Math 55 is a two-semester long first-year undergraduate mathematics course at Harvard University. The official titles of the course are Honors Abstract Algebra and Honors Real and Complex Analysis. (The official title was previously Honors Advanced Calculus and Linear Algebra.)
The Harvard math department claims, "This is probably the most difficult undergraduate math class in the country." Students used to begin the year by enrolling in Math 25, created in 1983 as a lower-level Math 55, and after 3 weeks of point-set topology and special topics (in 1994, p-adic analysis taught by Wilfried Schmid), students take a quiz. As of 2012, students may choose to enroll in either Math 25 or Math 55, but are advised to shop both courses and have five weeks to decide on one. Depending on the professor teaching the class, the diagnostic exam may still be given after three weeks to help students with their decision.
In 1994, 89 students took the test given after 3 weeks: students scoring more than 50% on the quiz could enroll in Wilfried Schmid's Math 55 (15 students), students scoring between 10 and 50% could stay in Benedict Gross's Math 25 (55 students), and students scoring less than 10% were advised to enroll in a lower-level course such as Math 21, multivariate calculus (19 students).
Historical retention rate
In 1970, this demanding course covered almost four years worth of mathematics classes in two semesters, and therefore drew only the most dedicated students. In the class of 1970, only 20 of the 75 students who began the class finished it due to its difficulty. Similar drop-out rates were true for the class of 1976: "Seventy started it, 20 finished it, and only 10 understood it." and for the class of 2009: "51 students the first day, 31 students the second day, 24 for the next four days, 23 for two more weeks, and then 21 for the rest of the first semester."
Problem sets are expected to take from 24 to 60 hours per week to complete. Of those students who could handle the workload, some became math or physics professors, including members of the Harvard Math Department such as Benedict Gross and Joe Harris; also, Harvard physics professor Lisa Randall '84 and Harvard economics professor Andrei Shleifer '82. Although a 2006 Harvard Crimson article alleged that only 17 women completed the class between 1990 and 2006, in fact 39 women completed 55a (the first of the two semesters), and 26 completed 55b. Math 25 has more women: in 1994–95, Math 55 had no women, while Math 25 had about 10 women in the 55 person course.
In addition to these professors, past students of Math 55 include Bill Gates and Richard Stallman. Gates said the experience of taking a class "where everybody had an 800 on their SAT and 5 on their AP" was a "neat experience".
Demographics of students taking this course over the years has been used to study causes of gender and race differences in the fields of mathematics and technology.
Through 2006, the instructor had broad latitude in choosing the content of the course, and the course bore the title "Honors Advanced Calculus and Linear Algebra". For example, in 1970 by the second semester, students were learning about the differential geometry of Banach manifolds. In 1994, students of Math 25 learned differential geometry using DoCarmo's book and calculus of variations using C.H. Edwards's book.
From 2007 onwards, the scope of the course was changed to more strictly cover the content of four existing semester-long courses in two semesters: Math 25a (linear algebra) and Math 122 (group theory) in Math 55a; and Math 25b (calculus, real analysis) and Math 113 (complex analysis) in Math 55b. The name was changed to "Honors Abstract Algebra" and "Honors Real and Complex Analysis" to reflect this.
- "Harvard Mathematics Department 21, 23, 25, or 55?", 2009
- Chen, Susan A. "In Math Department, It's Mostly Male".
- Williams, Sam (2002). Free as in Freedom: Richard Stallman's Crusade for Free Software. O'Reilly. ISBN 0-596-00287-4.
- Ury, Logan (2006). "Burden of Proof".
- "Class of 1984: Lisa Randall". June 2, 2009. "As a college freshman, Lisa J. Randall '84 stood out for many reasons. In her first semester, she enrolled in Math 55 and Physics 55, the most difficult freshman math and physics classes offered."
- "Andrei Shleifer and J. Bradford DeLong". June 4, 2007. "“Math 55 permanently disabused me of the idea of becoming a mathematician,” Shleifer says. Though he would tough the class out and remain a math major, he says he became drawn to economics—a subject he knew nothing of in high school—after taking some introductory courses in the field."
- "Registrar data for Math 55"., hosted by Lauren Williams
- Manes, Stephen; Paul Andrews (1993). Gates: how Microsoft's mogul reinvented an industry--and made himself the richest man in America. Doubleday. p. 58. ISBN 0-385-42075-7.
- Lev Grossman (February 29, 2004). "10 Questions For Bill Gates". Time. Retrieved 2009-03-12. "I certainly learned that. I took this class, Math 55, where everybody had an 800 on their SAT and 5 on their AP, and that was quite a neat experience."
- Christina Hoff Sommers (March/April 2008). "Why Can’t a Woman Be More Like a Man?". The American. Retrieved 2009-08-13. "Math 55 is advertised in the Harvard catalog as “probably the most difficult undergraduate math class in the country.” It is legendary among high school math prodigies, who hear terrifying stories about it in their computer camps and at the Math Olympiads. Some go to Harvard just to have the opportunity to enroll in it. Its formal title is “Honors Advanced Calculus and Linear Algebra,” but it is also known as “math boot camp” and “a cult.” The two-semester freshman course meets for three hours a week, but, as the catalog says, homework for the class takes between 24 and 60 hours a week."
- Compare Elkies course page (2005) and McMullen course page (2008).
- do Carmo, Manfredo P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall. ISBN 0-13-212589-7.
- Edwards, Charles Henry. Advanced calculus of several variables. Dover Press.
- Harvard Mathematics Department 21, 23, 25, or 55? – Harvard Math Freshmen Guide pamphlet
- Problem sets from Math 55 from 1999-2006
- 1983: Harvard Crimson, Feb 14, 1983
- 1994: In Math Department, It's Mostly Male, Harvard Crimson, Oct. 20, 1994
- 1999: 
- 2000: Math 55 student gets laid, considered for Fields Medal, Satire V (humor magazine), 2000
- 2006: Burden of Proof, December 2006, Harvard Crimson
- 2005: Math 55a – Noam Elkies
- 2009/10: Math 55a, Math 55b – Curtis McMullen
- 2010/11: Math 55a, Math 55b – Noam Elkies