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Math 55
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<h2 id="description">Description</h2> <p>In the past, Harvard University's Department of Mathematics had described Math 55 as "probably the most difficult undergraduate math class in the country."<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> More recently, the Math 55 lecturer in the year 2022, Professor <a href="/facts/Denis_Auroux/Ku4Wbbom">Denis Auroux</a>, 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."<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>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 <a href="/facts/Point-set_topology/ezC7B0yP">point-set topology</a> and special topics (for instance, in 1994, <a href="/facts/P-adic_analysis/VrdVVN43"><i>p</i>-adic analysis</a> was taught by <a href="/facts/Wilfried_Schmid/HaBoRhbu">Wilfried Schmid</a>), 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.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>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 <a href="/facts/Benedict_Gross/XS6uAajW">Benedict Gross</a>'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).<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>In the past, <a href="/facts/Problem_set/LKaWaeaj">problem sets</a> were expected to take from 24 to 60 hours per week to complete,<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> although some claim that it is closer to 20 hours.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> In 2022, on average, students spend a total of 20 to 30 hours per week on this class, including homework.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> Taking many other challenging courses and extracurricular activities in the same semester is ill-advised.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p><p>Students typically typeset their homework in <a href="/facts/LaTeX/c1lvjF0A">LaTeX</a> and essentially write their own textbook for the class,<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> which ends with a <a href="/facts/Take-home_final/Xko3ohe2">take-home final exam</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h3>Historical retention rate</h3> <p><a href="/facts/Richard_Stallman/hhoPN0Mw">Richard Stallman</a> 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.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> <a href="/facts/David_Harbater/5gqzGAbR">David Harbater</a>, a mathematics professor at the <a href="/facts/University_of_Pennsylvania/LBBBULox">University of Pennsylvania</a> 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).<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p><p>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.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> Even those who passed <a href="/facts/AP_Calculus/gNjt3l5r">Advanced Placement Calculus</a> and were veterans of the <a href="/facts/United_States_of_America_Mathematical_Olympiad/HQ4EI4Lx">USA Mathematical Olympiad</a> might feel that Math 55 was too much to handle.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> <h2 id="course-content">Course content</h2> <p>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.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Through 2006, the instructor had broad latitude in choosing the content of the course.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> Though Math 55 bore the official title "Honors Advanced Calculus and Linear Algebra," advanced topics in <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, point-set topology, <a href="/facts/Group_theory/NfowcI5H">group theory</a>, and differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable <a href="/facts/Real_analysis/12WjlXCW">real analysis</a> as well as abstract <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>. In 1970, for example, students studied the <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> of <a href="/facts/Banach_manifold/EySHBrBz">Banach manifolds</a> in the second semester of Math 55.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> 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 <a href="/facts/Generalized_Stokes_theorem/tNBH6MKK">generalized Stokes theorem</a>, though, time permitting, other relevant topics (e.g. <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, <a href="/facts/De_Rham_cohomology/4qGKeJq0">de Rham cohomology</a>) might also be covered.