Foreword It is a pleasure to bring you our first issue of m3, the Math Majors Magazine. Over twenty m 3 years ago, the UMA used to publish a magazine also called m3, although back then, the initials stood for “math majors monthly”. After a couple of issues, it mysteriously disappeared, with the reasons still under investigation. Now, after over twenty years, it has finally come back to life in a different form! We put together this magazine in an effort to entertain, inform, and connect, for there are so many enjoyable, fulfilling, and educational aspects of life as a math major at MIT, and there are many things we can learn from each other. We have invested a lot of time and math majors magazine effort bringing together this magazine, which has been an arduous task. Nonetheless, we plan on producing more issues in the future with hopefully even more interesting content. Volume 1, Issue 1 For a magazine dealing with what MIT undergraduates are doing in math, we urge you to check out MURJ, the MIT Undergraduate Research Journal. For a magazine dealing with December 2008 interesting math research, we urge you to check out the American Mathematical Monthly. We hope that you find the contents entertaining and useful, because we certainly did. Please do not hesitate to send us feedback or articles that you have written! Sincerely, The m3 Team Brought to you by the Undergraduate Math Association Hyun Soo Kim, Editor In Chief Daniela Çako, Managing Editor Basant Sagar, Managing Editor December 4, 2008 Table of Contents Current Events What’s New The MIT Math Department’s website www‐math.mit.edu now Current Events 3 has a snazzy new design. Meet the UMA 4 What We Did This Term 7 Fun with Math Tried and True 10 Compiled by Hyun Soo Kim Random Tidbits 11 Compiled by Daniela Çako Math Fail 14 Artinian Rings 15 Compiled by Basant Sagar Zero – A Poem 16 Written by Maria Monks Research Experiences My Experiences at Duluth 19 Written by Maria Monks MIT’s team for this year’s Putnam comprises Qingchun Ren ’10, Interview with Doris Dobi 21 Interview by Daniela Çako Xuancheng Shao ’09, and Yufei Zhao ’10. Grad Life 23 Interview by Daniela Çako Maria Monks, a junior majoring in math, has been chosen to Who’s Who at MIT receive the Alice T. Shafer Prize for Undergraduate Women in Interview with Professor Stanley 28 Interview by Hyun Soo Kim Mathematics. Doris Dobi, a senior, will receive an honorable Interview with Professor Artin 34 Interview by Hyun Soo Kim mention. Excerpt: What is algebra, and why is it important? 40 Compiled by Hyun Soo Kim 2 3 Meet the UMA Name Daniela Çako, ’09 Name Hyun Soo Kim, ’09 Title Treasurer – the money woman, Special Projects Co‐ Chair Title President Majors 18C, Minor Applied International Studies Majors 18, 6‐3 Math Interests Counting – it’s actually quite hard Math Interests Algebra, Topology Other Interests Traveling, learning foreign languages, Other Interests Starcraft always getting involved in more fun and interesting things than I can handle Favorite Number 17 Favorite Number I can’t say, all numbers are special in Name Cinjon Resnick, ’10 their own way… Title Vice President Name Brayden Ware, ’11 Majors 18 Title Publicity Co‐Chair Math Interests Folding and other interesting properties Majors 18, 8 (in that order) of pita bread Math Interests Geometry, Algebra, Mathematical Physics, Other Interests Unlocking value and the interplay between them Favorite Number Number 9 Other Interests Surviving MIT, Soccer, Cooking, Sigma Phi Epsilon Name Maria Monks, ’10 Favorite Number i*8 Title Secretary Majors 18, 8 Math Interests Combinatorics, Chaos Theory Other Interests Cross‐country, Piano Favorite Number 4 4 5 Name Emily Berger, ’11 What We Did This Term Title Publicity Chair September 16, 2008 UMA Talk 2‐102, 5pm Majors 18 Tuesday Prof. Ben Brubaker Math Interests Algebra, Number Theory, and Probability “Discovering a new L‐function” Other Interests Hogs September 23, 2008 Putnam Talk 2‐102, 5pm Favorite Number 2 and 17 Tuesday Daniel Kane Name Yufei Zhao, ’10 “Linear Algebra” September 24, 2008 Start‐of‐Term BBQ EC Courtyard, 6pm Title Webmaster Wednesday Majors 18, 6‐3 September 30, 2008 UMA Talk 2‐102, 5pm Math Interests Theoretical. Leaning towards algebra and combinatorics. Tuesday Prof. Bjorn Poonen Other Interests Huh? There is time for something other “Zero and the Empty Set” than math? October 14, 2008 Putnam Talk 2‐102, 5pm Favorite Number Not telling… Tuesday Thomas Belulovich Name Basant Sagar, ’11 October 21, 2008 UMA Talk 2‐102, 5pm Title Special Projects Co‐Chair Tuesday Prof. Manolis Kelis Majors 16, 18 “Interpreting the Human Genome” Math Interests Combinatorics, Topology (still getting a October 23, 2008 Liberty Mutual Presentation 4‐153, 5pm grip) Thursday Other Interests Space exploration, Music October 30, 2008 UMA Talk 2‐102, 5pm Favorite Number Anything prime! Thursday Prof. Scott Aaronson 6 7 “The Past and Future of Closed Timelike Curves” Vote Yes for Proposition 18. November 4, 2008 Putnam Talk 2‐102, 5pm Tuesday Rishi Gupta November 20, 2008 UMA Talk 2‐102, 5pm Thursday