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Home , David Mumford, David Vogan, Marston Morse, Norman Levinson

Winter 2013-2014 Volume 8 Institute of Technology 1ntegral n e w s f r o m t h e d e pa rt m e n t at m i t

Building E17/18. Our old home was a fourth-floor addition. Their design quickly enshrouded in a cocoon of scaf- beautifully marries a contemporary “glass folding and plastic sheets. Its interior is box” top with the classically styled lower being deconstructed and rebuilt, eventu- building. It solves the space shortage by ally to emerge better than ever. Target providing additional offices and meeting completion date: January, 2016. rooms. The additional cost was a concern, As you can see from the drawing on this but we reached out to key department page, a signature feature of the renovation supporters who appreciated the value of is the addition of a new fourth floor over this approach and contributed additional the south and east sections of the building. funds. For their timely help with this proj- This addition wasn’t in the original plan. ect we are deeply grateful to Tom Leigh- Here’s how it came to be. ton, Art Samberg, and David desJardins as well as to Jim and Marilyn Simons, Bob One of the renovation goals was to allevi- Dear Friends, and Lisa Reitano, Ted Kelly, Alex Morcos, ate our severe space shortage. During the and other friends for major gifts. I know, Integral is later than usual. So summer of 2012, extensive planning and much has been happening, we haven’t had design suggested a solution to this prob- Other Business time to stop and tell till now. Read on . . . lem: split the first floor horizontally into Dan Spielman and Ben Green are giving two floors and convert a few classrooms to Our extraordinary faculty are recognized this year’s Simons Lectures. William Law- offices—voilà, the space shortage is solved. with marvelous honors every year, and son from the Yale School of Public Health The heavily-used classrooms would be this year I’d like to call out two that make joined the department as our new Ad- missed, but overall this plan seemed good. us especially proud: Jacob Fox received ministrative Officer. Finally, Marc Kastner Later that fall, our Facilities department the first-ever Packard Award given to a stepped down from his position as Dean wisely commissioned a mockup of the member of our department while at MIT. of Science to become the next Director of split-level space to see how two floors with Mike Artin received the Wolf Prize for the Office of Science at the DOE, pend- ceilings 7' 6" high would compare with the his achievements in algebraic , a ing Senate confirmation. I’ve assumed his current magnificent 16-foot ceiling. truly monumental recognition. The mockup quickly clarified that In other major faculty news, three new even though this plan would gener- assistant professors have joined the de- ate lots of additional office space, no partment: Jörn Dunkel, physical applied one would want to be in those of- mathematics; Ankur Moitra, theoretical fices. We needed a different plan. computer science; and Charles Smart, Roll back the clock several months. probability theory. Steve Kleiman retired John Bush and others on the faculty in January, 2014. Following in his father’s were pressing me to investigate footsteps, Steve came to MIT in 1958 as building a roof deck, a concept that an undergraduate in EE, but then switched appealed to us, but not to those in to mathematics. He graduated with an SB the administration who would be in 1961, studying with Arthur Mattuck as responsible for that outdoor space. his academic advisor. Steve got his PhD Our facilities people were unen- under and thusiastic about pursuing the roof at Harvard in 1965 and joined our depart- deck idea. We argued for further study: duties as interim Dean, in addition to my ment in 1969. He has been active in alge- our renovation would be the template responsibilities as Head of Mathematics. braic geometry and commutative algebra for other main renovations, so this If I stay on as Dean next year, this will be since then, supervising 22 PhDs. Steve decision should be made only after care- my last letter to you as Head. Stay tuned. produced the MIT Undergraduate Journal ful consideration. Facilities Head Dick of Mathematics for eleven years. Have a good year! Amster agreed, and got administration Building 2 Renovation approval to study how we might use the The Building 2 renovation is underway! roof for a deck or other construction. The Mathematics Department moved Our master designers at Ann Beha Ar- Michael Sipser over the summer to temporary digs in chitects then produced this drawing for Department Head 2

