The Harvard College Mathematics Review Volume 3 Spring 2011 In this issue: CHRISTOPHER POLICASTRO Artin's Conjecture ZHAO CHEN, KEVIN DONAGHUE & ALEXANDER ISAKOV A Novel Dual-Layered Approach to Geographic Profiling in Serial Crimes MICHAEL J. HOPKINS The Three-Legged Theorem MLR 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 will be listed in the print edition of The HCMR and on a spe http://www.thehcmr.org/ fl) 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 Editor-in-Chief Rediet Abebe at hcmr@ hcs . Contributor $500-$1,999 harvard.edu. 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 ■ Jane Street Capital • The Harvard Uni uate mathematics or in related fields, including computer sci versity Mathematics Department ence, physics, applied mathematics, statistics, and mathemat Cover Image. The image on the cover depicts several ical economics. functions, whose common zero locus describes an algebraic Authors may submit articles electronically, in .pdf, .ps, or .dvi format, to [email protected], or in hard variety (which is in this case an elliptic curve). This issue's article "Hilberts Nullstellensatz and Schemes" by Miles Ed copy to wards (p. 17) describes one of the foundational theorems in algebraic geometry, relating an ideal in a polynomial ring to The Harvard College Mathematics Review its corresponding algebraic variety. This image was created in Student Organization Center at Hilles Asymptote by Graphic Artists Eric Larson and Katherine Box # 360 Banks. 59 Shepard Street Cambridge, MA 02138. Submissions should include an abstract and reference list. Fig ures, if used, must be of publication quality. If a paper is accepted, high-resolution scans of hand drawn figures and/or scalable digital images (in a format such as eps) will be re quired. Problems. The HCMR welcomes submissions of original problems in all mathematical fields, as well as solutions to previously proposed problems. Proposers should send problem submissions to Problems Editor Lucia Mocz at hemr-problems@hcs. harvard, edu or to the address above. A complete solution or a detailed sketch of the solution should be included, if known. Solutions should be sent to hcmr-solutions@hcs . harvard.edu or to the address above. Solutions should include the problem reference number. All correct solutions will be acknowledged in future issues, and the most outstand ing solutions received will be published. 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 hemr- advert ise@hcs .harvard, edu, addressed to Business Manager Gerishom Gimaiyo. ©2007-2011 The Harvard College Mathematics Review Subscriptions. One-year (two issue) subscriptions are Harvard College available, at rates of $5.00 for students, $7.50 for other indi Cambridge, MA 02138 viduals, and $15.00 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 Editor Rediet Abebe, Harvard University '13 Student Articles Artin's Conjecture Christopher Policastro, Massachusetts Institute of Technology '11 4 Hilbert's Nullstellensatz and Schemes M i l e s D i l l o n E d w a r d s , I n d i a n a U n i v e r s i t y ' 1 3 1 7 Toronto Spaces Manuel Rivera, Massachusetts Institute of Technology '10 24 An Introduction to Sieve Theory S e t h N e e l , T h e W h e e l e r H i g h S c h o o l ' 1 3 2 8 Faculty Feature Article 5 A Direct Geometric Proof of Gregory's series for \ P r o f . P a u l G . B a m b e r g , H a r v a r d U n i v e r s i t y 3 3 Features 6 Mathematical Minutiae -1 + 1 = 1? K a t r i n a E v t i m o v a , H a r v a r d U n i v e r s i t y ' 1 3 3 6 7 Statistics Corner • Random Walk Model For Dating G r e g Y a n g , H a r v a r d U n i v e r s i t y ' 1 4 3 8 8 Applied Mathematics Corner • A Novel Dual-Layered Approach to Geographic Profiling in Serial Crimes Zhao Chen, Kevin Donoghue, and Alexander Isakov, Harvard University '09 42 9 My Favorite Problem • Sums of Four Squares To n y F e n g a n d L u c i a M o c z , H a r v a r d U n i v e r s i t y ' 1 3 5 3 1 0 P r o b l e m s 5 9 1 1 S o l u t i o n s 6 1 12 Endpaper • The Three-Legged Theorem P r o f . M i c h a e l J . H o p k i n s , H a r v a r d U n i v e r s i t y 6 8 Editor-in-Chief Rediet Abebe'13 Business Manager Gerishom Gimaiyo '13 A r t i c l e s E d i t o r A s s o c i a t e D i r e c t o r F r a n c o i s G r e e r ' 1 1 E r i c L a r s o n ' 1 3 F e a t u r e s E d i t o r A s s i s t a n t A r t i c l e s E d i t o r K a t h e r i n e B a n k s ' 1 2 A k h i l M a t h e w ' 1 4 P r o b l e m s E d i t o r A s s i s t a n t P r o b l e m s E d i t o r s L u c i a M o c z ' 1 3 G e Ya n g ' 1 4 a n d L e v e n t A l p o g e ' 1 4 Issue Production Directors Eric Larson '13 and Katherine Banks '12 Graphic Artists Eric Larson '13 and Katherine Banks' 12 Editors Emeritus Zachary Abel' 10, Scott Kominers '09/AM' 10/PhD' 11, Ernest Fontes ' 10 Webmaster Rediet Abebe'13 B o a r d o f R e v i e w e r s B o a r d o f C o p y E d i t o r s Levent Alpoge ' 14 Rediet Abebe ' 13 A n i r u d h a B a l a s u b r a m a n i a n ' 1 4 J o h n C a s a l e ' 1 2 K a t h e r i n e B a n k s ' 1 2 A s h o k C u t k o s k y ' 1 3 John Casale ' 12 Yale Fan ' 14 A s h o k C u t k o s k y ' 1 3 F r a n c o i s G r e e r ' 1 1 Y a l e F a n ' 1 4 E r i c L a r s o n ' 1 3 F r a n c o i s G r e e r ' 1 1 G e f f o r y L e e ' 1 4 E r i c L a r s o n ' 1 3 A k h i l M a t h e w ' 1 4 G e o f f r e y L e e ' 1 4 L u c i a M o c z ' 1 3 A k h i l M a t h e w ' 1 4 Z a c h a r y Y o u n g ' 1 4 Lucia Mocz'13 Ge Yang'14 Zachary Young '14 Faculty Adviser Professor Peter Kronheimer, Harvard University o From the Editor Rediet Abebe Harvard University '13 Cambridge, MA 02138 [email protected] When I took on the role of Editor-in-Chief of the Harvard College Math Review in May 2010,1 was not sure how HCMR was going to turn out, or if we were even going to have an HCMR by the end of the school year.