Prof. Ken Ono “Coins of Ramanujan and Selberg” November 25, 2008 UMA Talk 2‐102, 5pm Tuesday Todd Kemp “Clifford Combed a Coconut” December 4, 2008 Putnam Talk 2‐102, 6pm Thursday Daniel Kane “Generating Functions” December 9, 2008 End‐of‐Term Dinner 9th Floor Green Building, Tuesday 6pm Expect Great Things Next Term • Kick‐off BBQ • e Day Be proud of your choice. • Valentine’s Day Buy our new t-shirt for only $10. • Pi Day Available in black, blue, and green. • More issues of m3 Please direct all inquiries to uma‐email@example.com. 8 Tried and True Random Tidbits 1. What did one mathematician say to another mathematician when he found the On Quotes Christmas tree he wanted to buy? “I use quotes reluctantly, quotes are an asinine way of denying responsibility for 2. An infinite crowd of mathematicians enters a bar. The first one orders a pint, the what you write; it’s like I didn’t write this, don’t blame me.” – Prof. Mattuck second one a half pint, the third one a quarter pint. “I understand,” says the The Power of Colored Chalk in Mathematics bartender, and pours two pints. “Purple for equations, green for – I don't know what. Orange for rules.” 3. New York (CNN). At John F. Kennedy International Airport today, a Caucasian male – Prof. Mattuck (later discovered to be a high school mathematics teacher) was arrested trying to “Real miracles deserve green chalk.” – Prof. Toomre board a flight while in possession of a compass, a protractor and a graphical calculator. “This must be in pink! Where's the pink? Or whatever the color is! It's called color‐ According to law enforcement officials, he is believed to have ties to the Al‐Gebra coding. The ideas in this course will be color‐coded for the second half of the course, network. He will be charged with carrying weapons of math instruction. since I have a brand‐new box of chalk!” – Prof. Mattuck “Now we're gonna pass the law of conservation of pink!” – Prof. Mattuck “I think I'd better have some colored chalk in my hand; otherwise, everything will be 4. Theorem: Every positive integer is interesting. in white and nothing will be intelligible.” – Prof. Mattuck Proof: Assume that there is an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. But being the smallest uninteresting positive Mattuck’s Words of Wisdom integer is interesting by itself, a contradiction. “Playing badly can be overcome by playing for a long time.” – when talking about the skills for playing an instrument “You've reduced a second order equation you can't solve to a first order equation you can't solve. And that's called progress.” “Giving mysterious values names is an acceptable mathematical procedure.” “The more I say, the less you’ll understand.” Answer to 1: isometry “There’s no way of learning that except by brute force.” 10 11 “All theorems have three names. A French name, a German name, and a Russian name, Inner Conviction/Mysticism: “My program is perfect. I know this to be true.” each nationality having claimed to discover it first. Once in a while there’s an English name too, but it’s always Newton.” “I don’t see why not...”: Claim something is true and then shift the burden of proof to anyone who disagrees with you. Other Professors are funny too… “Cogito ergo sum”: Proof by reasoning about undefined terms. This Latin quote translates as “I think, therefore I am.” It comes from the beginning of a famous essay by the “This is a chapter that can be extremely difficult, but if you see how simple it is, it 17th century mathematician/philosopher Rene ́ Descartes. It may be one of the most famous can be extremely easy.” – Prof. Hartley Rogers, Jr. quotes in the world. Deducing your existence from the fact that you’re thinking about your existence sounds like a pretty cool starting axiom. But it ain’t Math. In fact, Descartes goes “Logic: a frightening word for too many people in this lecture.” – Prof. Rogers on shortly to conclude that there is an infinitely beneficent God, so go figure. “Intuitively obvious, even to the most casual observer.” – Prof. Kleppner “Differentiation is easy, even monkeys can do it. But integration – ah, integration is an art. It's like black magic.” – Prof. Kedlaya, in 18.014 “Well class, we probably won't get to the end of infinity by the end of this lecture.” – Prof. Kelner, in 18.440 What is a Proof? Taken from OCW: http://ocw.mit.edu/NR/rdonlyres/Electrical‐Engineering‐and‐Computer‐Science/6‐ Compiled by Daniela Çako 042JMathematics‐for‐Computer‐ScienceFall2002/D7C42878‐108E‐4C45‐902A‐61783145EC0A/0/ln1.pdf A proof is a method of ascertaining truth. There are many ways to do this: Jury Trial Truth: It is ascertained by twelve people selected at random. Word of God Truth: It is ascertained by communication with God, perhaps via a third party. Word of Boss Truth: It is ascertained from someone with whom it is unwise to disagree. Experimental Science Truth: The truth is guessed and the hypothesis is confirmed or refuted by experiments. Sampling Truth: It is obtained by statistical analysis of many bits of evidence. For example, public opinion is obtained by polling only a representative sample. 