New Faculty

Ankur Moitra, Assistant Professor of Jörn Dunkel, Assistant Professor of Charles Smart, Assistant Professor Applied Mathematics in theoretical Applied Mathematics, comes to MIT of Mathematics, came to MIT as a computer science, received his PhD at from postdoctoral appointments at the CLE Moore Instructor in 2011. Smart MIT under Tom Leighton in 2011. Ankur Universities of Cambridge and Oxford. is an analyst who works primarily on has held joint postdoctoral appointments He received his PhD in statistical physics nonlinear PDEs arising in probabilistic at Princeton and IAS. His research under Peter Hänggi from the University settings, often as a scaling limit of a concentrates on algorithmic design of Augsburg in 2008. His dissertation ad- discrete stochastic process or game. He across a wide variety of areas, including dressed unresolved questions linking the completed his PhD at UC Berkeley under statistical inference, optimization, and laws of special relativity to thermodynam- Leo Harrington in 2010, followed by a learning theory. ics. His current program involves devel- year at Courant as an NSF fellow. oping models and mathematical tools for understanding biological phenomena.

Faculty Recognitions

Victor Kac and were elected members of the awards. Victor Kac gave the Hadamard Lecture Series at the National Academy of Sciences. Larry Guth was awarded the IHÉS, a series of eight lectures titled, “Algebraic structures Salem Prize for outstanding contributions in analysis. Larry arising in physics and applications.” also gave the 2013 Lectures at the IAS, titled Lie Wang was promoted to Associate Professor. “The codimension barrier in incidence geometry.”Michel Goemans was selected to be a SIAM fellow, and also received Research Staff Award the 2012 Farkas Prize of the INFORMS Optimization Society. Chelsea Walton, CLE Moore Instructor and NSF postdoctoral Gigliola Staffilani and Tom Leighton were elected fellows of fellow, received the School of Science Infinite Kilometer Award the Massachusetts Academy of Sciences. David Jerison for her work as Coordinator of the PRIMES Circle Program. received the 2012 Stefan Bergman Prize of the AMS with collaborator Jack Lee. The AMS named 29 of our faculty to Staff Distinctions their 2013 Class of Fellows. Bonnie Berger received the Cesar Duarte received the 2013 MIT Excellence Award in Alumni Achievement Award from Brandeis. the category of Innovative Solutions for creating many of the was given the Doctor Honoris Causa from Aalborg University department’s design projects and playing a major role in plan- in Denmark. Jonathan Kelner received the 2012 School of ning and coordinating our move to temporary space in E17/18. Science Teaching Prize for Undergraduate Education. Jacob Debbie Bower, Shirley Entzminger, Susan Ferguson, and Fox, Sug Woo Shin, Charles Smart, and Jared Speck each Avisha Lalla each received the School of Science Infinite Mile received a Sloan Research Fellowship. Jacob Fox, Laurent Award, for going “Above and Beyond” in their contributions to Demanet, and Gonçalo Tabuada received NSF CAREER the Mathematics Department. 3

Mike Artin honored with Wolf Prize Jacob Fox receives the Packard

Mike Artin received Jacob Fox received the 2013 the 2013 Wolf Prize Packard Fellowship in Science in Mathematics, and Engineering, the first math presented by Israeli faculty member to do so while President Shimon at MIT. Jacob’s program is at the Peres at the Knesset. interface between combinatorics “ is one and computer science, geometry, of the main architects analysis, and . Jacob of modern algebraic works on fundamental problems geometry,” the on structure and randomness in citation begins. “His fundamental contributions encompass extremal combinatorics, especially a bewildering number of areas in this field.” The citation lists involving the Szemerédi regularity numerous contributions, such as the development of the lemma and Ramsey theory. Jacob also received a Sloan Fellow- theory of étale cohomology with and ship and an NSF CAREER award. defining étale homotopy with ; advances to the theory of moduli and modern intersection theory, and seminal work on the theory of surface singularities and in deformation Clark Barwick and Laurent Demanet theory. According to Department Head Mike Sipser, Mike is selected for Career Development an extraordinary , a legendary teacher, and a wonderful colleague. We’re proud of him! Assistant Professorships