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> 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 <a href="/facts/Mathematical_maturity/C11NPyXT">mathematical sophistication</a>. </p><p>Loomis and Sternberg's textbook <i>Advanced Calculus</i>,<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> an abstract treatment of calculus in the setting of <a href="/facts/Normed_vector_spaces/rmDljfyE">normed vector spaces</a> and on <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifolds</a>, was tailored to the authors' Math 55 syllabus and served for many years as an assigned text. Instructors for Math 55<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a><a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> and Math 25<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> have also selected <a href="/facts/Walter_Rudin/9mxRAECY">Rudin</a>'s <i>Principles of Mathematical Analysis</i>,<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> <a href="/facts/Lars_Ahlfors/aukXlC2t">Ahlfors</a>' <i>Complex Analysis</i>,<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> <a href="/facts/Michael_Spivak/C67tKX7d">Spivak</a>'s <i><a href="/facts/Calculus_on_Manifolds_(book)/JENnXtEC">Calculus on Manifolds</a></i>,<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> <a href="/facts/Sheldon_Axler/GMKk6zwI">Axler</a>'s <i>Linear Algebra Done Right</i>,<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> <a href="/facts/Paul_Halmos/mR11lcap">Halmos</a>'s <i>Finite-Dimensional Vector Spaces,<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a></i> <a href="/facts/James_Munkres/Ybfa6fs4">Munkres</a>' <i>Topology</i>,<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> and <a href="/facts/Michael_Artin/Tfk0f6rK">Artin</a>'s <i>Algebra<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a></i> as textbooks or references. </p><p>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 <a href="/facts/Mathematical_proof/7ECDrU80">mathematical proofs</a> is listed as a course prerequisite for Math 55, while such experience is considered "helpful" but not required for Math 25.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> 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 <a href="/facts/United_States_of_America_Mathematical_Olympiad/HQ4EI4Lx">USAMO</a> or <a href="/facts/International_Mathematical_Olympiad/HWBMkwEq">IMO</a>) or attendees of research programs (such as <a href="/facts/Research_Science_Institute/zj7kC8zs">RSI</a>). Typical students of Math 25 have also had previous exposure to proof writing through mathematical contests or university-level mathematics courses. </p> <h2 id="notable-alumni">Notable alumni</h2> <p>Many students who complete the course become professors in quantitative fields.<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> Among those who took Math 55 were UC San Diego mathematician and former Harvard Dean <a href="/facts/Benedict_Gross/XS6uAajW">Benedict Gross</a>,<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> Harvard mathematician <a href="/facts/Joe_Harris_(mathematician)/XPzjGwRf">Joe Harris</a>,<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> Columbia mathematical physicist <a href="/facts/Peter_Woit/W7gTlflX">Peter Woit</a>,<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> Harvard physicist <a href="/facts/Lisa_Randall/3kp0wIlf">Lisa Randall</a>,<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a> Oxford geophysicist <a href="/facts/Raymond_Pierrehumbert/wNqYtMhV">Raymond Pierrehumbert</a>,<a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a> Harvard economists <a href="/facts/Andrei_Shleifer/arNm3L62">Andrei Shleifer</a> and <a href="/facts/Eric_Maskin/7DEYYL5G">Eric Maskin</a>, and UC Berkeley economist <a href="/facts/J._Bradford_DeLong/S2JWRYHr">Brad DeLong</a>.<a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a> Other alumni of Math 55 include business magnate and computer programmer <a href="/facts/Bill_Gates/z16t4m7s">Bill Gates</a>,<a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a><a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a> computer programmer and free-software promoter <a href="/facts/Richard_Stallman/hhoPN0Mw">Richard Stallman</a>,<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a> and television writer and executive producer <a href="/facts/Al_Jean/HxT9MQuz">Al Jean</a>.