12 13 Math Fail Artinian Rings “The group sells Artinian rings. I hear the field is the local favorite, perhaps because of their simplicity.” Submitted by Anand Deopurkar ’08 14 15 Zero Preserved by every morphism, By Maria Monks Destroyed by every pole ‐ In certain complex manifolds2 Once upon a number It's nothing but a hole... That lived within a field It was the only number Amidst these wild properties Taking part in both ideals. It lives, and still exists ‐ For no other identities Endlessly unchanging Are additive as this. When doubled; yet, in fact ‐ When added to another thing The other stayed intact. Always indivisible Regardless of domain It formed the whole nilradical Of every Affine plane!1 So unique, this number, 'Twas left out of the loop ‐ It could not be a member Of the multiplicative group. And yet, in categories, It often had a place... But can't be pointed to by any Map, in any case! 1 Whose base field is algebraically closed. 2 For instance, the punctured complex plane. 16 17 m This article is a personal account of an MIT student’s experience at the summer research 3 program in Duluth. The undergraduate math research program at Duluth is one of the most successful in the country. Since 1977, Joseph Gallian of the University of Minnesota has been inviting students to spend the summer with him to work on mathematical research problems. Over the years, his work with seventy‐five students has resulted in approximately seventy published research articles in professional mathematics journals. This year three students from MIT participated in this program. My Experiences at Duluth By Maria Monks Be part of the action. The majority of my time during the summers of 2007 and 2008 were spent hiking or jogging in the cool, pristine forests surrounding the beautiful lakeside city of Duluth, MN. But I was not simply exploring – I was thinking, thinking about the unsolved problem in combinatorics (in 2007) or number theory (in 2008) that I was to work on for Joe Gallian's mathematics REU. Nature was my mathematical laboratory. The problems at the Duluth REU are as beautiful as the surroundings. My first Make suggestions. summer at the program, I worked on the following problem: A partition of a positive integer is a way of writing that integer as a sum of other positive integers. For instance, Provide feedback. 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 Contribute. are the partitions of 4. Given a partition λ of n, consider its set of k‐minors: partitions of n – k whose summands, written in decreasing order, are less than or equal to the Please let us know at uma‐firstname.lastname@example.org. corresponding summands of λ. For instance, the partition 5+3+2 is a 3‐minor of the partition 5+5+2+1. The partition reconstruction problem asks: for which n and k can every partition Our website: http://web.mit.edu/uma/www/index.html. of n be uniquely reconstructed from its set of k‐minors? The problems given to the nine students at Duluth often have a combinatorial flavor such as this one. They are generally approachable without much background knowledge, and Joe has a knack for choosing a problem that suits each student. The partition reconstruction problem turned out to be a huge success in my case. I solved it completely by the end of the summer and published the results in the Journal of Combinatorial Theory. When I was not running, hiking, muttering to myself about partitions while waving my hands in the air, or writing up results on my laptop, I could usually be found in a lounge 19 in the program apartment. Here, the Duluthians would gather for games of Gluck (a fast‐ Interview with Doris Dobi paced card game), Scrabble, Set, “Duluthopoly,” or simply lively mathematical discussions. The lounges also had kitchens, and groups of students cooked dinner together on most A runner up for the Alice T. Schafer Prize for Undergraduate Women in Mathematics nights. for showing excellence in studies and research, Doris Dobi is one of the most inspiring The structure of the program is ideal for self‐motivated, enthusiastic students of students in the math department at MIT. “I have always been really passionate about math,” mathematics. Every Monday and Tuesday afternoons, the students give talks about what says she and explains how being at MIT made it natural to major and do research in they have discovered (or where they got stuck) over the past week, and on Wednesdays mathematics. She started out with the idea of double majoring in 18 and 6. Later on she there are mandatory field trips. The rest of the week is unstructured, so that students are changed and decided to focus in theoretical mathematics and taking classes in course 6 that free to set their own hours. In addition, all the work is done individually – there is no interested her rather than classes that fulfilled requirements. collaboration between the students on their problems. Despite the individual nature of the work, the sense of community and camaraderie At MIT during her freshman year she got involved into a UROP