Laurent Demanet was awarded the Class of 1954 Career Development Assistant Professorship. Laurent works in scientific computing, concentrating on problems in Toby Colding awarded wave-based imaging and inverse problems, with applications to Cecil and Ida Green Professorship seismology, medical imaging, and synthetic aperture radar. He Toby Colding was se- maintains a robust research group of UROPs, graduate lected by the Provost students, and postdocs and recently received an NSF for the Cecil and Ida CAREER award. His PhD is from Caltech in 2006. He was Green Distinguished a Szegö Assistant Professor at Stanford before joining our Professorship, effec- faculty in 2009. tive September 2013. Clark Barwick was selected He has also held the for the Cecil and Ida Green Norman Levinson Career Development Assistant Professorship of Math- Professorship. Clark is an algebraic ematics. Toby is one topologist whose program seeks of the world’s leading to connect K-theory to homotopy differential geometers. theory, higher category theory, Early in his career he studied the geometric structure of and . Among Riemannian manifolds, proving a number of long-standing his projects, he worked with Dan conjectures, and developing with a regularity Kan on relative categories before Dan’s passing this year theory for spaces with a lower bound on the Ricci curvature. (see p. 8). Clark joined our faculty in 2011. He received Working with long-time collaborator Bill Minicozzi (who his PhD from U Penn in 2005, followed by postdoctoral joined our faculty in 2012), Toby has produced groundbreak- appointments at Göttingen, Oslo, IAS, and was a Benjamin Peirce Assistant Professor at Harvard before ing work on minimal surfaces and the mean curvature flow. coming to MIT in 2010. For their series of papers on minimal surfaces, Toby and Bill jointly received the 2010 Veblen Prize in geometry. 4

Morgan Stanley supports mathematics education The Mathematics Department their application with lectures by A field trip to New York allowed has expanded its course offerings into practitioners. students to visit Morgan Stanley financial mathematics with the new Morgan Stanley has also contributed offices, tour the trading floors, and see subject, “Applications of Mathematics to the mathematics department how their lecturers work. in the Financial Industry.” Offered in other ways, supporting some of first as a pilot in fall 2012 and then our student research programs and For information on making in an expanded form in fall 2013, the faculty. “It is extremely important a gift to the Mathematics course was organized and taught by to have a solid mathematical Department, please contact MIT faculty and instructors Pavel education in order to successfully Director of Development for Etingof, Scott Sheffield, Lie Wang, navigate modern financial markets, Mathematics, Erin McGrath, Peter Kempthorne, Choongbum Lee, with their increasingly complex at [email protected] or and MIT alumni Jake Xia (formerly of products and sophisticated analytical 617-452-2807. Morgan Stanley) and Vasily Strela of tools,” Morgan Stanley’s Dr. Strela Morgan Stanley. observed. “I hope our class gives Graduate and undergraduate students a good overview of what kind of had the opportunity to learn the deep mathematical machinery is needed mathematical tools that underlie in finance and how interesting and modern finance theory and to see diverse the problems can be.”

Putnam triumphs continue The MIT team placed second and MIT undergraduates continued record par- ticipation and individual rankings in the 2012 William Lowell Putnam Math- ematical Competition. Team members Benjamin Gunby, Brian Hamrick, and Charles and Holly Housman Teaching Awardees: Nan Li, Ailsa Keating, and John Lesieutre, flanked by then–Associate Head Haynes Miller and Holly and Charles Housman Jonathan Schneider were mentored by Henry Cohn and Abhinav Kumar, with 2013 Student Awards 34/84 top individual scorers: Putnam Fellows: 3/5 Benjamin Gunby, Graduate students Nan Li, Ailsa Keating, and John Lesieutre received the Mitchell Lee, and Zipei Nie Charles and Holly Housman Award for excellence in undergraduate teach- Next twenty: 9/20 ing. Yin Tat Lee and Eric Marberg received the Charles W. and Jennifer C. Joshua Alman, Whan Ghang, Holden Johnson Prize for an outstanding paper accepted for publication. Undergradu- Lee, Sung Gi Park, Jeffrey Shen, Ka Yu ate Holden Lee ’13 received the Jon A. Bucsela Prize in Mathematics for dis- Tam, Szu-Po Wang, Tianyou Zhou, and tinguished scholastic achievement, professional promise, and enthusiasm for Alex Zhu mathematics. Holden also received a Gates Cambridge Scholarship. Yangzhou Honorable Mentions: 22/59 Hu ’13 received an Honorable Mention in the Alice T. Schafer Prize for excel- Robi Bhattacharjee, Justin Brereton, Lu- lence in mathematics by an undergraduate woman, given by the Association cas Camelo Sa, Kevin Chen, Alexander for Women in Mathematics. Fan Wei ’12 received the 2013 Frank and Bren- Cole, Michael Cohen, Zheng Fan, Brian Hamrick, Jiaoyang Huang, Hyun Sub nie Morgan Prize for outstanding research by an undergraduate, and Jon Hwang, Kuan-Yu Lin, Eric Mannes, Ofir ’13 received an Honorable Mention. ’13 was selected Schneider John Mikhael Nachum, Jonathan Schneider, Brandon for a Rhodes Scholarship. Kirin Sinha ’14, a double math and EECS major, Tran, Mark Velednitsky, Anderson received a Marshall Scholarship. Noam Angrist ’13, a double mathematics Wang, Anthony Wang, Michael Wu, and economics major, received a Fulbright Scholarship to work in Botswana. George Xing, Kerry Xing, and Dai Yang. 5

