Math 55

In the past, Harvard University's Department of Mathematics had described Math 55 as "probably the most difficult undergraduate math class in the country."{{cite web |title=Harvard Mathematics Department 21, 23, 25, or 55? |url=http://www.math.harvard.edu/pamphlets/freshmenguide.html |url-status=dead |archive-url=https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html |archive-date=October 19, 2017 |access-date=December 9, 2018}} More recently, the Math 55 lecturer in the year 2022, Professor Denis Auroux, said of the modern version, "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."{{Cite web |last=Yefremova |first=Anastasia |date=May 5, 2022 |title=Demystifying Math 55 |url=https://www.math.harvard.edu/demystifying-math-55/ |archive-url=https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/ |archive-date=August 8, 2022 |access-date=August 25, 2022 |website=Department of Mathematics, Harvard University}}

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.{{cite web |url=http://www.harvardindependent.com/2003/10/math-55-dont-try-this-at-home/ |title=Math + 55 = Don't Try This at Home |first=Steve |last=Lee |date=October 16, 2003|work=Harvard Independent |url-status=dead |archive-url=https://web.archive.org/web/20140121131045/http://www.harvardindependent.com/2003/10/math-55-dont-try-this-at-home/ |archive-date=January 21, 2014 |access-date=December 9, 2018}}

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).{{cite news |last=Chen |first=Susan A. |title=In Math Department, It's Mostly Male |url=https://www.thecrimson.com/article/1994/10/20/in-math-department-its-mostly-male/ |date=October 20, 1994 |access-date=December 9, 2018 |publisher=The Harvard Crimson}}

In the past, problem sets were expected to take from 24 to 60 hours per week to complete, although some claim that it is closer to 20 hours.{{cite news |last=Huang |first=Susie Y. |date=January 6, 1999 |title=Math 55: Rite of Passage for Dept.'s Elite Intimidates Many |publisher=The Harvard Crimson |url=https://www.thecrimson.com/article/1999/1/6/math-55-rite-of-passage-for/?page=4 |access-date=December 9, 2018}} In 2022, on average, students spend a total of 20 to 30 hours per week on this class, including homework.{{Cite web |last1=Arffa |first1=Leslie B. |last2=Pincus |first2=Liza E. |date=March 3, 2011 |title=Blocking at Harvard |url=https://www.thecrimson.com/article/2011/3/3/blocking-group-groups-students/ |archive-url=https://web.archive.org/web/20210516174854/https://www.thecrimson.com/article/2011/3/3/blocking-group-groups-students/ |archive-date=May 16, 2021 |access-date=August 25, 2022 |website=Harvard Crimson}} Taking many other challenging courses and extracurricular activities in the same semester is ill-advised.

Students typically typeset their homework in LaTeX and essentially write their own textbook for the class, which ends with a take-home final exam.{{cite web |title=Math 55a Syllabus |url=http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/syl.html |website=Math 55a: Honors Abstract Algebra |access-date=December 9, 2018}}

Richard Stallman estimated that, in 1970, Math 55 covered almost four years worth of department coursework in two semesters, and thus, 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.{{cite book |last=Williams |first=Sam |author-link=Sam Williams (American journalist) |title=Free as in Freedom: Richard Stallman's Crusade for Free Software |publisher=O'Reilly |year=2002 |isbn=0-596-00287-4 |page=[https://archive.org/details/freeasinfreedomr00will/page/41 41] }} 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 [students] started 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. Even those who passed Advanced Placement Calculus and were veterans of the USA Mathematical Olympiad might feel that Math 55 was too much to handle.