in plasma physics in that forms among Duluthians is incredible, and is one of the best aspects of the REU. Joe which she used differential equations to model physical systems giving her a wide range of invites past participants to come as visitors to the program, who give students pointers and involvement and a taste of various fields in mathematics. However, she has been involved ideas for their problems and also pass on the myriad of Duluth traditions that have almost every summer in various research opportunities through Research Experience for accumulated over the years. Undergraduates (REU). During her freshman summer, she worked in finding stable periodic To mention just a few of these traditions: watching the sunrise from Lake Superior, billiard trajectories in polytopes, which are merely higher dimensional generalization of the Duluth Mile (a mile race on the track), trips to the famous Malt Shop by the lake, and polyhedra. Then she moved on to research with elliptic curves in Drinfel’d modules in never‐ending games of Gluck. Every year, on one of the Wednesday field trips, the REU goes number theory in the summer before her junior year. This past summer she was taking Alpine Sliding, in which you ride down a concrete chute on the side of a mountain on a little classes through the Princeton Analysis and Geometry Program where she dealt with a lot of wheeled cart with a hand‐controlled brake/accelerator. It is like a long roller coaster in differential geometry and Navier Stokes equations. which you can control your own speed! My experiences at the Duluth REU truly inspired me to pursue mathematics to the “I am really honored,” she expressed humbly her feelings about the recent award best of my ability. I had come into the program knowing that I loved mathematics, but I was she is receiving. She “feels a sense of responsibility along with it which is the responsibility not certain that I wanted to be a professional mathematician – I doubted my abilities and that comes from living up to all the price entails and wanting to share my passion for also my desire to do mathematics for the rest of my life. mathematics with other people and having something come out of it.” When one looks at By the end of last summer, however, it was all but settled. I had found my calling in her various achievements we can be sure that she will live up to it. the enchanted forests of pure mathematics. Doris elaborated on how she got help from the entire math faculty at MIT who were willing to ease her doubts and questions during her undergraduate years. “My advisor Richard Stanley has been open to all of my questions” explained Doris, and “this term I am working closely with Prof. Kleiman who has been very helpful with whole grad school process”. The next step for Doris is to go to mathematics graduate school. Afterwards, she wants to see how she can apply her math skills to the world around her. One of the interesting topics in mathematics for her is probability theory, which she feels is a part of math that can be applied to various fields from finance to computation biology. Her interests are seeing and understanding how the world works, “predicting the future and probability theory answers some of those questions” she explains. 20 21 Mathematics is not the only thing that Doris is passionately involved in. She Grad Life participates in many activities such as capoeira (a Brazilian martial arts) and enjoys playing basketball, watching good movies and listening to music. So how does she manage to MIT’s Endless Activities balance all of this? “I prioritize work above all and I am very exact in planning things out and following my plan closely” Doris reveals her secret, and in the process emphasizing the “You don't have to be a freak to be very smart” is one of the many things that importance of being structured and using time to its maximum. Furthermore, she Martin Frankland noticed about MIT. The environment is filled with very talented and encourages the incoming students to get involved in many UROPs in the math department friendly students which was somewhat surprising to him. Martin Frankland always knew and to apply for various math programs that are to their level. Most importantly though that he loved science and math but after going to math camp twice his passion was clear. “It another important aspect of Doris's success is the fact that she enjoys everything she does. was math above all,” he said. He was involved in math camp in the Quebec AMQ at first as a Having fun and enjoying mathematics is an important part of learning and succeeding. student and then as a mentor. His teachers and professors helped him with applying, preparing, and getting funding for getting into excellent graduate schools. A couple of years later, Martin is a graduate student at MIT singing in the concert choir, doing research under the guidance of Professor Miller and teaching linear algebra to undergraduates. Once he arrived on campus Martin had initial concerns about “the ultra‐competitive graduate students” but he found them to be very friendly