From Associate Head Gigliola Staffilani

workshop that prepares our graduate to include more advanced concepts that students for their future as teachers of may better prepare our students for mathematics. Haynes will be a difficult their future studies at MIT. The number act to follow! of math majors has also increased sub- Responsibility for making sure that our stantially, and they are quite a diverse department’s academic component group, often double majoring. continues to run smoothly now falls We believe it is time to take a good to me. I am also embarking on a major look at these forces acting on our ma- review of our curriculum and teaching jor, and to make sensible and much- This last July I became the new resources. Several new professors have needed changes. Working with Mike Associate Head, the person responsi- recently joined our faculty; we feel that Sipser, we have created a Committee ble for the academic aspects of our de- their expertise should be better repre- on Curriculum and Resources that will partment. This position, created three sented in the classes we offer to both work through the spring semester and years ago, was first filled by Haynes graduates and undergraduates. The new write a report with recommendations Miller. Haynes was named a MacVicar faculty members may want to intro- to the head of the department before Faculty Fellow and received several duce new topics, merge some existing the summer. grants in recognition of his deep in- courses, change the syllabus of others, Finally, the class, “Applications volvement with teaching and course or propose interdisciplinary courses. of Mathematics in the Financial development. He is still working on The students have changed as well. The Industry,” taught for the past two years several projects: the future develop- mathematical sophistication of fresh- as a special class, is now ready to take ment of courses for MITx, revision of men arriving at MIT is much higher its place in our Applied Mathematics the 18.03 curriculum (helped by David than in even the recent past, as seen in degree. The MIT Committee on Jerison and 2010 math PhD Jennifer the surge of students going directly to Curricula has recently approved this French), supervision of the new ver- multivariable calculus. We see an op- class as a standard subject to be of- sion of 18.05, and the micro-teaching portunity to revise our calculus classes fered on a regular basis.

Martin Luther King Visiting Faculty

Erika Camacho and Stephen Wirkus have come with their family as MLK Visiting Faculty for the year. Erika and Stephen are applied from Arizona State University and have been active nationally serving on both outreach and research boards, committees, and conferences to promote student and faculty diversity in mathematics. Erika’s interest in mathematics was inspired by her teacher Jaime Escalante, subject of the film Stand and Deliver. She works on the nonlinear modeling of biological phenomena with a focus on retinal degeneration, as well as on math- ematical sociology. Stephen concentrates on biological systems and patch pop- ulation dynamics, and is finishing the second edition of his textbook on ODEs. They are here to build research collaborations and work with faculty to increase our underrepresented and women ma- jors, identify potential PhD candidates, and build a network of outside scholars to explore possible partnering leads. 6

Our Temporary Home Years from now we will look back on this period as a time of transition, with retirements and new hires complementing our nomadic existence. Here, for the record, are some pictures of our temporary quarters, E17/18, as we await the completion of a more spacious and beautiful Building 2. We’ve included some architectural drawings of the new construction to give a sense of what we have to look forward to! Department Retreat 2013

Headquarters hallway

Common room

Academic office

Math enclave Food for thought “The helm”

Our Future Home

Graduate student tower Hard Math Café 7

Department Retreat 2013

Headquarters hallway

Many thanks to our graduate students of the Retreat Committee for planning and coordinating the 2013 retreat: Chief Organizer John Binder; Hans Liu – transportation, bus sign-up, rooming, venue search; Alisa Knizel – activities coordinator, BBQ planner, venue search; Yasha Berchenko-Kogan – activities coordinator; Sean Simmons – advertising, venue search; Padma Srinivasan – venue search; Michael Donovan, Dana Mendelson, and Saul Glasman – food, drinks and supplies. Activity planners: Ben Elias – board games, Pavel Etingof, David Rolnick, and Susan Ruff – hikes, Alisa Knizel – trivia games, Carlos Sauer – soccer, Dan Thompson – basketball. 2013 Doctorates

