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Kummer, Regular Primes, and Fermat's Last Theorem e H a r v a e Harvard College r d C o l l e Mathematics Review g e M a t h e m Vol. 2, No. 2 Fall 2008 a t i c s In this issue: R e v i ALLAN M. FELDMAN and ROBERTO SERRANO e w , Arrow’s Impossibility eorem: Two Simple Single-Profile Versions V o l . 2 YUFEI ZHAO , N Young Tableaux and the Representations of the o . Symmetric Group 2 KEITH CONRAD e Congruent Number Problem A Student Publication of Harvard College Website. Further information about The HCMR can be Sponsorship. Sponsoring The HCMR supports the un- found online at the journal’s website, dergraduate mathematics community and provides valuable high-level education to undergraduates in the field. Sponsors http://www.thehcmr.org/ (1) will be listed in the print edition of The HCMR and on a spe- cial page on the The HCMR’s website, (1). Sponsorship is available at the following levels: Instructions for Authors. All submissions should in- clude the name(s) of the author(s), institutional affiliations (if Sponsor $0 - $99 any), and both postal and e-mail addresses at which the cor- Fellow $100 - $249 responding author may be reached. General questions should Friend $250 - $499 be addressed to Editors-In-Chief Zachary Abel and Ernest E. Contributor $500 - $1,999 Fontes at [email protected]. Donor $2,000 - $4,999 Patron $5,000 - $9,999 Articles. The Harvard College Mathematics Review invites Benefactor $10,000 + the submission of quality expository articles from undergrad- uate students. Articles may highlight any topic in undergrad- Contributors · The Harvard Undergraduate Council · uate mathematics or in related fields, including computer sci- American Mathematical Society · Patrons · The Harvard Uni- ence, physics, applied mathematics, statistics, and mathemat- versity Mathematics Department ical economics. The image on the cover depicts a Young Authors may submit articles electronically, in .pdf, .ps, Cover Image. tableaux with towers over each box to illustrate the defining or .dvi format, to [email protected], or in hard property that values increase (equivalently, heights decrease) copy to in the rightward and downward directions. This issue’s article “Young Tableaux and the Representations of the Symmetric The Harvard College Mathematics Review Group” by Yufei Zhao (p. 33) studies combinatorial properties Student Organization Center at Hilles and applications of Young tableaux. The image was created TM Box # 360 in Asymptote by Graphic Artist Zachary Abel. 59 Shepard Street Cambridge, MA 02138. 1 Submissions should include an abstract and reference list. Fig- 2 3 ures, if used, must be of publication quality. If a paper is 5 accepted, high-resolution scans of hand drawn figures and/or 4 6 scalable digital images (in a format such as .eps) will be re- 7 quired. 8 13 Problems. The HCMR welcomes submissions of original 9 10 problems in all mathematical fields, as well as solutions to 15 11 14 previously proposed problems. 18 Proposers should send problem submissions to Problems12 17 Editor Zachary Abel at [email protected]. 16 20 22 edu or to the address above. A complete solution or a detailed 19 21 sketch of the solution should be included, if known. 24 25 Solutions should be sent to hcmr-solutions@hcs. 23 harvard.edu or to the address above. Solutions should 26 include the problem reference number. All correct solutions 27 28 will be acknowledged in future issues, and the most outstand- ing solutions received will be published. 29 30 Advertising. Print, online, and classified advertisements are available; detailed information regarding rates can be found on The HCMR’s website, (1). Advertising inquiries should be directed to [email protected]. edu, addressed to Business Manager Oluwadara Johnson. c 2007–2009 The Harvard College Mathematics Review Subscriptions. One-year (two issue) subscriptions are Harvard College available, at rates of $10.00 for students, $15.00 for other in- Cambridge, MA 02138 dividuals, and $30.00 for institutions. Subscribers should mail checks for the appropriate amount to The HCMR’s postal ad- The Harvard College Mathematics Review is produced and dress; confirmation e-mails or queries should be directed to edited by a student organization of Harvard College. [email protected]. -2 Contents 0 From the Editors Zachary Abel ’10 and Ernest E. Fontes ’10 3 Student Articles 1 Error-Correcting Codes and Sphere Packings Franc¸ois Greer ’11 and Xiaoqi Zhu ’11 4 2 Kummer, Regular Primes, and Fermat’s Last Theorem Ila Varma, California Institute of Technology ’09 12 3 Securing Your Hair Grant Dasher ’09 25 4 Young Tableaux and the Representations of the Symmetric Group Yufei Zhao, Massachusetts Institute of Technology ’10 33 Faculty Feature Article 5 Arrow’s Impossibility Theorem: Two Simple Single-Profile Versions Prof. Allan M. Feldman, Brown University and 46 Prof. Roberto Serrano, Brown University 6 The Congruent Number Problem Prof. Keith