<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a> </p> <h2 id="demographics">Demographics</h2> <p>A 2006 article in <i><a href="/facts/The_Harvard_Crimson/g3Laj9wU">The Harvard Crimson</a></i> reported that only 17 women completed the class between 1990 and 2006,<a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> and a 2017 article said that enrollment had been less than 7% female in the previous five years.<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a> 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.<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a> In 2006, the class was 45 percent Jewish (5 students), 18 percent Asian (2 students), 100 percent male (11 students).<a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a> </p> <h2 id="in-popular-culture">In popular culture</h2> <p>Math 55 is mentioned in Season 10, Episode 21 ("Mr. Scratch") of the <a href="/facts/Police_procedural/vylKjcPX">police procedural</a> <a href="/facts/Crime_drama/IbSVYSme">crime drama</a> <i><a href="/facts/Criminal_Minds/HwA2dywG">Criminal Minds</a></i>.<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a> </p> <h2 id="instructors">Instructors</h2> <ul><li>1996–1997: Alexander Polishchuk</li> <li>1997–1999: <a href="/facts/Pavel_Etingof/pvp2GUaa">Pavel Etingof</a></li> <li>1999–2000: <a href="/facts/Noam_Elkies/t1Uerz9E">Noam Elkies</a></li> <li>2000–2001: <a href="/facts/Wilfried_Schmid/HaBoRhbu">Wilfried Schmid</a></li> <li>2002–2003: Noam Elkies<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a><a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a></li> <li>2004–2005: <a href="/facts/Yum-Tong_Siu/uVglzjqW">Yum-Tong Siu</a></li> <li>2005–2006: Noam Elkies<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a></li> <li>2008–2010: <a href="/facts/Curtis_T._McMullen/WKEXxuc1">Curtis T. McMullen</a><a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a><a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a></li> <li>2010–2011: Noam Elkies<a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a><a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a></li> <li>2011–2012: Yum-Tong Siu</li> <li>2013–2015: <a href="/facts/Dennis_Gaitsgory/QrYkT2Kz">Dennis Gaitsgory</a></li> <li>2015–2016: Yum-Tong Siu</li> <li>2016–2018: Noam Elkies<a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a><a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a></li> <li>2018–2020: <a href="/facts/Joe_Harris_(mathematician)/XPzjGwRf">Joe Harris</a></li> <li>2020–2022: <a href="/facts/Denis_Auroux/Ku4Wbbom">Denis Auroux</a><a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a><a class="footnote-ref" id="fnref:61" href="#fn:61"><sup>61</sup></a></li> <li>2022–2024: Joe Harris</li> <li>2024-2025: <a href="/facts/Denis_Auroux/Ku4Wbbom">Denis Auroux</a></li></ul> <h2 id="see-also">See also</h2> <ul> <li>Education portal</li><li>Mathematics portal</li><li>United States portal</li></ul> <ul><li><a href="/facts/Mathematics_education_in_the_United_States/Gnf59Hfq">Mathematics education in the United States</a></li> <li><a href="/facts/William_Lowell_Putnam_Mathematical_Competition/wTJSGQHM">William Lowell Putnam Mathematical Competition</a></li> <li><a href="/facts/Part_III_of_the_Mathematical_Tripos/qGdwtR4R">Part III of the Mathematical Tripos</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Lynn_Loomis/P3yIpycg">Loomis, Lynn</a>; <a href="/facts/Shlomo_Sternberg/IsGM8VzY">Sternberg, Shlomo</a> (1990) [1968]. <a href="https://archive.org/details/advancedcalculus0000loom"><i>Advanced Calculus</i></a> (Revised ed.). Jones and Bartlett Publishers, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-86720-122-3. Retrieved December 9, 2018.</li> <li><a href="https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html">"Harvard Mathematics Department 21, 23, 25, or 55?"</a>. Archived from <a href="http://www.math.harvard.edu/pamphlets/freshmenguide.html">the original</a> on October 19, 2017. Retrieved December 9, 2018.</li> <li><a href="https://web.archive.org/web/20090722084436if_/http://www.ajorza.org/courses/math/m55/">Problem sets from Math 55 from 1999-2006</a></li> <li>1983: <a href="https://www.thecrimson.com/article/1983/2/14/math-faculty-committee-proposes-curriculum-revision/">"Math Faculty Committee Proposes Curriculum Revision"</a>. <i>The Harvard Crimson</i>. Retrieved 2024-09-13.</li> <li>1999: <a href="https://www.thecrimson.com/article/1999/1/6/math-55-rite-of-passage-for/">"Math 55: Rite of Passage for Dept.'s Elite Intimidates Many"</a>. <i>The Harvard Crimson</i>. Retrieved 2024-09-13.