Math 55

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Math 55 is a two-semester long first-year undergraduate mathematics course at Harvard University, founded by Lynn Loomis and Shlomo Sternberg. The official titles of the course are Honors Abstract Algebra (Math 55a) and Honors Real and Complex Analysis (Math 55b). Previously, the official title was Honors Advanced Calculus and Linear Algebra. Some claim it is the most difficult undergraduate math course in the United States.


The Harvard University Department of Mathematics claims that "This [Math 55] is probably the most difficult undergraduate math class in the country".[1] Formerly, students would begin the year in Math 25 (which was created in 1983 as a lower-level Math 55) and, after three weeks of point-set topology and special topics (for instance, in 1994, p-adic analysis was taught by Wilfried Schmid), students would 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.[2] 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 three 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).[3]

Historical retention rate[edit]

In 1970, this demanding course covered almost four years worth of mathematics classes in two semesters, and therefore drew only the most dedicated students.[4][page needed] In the class of 1970, only 20 of the 75 students who began the class finished it due to its difficulty.[4] 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: "[...] we had 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 after the fifth Monday," said Scott Duke Kominers.[5]

Course content[edit]

Through 2006,[6] 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.[4] In 1994, students of Math 25 learned differential geometry using Manfredo do Carmo's book[7] and calculus of variations using C.H. Edwards' book.[8]

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.[citation needed]

Notable alumni[edit]

Problem sets are expected to take from 24 to 60 hours per week to complete,[1] although some claim that it is closer to 20 hours.[9] Of those students who could handle the workload, some became math or physics professors,[4] including members of the Harvard Math Department such as Benedict Gross and Joe Harris; also, Harvard physics professor Lisa Randall '84[10] and Harvard economics professor Andrei Shleifer '82.[11] Although a 2006 Harvard Crimson article alleged that only 17 women completed the class between 1990 and 2006,[5] in fact 39 women completed 55a (the first of the two semesters), and 26 completed 55b.[12] 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.[3]

There are also Math 55 alumni who went on to be professors in other universities. These include Fields-medalist Manjul Bhargava, who is now a professor at Princeton University, as well as Kiran Kedlaya, now at the University of California, San Diego.

In addition to these professors, past students of Math 55 include Bill Gates[13] and Richard Stallman.[4]

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.[14]

Historical instances of Math 55[edit]

Year Instructor Course materials
1996–1997 Alexander Polishchuk
1997–1998 Pavel Etingof
1998–1999 Pavel Etingof
2000–2001 Wilfried Schmid
2002–2003 Noam Elkies 55a[15] b[16]
2003–2004 Yum-Tong Siu 55a[17]
2005–2006 Noam Elkies 55a[18] b
2008–2009 Curtis T. McMullen 55a[19] b[20]
2009–2010 Curtis T. McMullen 55a[21] b[22]
2010–2011 Noam Elkies 55a[23] b[24]
2014–2015 Dennis Gaitsgory
2015–2016 Yum-Tong Siu
2016-2017 Noam Elkies 55a[25]

Fictional references[edit]

Math 55, along with several other high-level mathematics courses, were brought up by Dr. Spencer Reid in a 2015 episode of Criminal Minds entitled "Mr. Scratch."[26]


  1. ^ a b "Harvard Mathematics Department 21, 23, 25, or 55?", 2009
  2. ^ Lee, Steve (16 October 2003). "Math + 55 = Don't Try This at Home". Harvard Independent. 
  3. ^ a b Chen, Susan A. "In Math Department, It's Mostly Male". 
  4. ^ a b c d e Williams, Sam (2002). Free as in Freedom: Richard Stallman's Crusade for Free Software. O'Reilly. ISBN 0-596-00287-4. [page needed]
  5. ^ a b Ury, Logan (2006). "Burden of Proof". 
  6. ^ Compare Elkies course page (2005) and McMullen course page (2008).
  7. ^ do Carmo, Manfredo P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall. ISBN 0-13-212589-7. 
  8. ^ Edwards, Charles Henry. Advanced calculus of several variables. Dover Press. 
  9. ^
  10. ^ "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. 
  11. ^ "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. 
  12. ^ "Registrar data for Math 55". , hosted by Lauren Williams
  13. ^ 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. 
  14. ^ Sommers, Christina Hoff (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. 
  15. ^ "Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2002)". 
  16. ^ "Math 55b: Honors Advanced Calculus and Linear Algebra (Spring 200[2-]3)". 
  17. ^ "Mathematics 55a Syllabus". 
  18. ^ "Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2005)". 
  19. ^ "Math 55a". 
  20. ^ "Math 55b". 
  21. ^ "Math 55a". 
  22. ^ "Math 55b". 
  23. ^ "Math 55a: Honors Abstract Algebra (Fall 2010)". 
  24. ^ "Math 55b: Honors Real and Complex Analysis (Spring [2010–]2011)". 
  25. ^ "Math 55a: Honors Abstract Algebra (Fall 2016)". 
  26. ^ "Criminal Minds Season 10 Episode 21: "Mr. Scratch" Quotes". TV Fanatic. Retrieved 16 December 2015.