Conrad, University of Connecticut 58 Features 7 Mathematical Minutiae · Quadratic Reciprocity by Group Theory Tim Kunisky, Livingston High School ’10 75 Statistics Corner · Conformal Invariance in the Scaling Limit of 8 Critical Planar Percolation Nike Sun ’09 77 9 Applied Mathematics Corner · DNA Computation and Algorithm Design Shrenik Shah ’09 83 10 My Favorite Problem · An Unconventional Inequality Ameya Velingker ’10 90 11 Problems 95 12 Solutions 97 13 Endpaper · Hunting for Perfect Euler Bricks Prof. Oliver Knill, Harvard University 102 1 -1 Staff Editors-In-Chief Zachary Abel ’10 Ernest E. Fontes ’10 Business Manager Oluwadara Johnson ’10 Articles Editor Features Editor Menyoung Lee ’10 Franc¸ois Greer ’11 Problems Editor Graphic Artist Zachary Abel ’10 Zachary Abel ’10 Issue Production Directors Zachary Abel ’10 Ernest E. Fontes ’10 John Lesieutre ’09 Editor Emeritus Scott D. Kominers ’09 Webmasters Brett Harrison ’10 Sean Li ’09 Board of Reviewers Board of Copy Editors Zachary Abel ’10 Eleanor Birrell ’09 Jeremy Booher ’10 Jeremy Booher ’10 John Casale ’12 Jannis R. Brea ’10 Ernest E. Fontes ’10 John Casale ’12 Jeffrey Kalmus ’12 Franc¸ois Greer ’11 Paul Kominers, MIT ’12 Kelley Harris ’09 Scott D. Kominers ’09 Nathan Kaplan, G1 Menyoung Lee ’10 Paul Kominers, MIT ’12 Sam Lichtenstein ’09 Scott D. Kominers ’09 Daniel Litt ’10 Philip Mocz ’12 Philip Mocz ’12 Menyoung Lee ’10 Aaron Szasz ’12 Daniel Litt ’10 Rachel Zax ’12 Shrenik N. Shah ’09 Xiaoqi Zhu ’11 Aaron Silberstein, G2 Xiaoqi Zhu ’11 Faculty Advisers Professor Benedict H. Gross ’71, Harvard University Professor Peter Kronheimer, Harvard University 2 0 From the Editors Zachary Abel Harvard University ’10 Cambridge, MA 02138 [email protected] Ernest E. Fontes Harvard University ’10 Cambridge, MA 02138 [email protected] This fall has been marked by the first major transition of leadership for The Harvard College Mathematics Review (HCMR). And with this transition, The HCMR has blossomed from an up- start student publication to an established organization. Many of our founding members are now applying to graduate schools and will soon be leaving the journal’s front-line operations. Guided by their precedent, we welcome and look forward to the contributions of the many new members that have joined the organization, with whose help we may continue serving as a resource to the undergraduate mathematical community. By our combined involvement in various roles in The HCMR’s staff including Problems Editor and Issue Production Director, we two have been privileged to witness and help the journal grow into its current state and are honored to guide the organizations for this exciting academic year. Our optimism springs from the tireless ingenuity of our contributing student and faculty authors, the continued devotion of our reviewing and editing staffs, and the indefatigable zeal of our produc- tion directors. These first four issues are a testament to your support, without which the journal could never have come so far. We also owe deep gratitude to Professor Peter Kronheimer and Professor Benedict H. Gross ’71 for their advice and guidance, Professor Clifford H. Taubes for continued encouragement, and the rest of The HCMR’s advisors and sponsors, whose profound contributions have been a foun- dation for the journal’s success. Our executive board owes much to the administrative assistance of Dean Paul J. McLoughlin II, David R. Friedrich and the rest of the staff at the Student Or- ganization Center at Hilles, and to the unceasing, generous support of the Harvard Mathematics Department. Finally, the two of us would like to express our gratitude to Editor Emeritus Scott D. Kominers ’09, who continues to offer invaluable expertise and guidance to The HCMR even though he has stepped down from his position as Editor-In-Chief. Scott was an inspirational leader and founder of The HCMR, and he will always be remembered fondly by our staff, authors, and countless readers who have been touched by his work in the journal. Thank you, Scott, and farewell. Zachary Abel ’10 and Ernest E. Fontes ’10 Editors-In-Chief, The HCMR 3 STUDENT ARTICLE 1 Error-Correcting Codes and Sphere Packings Franc¸ois Greery Harvard University ’11 Cambridge, MA 02138 [email protected] Xiaoqi Zhuz Harvard University ’11 Cambridge, MA 02138 [email protected] Abstract The study of arrangements of non-overlapping spheres in space, known as sphere packings, has given rise to numerous questions such as finding the densest sphere packing and the kissing number problem. Aside from these theoretical considerations, sphere packings is also closely related to the theory of codes. This paper introduces the basic problem of finding efficient error-correcting codes and discusses its geometric interpretation as a sphere packing problem. The paper focuses on certain families of codes with special properties and explores these properties in connection with two famous codes.
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