In short, Math 55 gives a survey of the entire undergraduate curriculum of mathematics in just two semesters and might even include graduate-level topics. Through 2006, the instructor had broad latitude in choosing the content of the course.Compare Elkies course page (2005) and McMullen course page (2008). 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.{{cite web |url=https://archive.org/details/MATH25abHonorsMultivariableCalculusAndLinearAlgebraHarvard20042005TextsRudinHalmosSpivak/page/n9 |title=Honors Multivariable Calculus and Linear Algebra, Spring 2005, texts, homework, course outline |editor1-first=Walter |editor1-last=Rudin |editor2-first=Paul R. |editor2-last=Halmos |editor3-first=Michael |editor3-last=Spivak |editor4-first=Tom |editor4-last=Coates |access-date=December 9, 2018}} 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,{{Cite book |last=Loomis |first=Lynn Harold |title=Advanced Calculus |last2=Sternberg |first2=Shlomo Zvi |publisher=World Scientific |year=2014 |isbn=978-9-814-58393-0 |edition=revised |author-link=Lynn Harold Loomis |author-link2=Shlomo Sternberg}} 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{{cite web |last=Auroux |first=Denis |title=Math 55A Course Syllabus (Fall 2020) |url=https://canvas.harvard.edu/courses/75936/assignments/syllabus |access-date=August 3, 2020}}{{cite web |last=Auroux |first=Denis |title=Math 55B Course Syllabus (Spring 2021) |url=https://canvas.harvard.edu/courses/79090/assignments/syllabus |access-date=March 2, 2020}} and Math 25 have also selected Rudin's Principles of Mathematical Analysis,{{Cite book |last=Rudin |first=Walter |title=Principles of Mathematical Analysis |publisher=McGraw Hill |year=1976 |isbn=978-0-070-54235-8 |edition=3rd |author-link=Walter Rudin}} Ahlfors' Complex Analysis,{{Cite book |last=Ahlfors |first=Lars Valerian |title=Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable |publisher=McGraw-Hill Higher Education |year=1978 |isbn=978-0-070-00657-7 |author-link=Lars Valerian Ahlfors}} Spivak's Calculus on Manifolds,{{Cite book |last=Spivak |first=Michael |title=Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus |title-link=Calculus on Manifolds (book) |publisher=CRC Press |year=1965 |isbn=978-0-367-09190-3 |author-link=Michael Spivak}} Axler's Linear Algebra Done Right,{{Cite book |last=Axler |first=Sheldon |title=Linear Algebra Done Right |publisher=Springer |year=2014 |isbn=978-3-319-11079-0 |edition=3rd |author-link=Sheldon Axler}} Halmos's Finite-Dimensional Vector Spaces,{{Cite book |last=Halmos |first=Paul |title=Finite-Dimensional Vector Spaces |publisher=Dover Publications |year=2017 |isbn=978-0-486-81486-5 |edition=2nd |author-link=Paul Halmos}} Munkres' Topology,{{Cite book |last=Munkres |first=James R. |title=Topology |publisher=Pearson |year=2000 |isbn=978-0-131-81629-9 |edition=2nd |author-link=James Munkres}} and Artin's Algebra{{Cite book |last=Artin |first=Michael |title=Algebra |publisher=Pearson |year=2017 |isbn=978-0-134-68960-9 |edition=2nd |author-link=Michael Artin}} 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) or attendees of research programs (such as RSI). Typical students of Math 25 have also had previous exposure to proof writing through mathematical contests or university-level mathematics courses.