Elette Boyle, “Secure Multi-Party Protocols Under a Modern Lens,” under Aramco Research Center in Cambridge. Shafi Goldwasser (EECS). Elette is now a postdoc at Technion. Po-Ru Loh, “Algorithms for Genomics and Genetics: Compression-Acceler- Jiawei Chiu, “Matrix Probing, Skeleton Decomposition, Sparse Fourier Trans- ated Search and Admixture Analysis,” under Bonnie Berger. Po-Ru is now a form,” under Laurent Demanet. Jiawei is now at Google. postdoc at the Harvard School of Public Health. Dustin Clausen, “Arithmetic Duality in Algebraic K-Theory,” under Jacob Eric Marberg, “Coxeter Systems, Multiplicity Free Representations, and Lurie (Harvard). Dustin is now a postdoc at the University of Copenhagen. Twisted Kazhdan-Lusztig Theory,” under David Vogan. Eric is now a postdoc Giorgia Fortuna, “The Beilinson-Bernstein Localization Theorem for the at Stanford. Affine Grassmannian,” under Dennis Gaitsgory (Harvard). Giorgia is now a Gregory Minton, “Computer-Assisted Proofs in Geometry and Physics,” un- postdoc at ETH Zürich. der Abhinav Kumar. Gregory is now a postdoc at Microsoft Research. Höskuldur Halldorsson, “Self-Similar Solutions to the Mean Curvature Flow Jan Moláček, “Bouncing and Walking Droplets: Towards a Hydrodynamic in Euclidean and Minkowski Space,” under Toby Colding. Höskuldur is now at Pilot-Wave Theory,” under John Bush. Jan is now a postdoc at the Max Planck Jane Street Financial. Institute in Göttingen. Rune Haugseng, “Weakly Enriched Higher Categories,” under Haynes Luis Pereira, “Goodwillie Calculus and Algebras over a Spectral Operad,” Miller. Rune is now a postdoc at the Max Planck Institute in Bonn. under Mark Behrens. Luis is now a postdoc at the University of Virginia. Geoffroy Horel, “Operads, Modules and Higher Hochschild Cohomology,” Bhairav Singh, “Some Results Related to the Quantum Geometric Langlands under Haynes Miller. Geoffroy is now a postdoc at the University of Münster. Program,” under Roman Bezrukavnikov. Bhairav is an instructor at East Los Tirasan Khandhawit, “Twisted Manolescu-Floer Spectra for Seiberg-Witten Angeles College. Monopoles,” under Tom Mrowka. Tirasan is now a postdoc at the Kavli Insti- Uhi Rinn Suh, “Structure of Classical W-Algebras,” under Victor Kac. Uhi tute at the University of Tokyo. Rinn is now a postdoc at the Seoul National University. Alexander Levin, “Graphs, Matrices and Populations: Linear Algebraic John Ullman, “On the Regular Slice Spectral Sequence,” under Mark Beh- Techniques in Theoretical Computer Science and Population Genetics,” under rens. John is now a postdoc at Stanford. Bonnie Berger. Alexander is now a data analyst at Yelp. Taedong Yun, “Diagrams of Affine Permutations and their Labellings,” under Nan Li, “Combinatorial Aspects of Polytope Slices,” under Richard Stanley. Richard Stanley. Taedong is now at Oracle. Nan is now a research scientist at Raytheon. Yan Zhang, “The Combinatorics of Adinkras,” under Richard Stanley. Yan is Xiangdong Liang, “Modeling of Fluids and Waves with Analytics and Nu- now a postdoc at U.C. Berkeley. merics,” under Steve Johnson. Xiangdong is now a research scientist at the 8

