Math 55

Math 55 is a two-semester long freshman 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.

Description

In the past, Harvard University's Department of Mathematics had described Math 55 as "probably the most difficult undergraduate math class in the country." But Math 55 lecturer for 2022 Professor Denis Auroux clarified that "if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you."

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.

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 diagnostic exam: students scoring more than 50% on the quiz could enroll in Schmid's Math 55 (15 students), students scoring between 10 and 50% could enroll in Benedict Gross's Math 25: Theoretical Linear Algebra and Real Analysis (55 students), and students scoring less than 10% were advised to enroll in a course such as Math 21: Multivariable Calculus (19 students).

In 2022, on average, students spend a total of 20 to 30 hours per week on this class, including homework. In addition to visiting their professor during office hours, students are encouraged to work together on homework assignments. Many have spent much time together working in the "war room" (a place in the Grays Basement) and have developed long-lasting friendships. Taking many other challenging courses and extracurricular activities in the same semester is ill-advised.

A take-home final ends the class.

Historical retention rate

In 1970, Math 55 covered almost four years worth of department coursework in two semesters, and subsequently, it drew only the most diligent of undergraduates. Of the 75 students who enrolled in the 1970 offering, by course end, only 20 remained due to the advanced nature of the material and time-constraints under which students were given to work. David Harbater, a mathematics professor at the University of Pennsylvania and student of the 1974 Math 55 section at Harvard, recalled of his experience, "Seventy tudentsstarted it, 20 finished it, and only 10 understood it." Scott D. Kominers, familiar with the stated attrition rates for the course, decided to keep an informal log of his journey through the 2009 section: "...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" (the beginning of the fifth week being the drop deadline for students to decide whether to remain in Math 55 or transfer to Math 25).

Numbers of students dropping are due in part to the tendency of undergraduates to "shop around" for appropriate courses at the start of each semester.

Course content

Through 2006, the instructor had broad latitude in choosing the content of the course. Though Math 55 bore the official title "Honors Advanced Calculus and Linear Algebra," advanced topics in complex analysis, point-set topology, group theory, and differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable real analysis as well as abstract linear algebra. In 1970, for example, students studied the differential geometry of Banach manifolds in the second semester of Math 55. In contrast, Math 25 was more narrowly focused, usually covering real analysis, together with the relevant theory of metric spaces and (multi)linear maps. These topics typically culminated in the proof of the generalized Stokes theorem, though, time permitting, other relevant topics (e.g. category theory, de Rham cohomology) might also be covered. Although both courses presented calculus from a rigorous point of view and emphasized theory and proof writing, Math 55 was generally faster paced, more abstract, and demanded a higher level of mathematical sophistication.

Loomis and Sternberg's textbook ''Advanced Calculus'', an abstract treatment of calculus in the setting of normed vector spaces and on differentiable manifolds, was tailored to the authors' Math 55 syllabus and served for many years as an assigned text. Instructors for Math 55 and Math 25 have also selected Rudin's ''Principles of Mathematical Analysis'', Spivak's '' Calculus on Manifolds'', Axler's ''Linear Algebra Done Right'', and Halmos's ''Finite-Dimensional Vector Spaces'' as textbooks or references.

From 2007 onwards, the scope of the course (along with that of Math 25) was changed to more strictly cover the contents of four semester-long courses in two semesters: Math 25a (linear algebra and real analysis) and Math 122 (group theory and vector spaces) in Math 55a; and Math 25b (real analysis) and Math 113 (complex analysis) in Math 55b. The name was also changed to "Honors Abstract Algebra" (Math 55a) and "Honors Real and Complex Analysis" (Math 55b). Fluency in formulating and writing mathematical proofs is listed as a course prerequisite for Math 55, while such experience is considered "helpful" but not required for Math 25. In practice, students of Math 55 have usually had extensive experience in proof writing and abstract mathematics, with many being the past winners of prestigious national or international mathematical Olympiads (such as USAMO or IMO). Typical students of Math 25 have also had previous exposure to proof writing through mathematical contests or university-level mathematics courses.

Notable alumni

Problem sets are expected to take from 24 to 60 hours per week to complete, although some claim that it is closer to 20 hours. Many of those who are able to handle the workload and complete the course become professors in quantitative fields; alumni of Math 55 include Harvard mathematicians Benedict Gross and Joe Harris, Columbia mathematical physicist Peter Woit, Harvard physicist Lisa Randall, Harvard economists Andrei Shleifer and Eric Maskin, and UC Berkeley economist Brad DeLong..

Past students of Math 55 also include business magnate and computer programmer Bill Gates, computer programmer and free-software promoter Richard Stallman, and television writer and executive producer Al Jean.

Demographics

Contrary to a 2006 article in ''The Harvard Crimson'' which alleged that only 17 women completed the class between 1990 and 2006, 39 women completed 55a 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 2006, the class was 45 percent Jewish (5 students), 18 percent Asian (2 students), 100 percent male (11 students).Burden_of_Proof",_Logan_R._Ury,_Dec_6,_The_Crimson
,_2006._"The_final_course_drop_forms_are_dutifully_submitted,_finalizing_the_class_roster:_45_percent_Jewish,_18_percent_Asian,_100_percent_male._The_tribe_has_spoken."

Demographics_of_students_taking_this_course_over_the_years_have_been_used_to_study_the_causes_of_Achievement_gaps_in_the_United_States#Math_and_science.html" ;"title="The Crimson">Burden of Proof", Logan R. Ury, Dec 6, The Crimson
, 2006. "The final course drop forms are dutifully submitted, finalizing the class roster: 45 percent Jewish, 18 percent Asian, 100 percent male. The tribe has spoken."

Demographics of students taking this course over the years have been used to study the causes of Achievement gaps in the United States#Math and science">gender and race differences in STEM.

Historical instances of Math 55

See also

* Mathematics education in the United States

References

*
*

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:
* 2006
Burden of Proof, December 2006, Harvard Crimson
* 2005:
* 2009/10

– Curtis McMullen
* 2010/11
Math 55aMath 55b
– Noam Elkies

{{Harvard, state=collapsed
Harvard University
Mathematics education in the United States
Undergraduate education in the United States