Many students who complete the course become professors in quantitative fields. Among those who took Math 55 were UC San Diego mathematician and former Harvard Dean Benedict Gross, Harvard mathematician Joe Harris, Columbia mathematical physicist Peter Woit,{{Cite web |last=Woit |first=Peter |date=May 25, 2022 |title=This and That |url=https://www.math.columbia.edu/~woit/wordpress/?p=12898 |access-date=December 27, 2022 |website=Not Even Wrong}} Harvard physicist Lisa Randall,{{cite news |url=https://www.thecrimson.com/article/2009/6/2/class-of-1984-lisa-randall-as/ |title=Class of 1984: Lisa Randall |quote=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. |date=June 2, 2009 |first=Evan T.R. |last=Robinson |publisher=The Harvard Crimson |access-date=December 9, 2018}} Oxford geophysicist Raymond Pierrehumbert, Harvard economists Andrei Shleifer and Eric Maskin, and UC Berkeley economist Brad DeLong.{{cite news |url=https://www.thecrimson.com/article/2007/6/4/andrei-shleifer-and-j-bradford-delong/ |title=Andrei Shleifer and J. Bradford DeLong |quote="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. |date=June 4, 2007 |first=Paras D. |last=Bhayani |access-date=December 9, 2018 |publisher=The Harvard Crimson}}. Other alumni of Math 55 include business magnate and computer programmer Bill Gates,{{cite book |last1=Manes |first1=Stephen |last2=Andrews |first2=Paul Andrews |title=Gates: How Microsoft's Mogul Reinvented an Industry -- and Made Himself the Richest Man in America |publisher=Doubleday |year=1993 |pages=58 |isbn=0-385-42075-7}}{{Cite magazine |last=Grossman |first=Lev |date=February 29, 2004 |title=10 Questions For Bill Gates |magazine=Time |url=http://www.time.com/time/magazine/article/0,9171,1101040308-596122,00.html |access-date=August 25, 2022 |archive-url=https://archive.today/20120912075312/http://www.time.com/time/magazine/article/0,9171,1101040308-596122,00.html |archive-date=September 12, 2012}} computer programmer and free-software promoter Richard Stallman, and television writer and executive producer Al Jean.{{cite tweet|number=1090017465751367680|user=AlJean|title=.@TheSimpsons I actually took this class |date=28 January 2019}}

A 2006 article in The Harvard Crimson reported that only 17 women completed the class between 1990 and 2006, and a 2017 article said that enrollment had been less than 7% female in the previous five years.{{Cite news |last=Natanson |first=Hannah |date=October 20, 2017 |title='A Sort of Everyday Struggle' |work=Harvard Crimson |url=https://www.thecrimson.com/article/2017/10/20/everyday-struggle-women-math/ |access-date=April 17, 2023}} 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).

Math 55 is mentioned in Season 10, Episode 21 ("Mr. Scratch") of the police procedural crime drama Criminal Minds.{{Citation |last=Gubler |first=Matthew Gray |title=Mr. Scratch |date=2015-04-22 |url=https://www.imdb.com/title/tt4524996/ |access-date=2024-05-23 |series=Criminal Minds |others=Joe Mantegna, Shemar Moore, Matthew Gray Gubler}}

• 1996–1997: Alexander Polishchuk
• 1997–1999: Pavel Etingof
• 1999–2000: Noam Elkies
• 2000–2001: Wilfried Schmid
• 2002–2003: Noam Elkies{{cite web |last=Elkies |first=Noam D. |title=Lecture notes for Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2002) |url=http://www.math.harvard.edu/~elkies/M55a.02/index.html |access-date=December 9, 2018}}{{cite web |last=Elkies |first=Noam D. |title=Lecture notes, etc., for Math 55b: Honors Advanced Calculus and Linear Algebra (Spring 200[2-]3) |url=http://www.math.harvard.edu/~elkies/M55b.02/index.html |access-date=December 9, 2018}}
• 2004–2005: Yum-Tong Siu
• 2005–2006: Noam Elkies{{cite web |last=Elkies |first=Noam D. |title=Lecture notes for Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2005) |url=http://www.math.harvard.edu/~elkies/M55a.05/index.html |access-date=December 9, 2018}}
• 2008–2010: Curtis T. McMullen{{cite web |last=McMullen |first=Curtis T. |title=Math 55a: Honors Abstract Algebra |url=http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/index.html |access-date=December 9, 2018}}{{cite web |last=McMullen |first=Curtis T. |title=Math 55b: Honors Real and Complex Analysis |url=http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/index.html |access-date=December 9, 2018}}
• 2010–2011: Noam Elkies{{cite web |last=Elkies |first=Noam D. |title=Lecture notes for Math 55a: Honors Abstract Algebra (Fall 2010) |url=http://www.math.harvard.edu/~elkies/M55a.10/index.html |access-date=December 9, 2018}}{{cite web |last=Elkies |first=Noam D. |title=Lecture notes, etc., for Math 55b: Honors Real and Complex Analysis (Spring [2010-]2011) |url=http://www.math.harvard.edu/~elkies/M55b.10/index.html |access-date=December 9, 2018}}
• 2011–2012: Yum-Tong Siu
• 2013–2015: Dennis Gaitsgory
• 2015–2016: Yum-Tong Siu
• 2016–2018: Noam Elkies{{cite web |last=Elkies |first=Noam D. |title=Lecture notes for Math 55a: Honors Abstract Algebra (Fall 2016) |url=http://www.math.harvard.edu/~elkies/M55a.16/index.html |access-date=December 9, 2018}}{{Cite web |last=Elkies |first=Noam D. |title=Lecture notes, etc., for Math 55b: Honors Real and Complex Analysis (Spring [2016-]2017) |url=https://people.math.harvard.edu/~elkies/M55b.16/index.html |access-date=2022-02-21}}
• 2018–2020: Joe Harris
• 2020–2022: Denis Auroux
• 2022–2024: Joe Harris
• 2024-2025: Denis Auroux