RSI, PRIMES, SPUR: In Remembrance: Chia-Chiao Lin and Daniel Kan High School and The Department lost two of its beloved emeritus faculty in 2013: Undergraduate Research Institute Professor Chia-Chiao Lin, at the age of 96, and Daniel Kan, who was 86. Both had served on the faculty for over 35 years. High school students mentored by our grad- uate students earned multiple awards for their research projects. RSI students Kath- erine Cordwell and Simanta Gautam won first and second place at Intel ISEF 2013; Joshua Brakensiek received the Davidson Fellowship; and Surya Bhupatiraju, Kevin Garbe, and Katherine Cordwell won final- ist awards at Intel STS 2013. PRIMES-Math students Kavish Gandhi and C.C. Lin was born in Beijing in 1916. He Daniel Kan played a key role in establish- Noah Golowich won 2nd place at Siemens received degrees from Tsinghua Univer- ing the foundations for the combinatorial 2013; Rohil Prasad and Jonathan Tidor sity and the University of Toronto be- reinterpretation of topology, or homotopy received 5th place at Siemens 2012; Sahana fore earning a PhD in aeronautics from theory, that led to the integration of topo- Vasudevan won 10th place at Intel STS Caltech in 1944. He joined the MIT fac- logical methods into many mathematical 2013; and William Kuszmaul received 3rd place at Intel STS 2014, the 2014 Glenn T. ulty in 1947. fields. His insights have proven so funda- Seaborg award, and a Davidson Fellowship. C.C. was instrumental in the develop- mental and natural that they have now Three research papers based on PRIMES ment of applied mathematics at MIT and become part of the universal language of projects were published in high-level math- the U.S., serving as our first faculty chair mathematics. He published two highly in- ematics journals. Over 50% of PRIMES of the applied math committee, 1961–66, fluential books with former students and alumni attended MIT. and as President of SIAM, 1973–74. He supervised fifteen PhDs, all at MIT. He reached many more through his unique In 2013 PRIMES-USA, our new section for contributed seminal work to hydrody- seminar in algebraic topology, known to out-of-state students, five juniors completed namics stability and turbulence, and to all as the Kan Seminar. research projects, mentored by our graduate astrophysics. He was appointed Institute students via weekly telecon sessions, with Professor in 1966 and received the Killian Born in Amsterdam in 1927, Dan received two to three visits to MIT per year. PRIMES- Faculty Award in 1981. Following retire- the BSc and MS from the University of USA students Jeffrey Cai, regional finalist ment from MIT in 1987, he returned to Amsterdam and the PhD from Hebrew at Siemens 2013, and Ritesh Ragavender, China in 2002 as Distinguished Profes- University in 1955. He joined our math- Bronze Award winner in the Yau HS Math sor and Honorary Director of the newly ematics faculty in 1959 and retired in Competition, demonstrated the success of established Zhou Pei-Yuan Center for 1993. In 1982 he was elected member of distance mentoring. In 2014 PRIMES-USA Applied Mathematics at Tsinghua Uni- the Royal Netherlands Academy of Arts expanded to thirteen students. versity. He was a member of the National and Sciences. PRIMES Circle, another new initiative in Academy of Sciences, academician of Dan continued working till the end of his 2013, offered a math enrichment curriculum Academia Sinica, foreign member of life meeting with colleagues and contrib- to sophomores and the Chinese Academy, and fellow of the uting to abstract homotopy theory. juniors from local American Academy of Arts and Sciences, photo credit: Jonah Kan urban public high among numerous distinctions. schools. Chelsea Walton served as Is Singer Richard Stanley Program Coor- 90th Birthday Conference 70th Birthday Conference dinator. Working May 2, 2014 June 23–27, 2014 in pairs under Medgine Joseph MIT and Harvard PRIMES Circle student UPCOMING undergraduates, David Vogan Michael Sipser Circle students studied geometry, probability, 60th Birthday Conference 60th Birthday Conference combinatorics, and theory, and made May 19–23, 2014 October 26, 2014 excellent presentations at a mini-conference at MIT. Pavel Etingof continues as Chief Research m i t d e p a r t m e n t o f mathematics Advisor, Slava Gerovitch as Director, and Tanya Khovanova as Head Mentor of both the RSI and PRIMES programs. Department of Mathematics The Hartley Rogers Jr. Prize for the best SPUR paper was shared between under- graduate Fan Zheng and his mentor, gradu- Massachusetts Institute of Technology ate student Chenjie Fan, and undergradu- Building E18, Room 369 Telephone: 617-253-4381 ates Zipei Nie and Anthony Wang and 77 Massachusetts Ave. Fax: 617-253-4358 their mentor, graduate student Ben Yang. Cambridge, MA 02139-4307 Web: math.mit.edu Editor Joel Segel Designer∫ ByteGraphics Photographer Bryce Vickmark Printed on recycled paper by Puritan Capital