</li> <li>2000: <a href="https://web.archive.org/web/20020618165708if_/http://www.hcs.harvard.edu/~satirev/old/vol1no2-math.html">"Math 55 student gets laid, considered for Fields Medal"</a>. <i>Satire V</i>. Archived from <a href="http://www.hcs.harvard.edu/~satirev/old/vol1no2-math.html">the original</a> on 2002-06-18.</li> <li>2005: <a href="/facts/Noam_Elkies/t1Uerz9E">Noam Elkies</a>. <a href="http://www.math.harvard.edu/~elkies/M55a.05/">"Math 55a"</a>.</li> <li>2009/10: <a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/09/html/index.html">Math 55a</a>, <a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/10/html/index.html">Math 55b</a> – <a href="/facts/Curtis_McMullen/WKEXxuc1">Curtis McMullen</a></li> <li>2010/11: <a href="http://www.math.harvard.edu/~elkies/M55a.10/">Math 55a</a>, <a href="http://www.math.harvard.edu/~elkies/M55b.10/">Math 55b</a> – <a href="/facts/Noam_Elkies/t1Uerz9E">Noam Elkies</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Mathematics 55a". Courses. Department of Mathematics, Harvard University. 2023. Retrieved February 9, 2023. <a href="https://www.math.harvard.edu/course/mathematics-55a/" target="_blank">https://www.math.harvard.edu/course/mathematics-55a/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Mathematics 55b". Courses. Department of Mathematics, Harvard University. 2023. Retrieved February 9, 2023. <a href="https://www.math.harvard.edu/course/mathematics-55b/" target="_blank">https://www.math.harvard.edu/course/mathematics-55b/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Harvard Mathematics Department 21, 23, 25, or 55?". Archived from the original on October 19, 2017. Retrieved December 9, 2018. <a href="https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html" target="_blank">https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Yefremova, Anastasia (May 5, 2022). "Demystifying Math 55". Department of Mathematics, Harvard University. Archived from the original on August 8, 2022. Retrieved August 25, 2022. <a href="https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/" target="_blank">https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Lee, Steve (October 16, 2003). "Math + 55 = Don't Try This at Home". Harvard Independent. Archived from the original on January 21, 2014. Retrieved December 9, 2018. <a href="https://web.archive.org/web/20140121131045/http://www.harvardindependent.com/2003/10/math-55-dont-try-this-at-home/" target="_blank">https://web.archive.org/web/20140121131045/http://www.harvardindependent.com/2003/10/math-55-dont-try-this-at-home/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Chen, Susan A. (October 20, 1994). "In Math Department, It's Mostly Male". The Harvard Crimson. Retrieved December 9, 2018. <a href="https://www.thecrimson.com/article/1994/10/20/in-math-department-its-mostly-male/" target="_blank">https://www.thecrimson.com/article/1994/10/20/in-math-department-its-mostly-male/</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"Harvard Mathematics Department 21, 23, 25, or 55?". Archived from the original on October 19, 2017. Retrieved December 9, 2018. <a href="https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html" target="_blank">https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Huang, Susie Y. (January 6, 1999). "Math 55: Rite of Passage for Dept.'s Elite Intimidates Many". The Harvard Crimson. Retrieved December 9, 2018. <a href="https://www.thecrimson.com/article/1999/1/6/math-55-rite-of-passage-for/?page=4" target="_blank">https://www.thecrimson.com/article/1999/1/6/math-55-rite-of-passage-for/?page=4</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Yefremova, Anastasia (May 5, 2022). "Demystifying Math 55". Department of Mathematics, Harvard University. Archived from the original on August 8, 2022. Retrieved August 25, 2022. <a href="https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/" target="_blank">https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Arffa, Leslie B.; Pincus, Liza E. (March 3, 2011). "Blocking at Harvard". Harvard Crimson. Archived from the original on May 16, 2021. Retrieved August 25, 2022. <a href="https://web.archive.org/web/20210516174854/https://www.thecrimson.com/article/2011/3/3/blocking-group-groups-students/" target="_blank">https://web.archive.org/web/20210516174854/https://www.thecrimson.com/article/2011/3/3/blocking-group-groups-students/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Yefremova, Anastasia (May 5, 2022). "Demystifying Math 55". Department of Mathematics, Harvard University. Archived from the original on August 8, 2022. Retrieved August 25, 2022. <a href="https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/" target="_blank">https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>"Math 55a Syllabus". Math 55a: Honors Abstract Algebra. Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/syl.html" target="_blank">http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/syl.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Williams, Sam (2002). Free as in Freedom: Richard Stallman's Crusade for Free Software. O'Reilly. p. 41. ISBN 0-596-00287-4. <a href="0-596-00287-4" target="_blank">0-596-00287-4</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Yefremova, Anastasia (May 5, 2022). "Demystifying Math 55". Department of Mathematics, Harvard University. Archived from the original on August 8, 2022. Retrieved August 25, 2022. <a href="https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/" target="_blank">https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Compare Elkies course page (2005) and McMullen course page (2008). <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Williams, Sam (2002). Free as in Freedom: Richard Stallman's Crusade for Free Software. O'Reilly. p. 41. ISBN 0-596-00287-4. <a href="0-596-00287-4" target="_blank">0-596-00287-4</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Rudin, Walter; Halmos, Paul R.; Spivak, Michael; Coates, Tom (eds.). "Honors Multivariable Calculus and Linear Algebra, Spring 2005, texts, homework, course outline". Retrieved December 9, 2018. <a href="https://archive.org/details/MATH25abHonorsMultivariableCalculusAndLinearAlgebraHarvard20042005TextsRudinHalmosSpivak/page/n9" target="_blank">https://archive.org/details/MATH25abHonorsMultivariableCalculusAndLinearAlgebraHarvard20042005TextsRudinHalmosSpivak/page/n9</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Loomis, Lynn Harold; Sternberg, Shlomo Zvi (2014). Advanced Calculus (revised ed.). World Scientific. ISBN 978-9-814-58393-0. <a href="978-9-814-58393-0" target="_blank">978-9-814-58393-0</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Auroux, Denis. "Math 55A Course Syllabus (Fall 2020)". Retrieved August 3, 2020. <a href="https://canvas.harvard.edu/courses/75936/assignments/syllabus" target="_blank">https://canvas.harvard.edu/courses/75936/assignments/syllabus</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Auroux, Denis. "Math 55B Course Syllabus (Spring 2021)". Retrieved March 2, 2020. <a href="https://canvas.harvard.edu/courses/79090/assignments/syllabus" target="_blank">https://canvas.harvard.edu/courses/79090/assignments/syllabus</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Rudin, Walter; Halmos, Paul R.; Spivak, Michael; Coates, Tom (eds.). "Honors Multivariable Calculus and Linear Algebra, Spring 2005, texts, homework, course outline". Retrieved December 9, 2018. <a href="https://archive.org/details/MATH25abHonorsMultivariableCalculusAndLinearAlgebraHarvard20042005TextsRudinHalmosSpivak/page/n9" target="_blank">https://archive.org/details/MATH25abHonorsMultivariableCalculusAndLinearAlgebraHarvard20042005TextsRudinHalmosSpivak/page/n9</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). McGraw Hill. ISBN 978-0-070-54235-8. <a href="978-0-070-54235-8" target="_blank">978-0-070-54235-8</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Ahlfors, Lars Valerian (1978). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill Higher Education. ISBN 978-0-070-00657-7. <a href="978-0-070-00657-7" target="_blank">978-0-070-00657-7</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Spivak, Michael (1965). Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. CRC Press. ISBN 978-0-367-09190-3. <a href="978-0-367-09190-3" target="_blank">978-0-367-09190-3</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Axler, Sheldon (2014). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0. <a href="978-3-319-11079-0" target="_blank">978-3-319-11079-0</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Halmos, Paul (2017). Finite-Dimensional Vector Spaces (2nd ed.). Dover Publications. ISBN 978-0-486-81486-5. <a href="978-0-486-81486-5" target="_blank">978-0-486-81486-5</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Munkres, James R. (2000). Topology (2nd ed.). Pearson. ISBN 978-0-131-81629-9. <a href="978-0-131-81629-9" target="_blank">978-0-131-81629-9</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Artin, Michael (2017). Algebra (2nd ed.). Pearson. ISBN 978-0-134-68960-9. <a href="978-0-134-68960-9" target="_blank">978-0-134-68960-9</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>"Harvard Mathematics Department 21, 23, 25, or 55?". Archived from the original on October 19, 2017. Retrieved December 9, 2018. <a href="https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html" target="_blank">https://web.archive.org/web/20171019122136/http://www.math.harvard.edu/pamphlets/freshmenguide.html</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Williams, Sam (2002). Free as in Freedom: Richard Stallman's Crusade for Free Software. O'Reilly. p. 41. ISBN 0-596-00287-4. <a href="0-596-00287-4" target="_blank">0-596-00287-4</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Yefremova, Anastasia (May 5, 2022). "Demystifying Math 55". Department of Mathematics, Harvard University. Archived from the original on August 8, 2022. Retrieved August 25, 2022. <a href="https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/" target="_blank">https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Yefremova, Anastasia (May 5, 2022). "Demystifying Math 55". Department of Mathematics, Harvard University. Archived from the original on August 8, 2022. Retrieved August 25, 2022. <a href="https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/" target="_blank">https://web.archive.org/web/20220808194309/https://www.math.harvard.edu/demystifying-math-55/</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li> <li id="fn:38"><p>Woit, Peter (May 25, 2022). "This and That". Not Even Wrong. Retrieved December 27, 2022. <a href="https://www.math.columbia.edu/~woit/wordpress/?p=12898" target="_blank">https://www.math.columbia.edu/~woit/wordpress/?p=12898</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li> <li id="fn:39"><p>Robinson, Evan T.R. (June 2, 2009). "Class of 1984: Lisa Randall". The Harvard Crimson. Retrieved December 9, 2018. 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. <a href="https://www.thecrimson.com/article/2009/6/2/class-of-1984-lisa-randall-as/" target="_blank">https://www.thecrimson.com/article/2009/6/2/class-of-1984-lisa-randall-as/</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li> <li id="fn:40"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li> <li id="fn:41"><p>Bhayani, Paras D. (June 4, 2007). "Andrei Shleifer and J. Bradford DeLong". The Harvard Crimson. Retrieved December 9, 2018. "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.. <a href="https://www.thecrimson.com/article/2007/6/4/andrei-shleifer-and-j-bradford-delong/" target="_blank">https://www.thecrimson.com/article/2007/6/4/andrei-shleifer-and-j-bradford-delong/</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li> <li id="fn:42"><p>Manes, Stephen; Andrews, 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. <a href="0-385-42075-7" target="_blank">0-385-42075-7</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li> <li id="fn:43"><p>Grossman, Lev (February 29, 2004). "10 Questions For Bill Gates". Time. Archived from the original on September 12, 2012. Retrieved August 25, 2022. <a href="https://archive.today/20120912075312/http://www.time.com/time/magazine/article/0,9171,1101040308-596122,00.html" target="_blank">https://archive.today/20120912075312/http://www.time.com/time/magazine/article/0,9171,1101040308-596122,00.html</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></p></li> <li id="fn:44"><p>Williams, Sam (2002). Free as in Freedom: Richard Stallman's Crusade for Free Software. O'Reilly. p. 41. ISBN 0-596-00287-4. <a href="0-596-00287-4" target="_blank">0-596-00287-4</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></p></li> <li id="fn:45"><p>@AlJean (28 January 2019). ".@TheSimpsons I actually took this class" (Tweet) – via Twitter. <a href="https://x.com/AlJean/status/1090017465751367680" target="_blank">https://x.com/AlJean/status/1090017465751367680</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></p></li> <li id="fn:46"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></p></li> <li id="fn:47"><p>Natanson, Hannah (October 20, 2017). "'A Sort of Everyday Struggle'". Harvard Crimson. Retrieved April 17, 2023. <a href="https://www.thecrimson.com/article/2017/10/20/everyday-struggle-women-math/" target="_blank">https://www.thecrimson.com/article/2017/10/20/everyday-struggle-women-math/</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></p></li> <li id="fn:48"><p>Chen, Susan A. (October 20, 1994). "In Math Department, It's