{{Portal|Education|Mathematics|United States}}

• Mathematics education in the United States
• William Lowell Putnam Mathematical Competition
• Part III of the Mathematical Tripos

{{clear}}

{{Reflist|2}}

{{refbegin|2}}

{{refend}}

• {{cite book|url=https://archive.org/details/advancedcalculus0000loom|title=Advanced Calculus|first1=Lynn|last1=Loomis|first2=Shlomo|author1-link=Lynn Loomis|last2=Sternberg|author2-link=Shlomo Sternberg|edition=Revised|publisher=Jones and Bartlett Publishers, Inc.|year=1990|orig-year=1968|isbn=0-86720-122-3|access-date=December 9, 2018|url-access=registration}}
• {{cite web|url=http://www.math.harvard.edu/pamphlets/freshmenguide.html|title=Harvard Mathematics Department 21, 23, 25, or 55?|url-status=dead|archive-url=https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html|archive-date=October 19, 2017|access-date=December 9, 2018}}
• [https://web.archive.org/web/20090722084436if_/http://www.ajorza.org/courses/math/m55/ Problem sets from Math 55 from 1999-2006]
• 1983: {{Cite web |title=Math Faculty Committee Proposes Curriculum Revision |url=https://www.thecrimson.com/article/1983/2/14/math-faculty-committee-proposes-curriculum-revision/ |access-date=2024-09-13 |website=The Harvard Crimson}}
• 1999: {{Cite web |title=Math 55: Rite of Passage for Dept.'s Elite Intimidates Many |url=https://www.thecrimson.com/article/1999/1/6/math-55-rite-of-passage-for/ |access-date=2024-09-13 |website=The Harvard Crimson}}
• 2000: {{Cite web|url=http://www.hcs.harvard.edu/~satirev/old/vol1no2-math.html|title=Math 55 student gets laid, considered for Fields Medal|website=Satire V|archive-url=https://web.archive.org/web/20020618165708if_/http://www.hcs.harvard.edu/~satirev/old/vol1no2-math.html|archive-date=2002-06-18|url-status=dead}}
• 2005: {{cite web|url=http://www.math.harvard.edu/~elkies/M55a.05/|title=Math 55a|author=Noam Elkies|author-link=Noam Elkies}}
• 2009/10: [http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/09/html/index.html Math 55a], [http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/10/html/index.html Math 55b] – Curtis McMullen
• 2010/11: [http://www.math.harvard.edu/~elkies/M55a.10/ Math 55a], [http://www.math.harvard.edu/~elkies/M55b.10/ Math 55b] – Noam Elkies

{{Harvard|state=collapsed}}

Category:Harvard University

Category:Mathematics education in the United States

Category:Undergraduate education in the United States