year seminar for 3 or 4.

9. Probability (Group A)

   Choose N

   Choose 1 of: IEOR E3658, IEOR E4150, STAT GU4203, MATH GU4155.

   One course from Group A. The Bulletin renders IEOR E4150 as an or-alternative to IEOR E3658; either satisfies this.

10. Applied Probability / Statistics (Group B)

    Choose N

    Choose 1 of: IEOR E3106, IEOR E4106, STAT GU4204, STAT GU4207, COMS W4771.

    One course from Group B, taken in semester VII.

11. Physics

    One sequence

    One sequence: Sequence 1 (PHYS UN1401, PHYS UN1402, PHYS UN1403), or Sequence 1, third term PHYS BC3001 (PHYS UN1401, PHYS UN1402, PHYS BC3001), or Sequence 2 (PHYS UN1601, PHYS UN1602, PHYS UN2601), or Sequence 3 (PHYS UN2801, PHYS UN2802).

    One complete physics sequence. Sequences 1 and 2 run three terms; sequence 3 is Accelerated Physics and the grid gives it a laboratory rather than a third lecture, which is the next requirement. Transfer students who did not finish the physics requirement before enrolling may substitute PHYS BC3001 for the third term of sequence 1, and that path is encoded.

12. Physics Laboratory

    Choose N

    Choose 1 of: PHYS UN1494, PHYS UN3081.

    One physics laboratory. PHYS UN1494 goes with sequences 1 and 2 and PHYS UN3081 with sequence 3 — the audit does not tie the laboratory to the sequence you chose, so check the pairing with your adviser. A footnote also allows "a lab course in Astronomy, Astrophysics, Biology, or Chemistry" in place of either, which names no courses and is not encoded.

13. Chemistry or Biology

    Choose N

    Choose 1 of: CHEM UN1403, BIOL UN2001, EEEB UN2001, BIOL UN2005.

    One lecture course in chemistry or biology, taken in semester I or II. The Bulletin allows "CHEM UN1403 or higher" and "BIOL UN2005 or higher"; only the named courses are matched, so a higher course will not go green automatically.

14. Introductory Computing

    All of these

    All of these: ENGI E1006.

    Required, with no alternative and no waiver on this page. The Applied Physics page carries a Python-proficiency waiver footnote on the same course; the Applied Mathematics page does not.

15. The Art of Engineering

    All of these

    All of these: ENGI E1102.

    Principles of Economics is also required, and is tracked on the Liberal Arts Core rather than repeated here.

16. Undergraduate Research

    All of these

    All of these: APMA E3900.

    3 points of undergraduate research in the final semester. The Bulletin allows an approved technical elective in its place with an adviser's permission, which names no course and is not checked.

17. MATH, APMA or STAT elective

    Points

    3 points matching MATH or APMA or STAT courses, not counting courses already used for another group in this program.

    3 points of MATH, APMA or STAT coursework beyond the courses that already satisfy your other requirements. The Bulletin states no level floor on this row, so a lower-level course counts here as printed.

18. Technical Electives

    Attested

    You confirm this yourself. 18 points of technical electives — 3 in the first two years and at least 15 of the 27 elective points in the third and fourth years. A technical elective is any science, math or engineering course at the 3000 level or above, approved by your adviser; courses in the minor in entrepreneurship and innovation do not count unless your adviser authorises them. "Science, math, or engineering" is a category of departments rather than a set of subject codes, so this one is yours to confirm.