Mostly Male". The Harvard Crimson. Retrieved December 9, 2018. <a href="https://www.thecrimson.com/article/1994/10/20/in-math-department-its-mostly-male/" target="_blank">https://www.thecrimson.com/article/1994/10/20/in-math-department-its-mostly-male/</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></p></li> <li id="fn:49"><p>Ury, Logan R. (December 6, 2006). "Burden of Proof". Harvard Crimson. Archived from the original on December 7, 2007. Retrieved January 17, 2023. <a href="/wiki/Logan_Ury" target="_blank">/wiki/Logan_Ury</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></p></li> <li id="fn:50"><p>Gubler, Matthew Gray (2015-04-22), Mr. Scratch, Criminal Minds, Joe Mantegna, Shemar Moore, Matthew Gray Gubler, retrieved 2024-05-23 <a href="https://www.imdb.com/title/tt4524996/" target="_blank">https://www.imdb.com/title/tt4524996/</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></p></li> <li id="fn:51"><p>Elkies, Noam D. "Lecture notes for Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2002)". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~elkies/M55a.02/index.html" target="_blank">http://www.math.harvard.edu/~elkies/M55a.02/index.html</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></p></li> <li id="fn:52"><p>Elkies, Noam D. "Lecture notes, etc., for Math 55b: Honors Advanced Calculus and Linear Algebra (Spring 200[2-]3)". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~elkies/M55b.02/index.html" target="_blank">http://www.math.harvard.edu/~elkies/M55b.02/index.html</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></p></li> <li id="fn:53"><p>Elkies, Noam D. "Lecture notes for Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2005)". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~elkies/M55a.05/index.html" target="_blank">http://www.math.harvard.edu/~elkies/M55a.05/index.html</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></p></li> <li id="fn:54"><p>McMullen, Curtis T. "Math 55a: Honors Abstract Algebra". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/index.html" target="_blank">http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/index.html</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></p></li> <li id="fn:55"><p>McMullen, Curtis T. "Math 55b: Honors Real and Complex Analysis". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/index.html" target="_blank">http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/index.html</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></p></li> <li id="fn:56"><p>Elkies, Noam D. "Lecture notes for Math 55a: Honors Abstract Algebra (Fall 2010)". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~elkies/M55a.10/index.html" target="_blank">http://www.math.harvard.edu/~elkies/M55a.10/index.html</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></p></li> <li id="fn:57"><p>Elkies, Noam D. "Lecture notes, etc., for Math 55b: Honors Real and Complex Analysis (Spring [2010-]2011)". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~elkies/M55b.10/index.html" target="_blank">http://www.math.harvard.edu/~elkies/M55b.10/index.html</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></p></li> <li id="fn:58"><p>Elkies, Noam D. "Lecture notes for Math 55a: Honors Abstract Algebra (Fall 2016)". Retrieved December 9, 2018. <a href="http://www.math.harvard.edu/~elkies/M55a.16/index.html" target="_blank">http://www.math.harvard.edu/~elkies/M55a.16/index.html</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></p></li> <li id="fn:59"><p>Elkies, Noam D. "Lecture notes, etc., for Math 55b: Honors Real and Complex Analysis (Spring [2016-]2017)". Retrieved 2022-02-21. <a href="https://people.math.harvard.edu/~elkies/M55b.16/index.html" target="_blank">https://people.math.harvard.edu/~elkies/M55b.16/index.html</a> <a href="#fnref:59" class="footnote-back-ref">↩</a></p></li> <li id="fn:60"><p>Auroux, Denis. "Math 55A Course Syllabus (Fall 2020)". Retrieved August 3, 2020. <a href="https://canvas.harvard.edu/courses/75936/assignments/syllabus" target="_blank">https://canvas.harvard.edu/courses/75936/assignments/syllabus</a> <a href="#fnref:60" class="footnote-back-ref">↩</a></p></li> <li id="fn:61"><p>Auroux, Denis. "Math 55B Course Syllabus (Spring 2021)". Retrieved March 2, 2020. <a href="https://canvas.harvard.edu/courses/79090/assignments/syllabus" target="_blank">https://canvas.harvard.edu/courses/79090/assignments/syllabus</a> <a href="#fnref:61" class="footnote-back-ref">↩</a></p></li> </ol>