and MIT in general a good community giving him all the resources needed to do research. Of course, he was expecting a lot of hard work and he got it. However he did not expect so many activities outside of Interviewed by Daniela Çako class. “It's incredible to see the breadth of activities that go on in campus” he said surprised and excited and added with a wow “they [students] go to class and do crazy things like building robots and playing music.” Martin was pleasantly surprised at this side of MIT and at noticing active social life. He is involved in activities that he enjoys such as concert choir, chamber chorus and Techiya, making him musically inclined. He is amazed that the math department is the only one that has an IAP music recital. Currently, Martin is a teacher assistant for 18.06, a class he took a long time ago. “Now it's most interesting when you know more math,” he says. “As you advance you see things differently. In hindsight, linear algebra makes more sense,” he adds. Furthermore, he had been a TA for many other classes such as 18.02A, has graded 18.901, and has been a mentor for 18.821. All the contents and volume of the materials covered had impressed him. He hopes he has been of help to the students by making the subjects easier to grasp. His main interaction with undergrads has been as a TA. However, he has also met a couple of undergrads who take graduate classes, which have amazed him with their hard work and intellect. His advice to seniors applying to grad schools? Look and apply to places that you really want to go to. Moreover, look up different professors in various schools and see what 22 23 kind of research they are involved in. Another important part of the application is strong letters of recommendations and being in touch with a mentor to guide you through the Move aside, Pikachu. application process. Besides applying, you should also find means of supporting yourself. You should look into teaching and research assistances and also funds like the NSF and ask individuals and professors who should know more about this. For Martin, taking topology in his last year of undergrad made him feel “a revelation” as he describes it. To him topology soon turned out to be his favorite topic. His research topic now is part of algebraic topology where he is working closely with Professor Miller. Further into his career he plans to get involved in academia and go for a post‐doc position after his graduate education and work his way up the ladder. Special thanks to Martin Frankland Interviewed by Daniela Çako Be proud of your choice. Buy our new t-shirt for only $10. Available in black, blue, and green. Please direct all inquiries to uma‐email@example.com. 24 From Philosophy to Mathematics different from other disciplines. One can sit and try to prove something for twelve hours and not get anywhere. However, the work of those twelve hours has not been in vain, and Craig Desjardins is not your typical mathematics graduate student. He started out regardless of the fact that there is no solution, it gives the mathematician a greater having a strong dislike of the subject in high school due to the teaching methods used. “It understanding of the problem. [mathematics] was taught in school as a tool for science rather than a subject on its own,” he says, as he explains his initial disliking. By fate, in college, he took a class in mathematics In his plans for the future, after having been a TA for various classes ranging from by Tom Banchoff which gave him a new perspective. Now, mathematics was not a tool for 18.03 to 18.821, he realized he enjoys teaching and wants to further pursue a career as a science but rather a subject on its own with “intrinsic aesthetics” as Craig describes it. Since professor. Craig's thesis is focused on combinatorial reinterpretation of objects in the then he immersed himself into the subject and changed his mind from philosophy to invariants of algebraic groups. “It's most fun to realize the connections between the fields,” mathematics. However, he felt that due to his initial major, he would not have to chance to he says about his work. Regardless of what one does though, “mathematics gives the ability go into a good graduate school and further pursue his quest for knowledge. Hence, he got to think logically and linearly exceptionally well which from a practical point of view gives involved as soon as he graduated into a math program in Budapest, Hungary. rise to a lot of possibilities of technical jobs,” says Craig. “But for me I do it for its pure aesthetics and for gaining a deeper understanding of it.” He depicts the experience as an important step into his graduate studies. A very essential aspect of graduate school is being able to see different ways and methods of doing mathematics, and Budapest enabled him to do that along with being completely engaged in the subject. At the same time the experience made him realize that with hard work anything is doable. “Europeans and other non‐American undergraduate math majors are always doing math and taking classes in it,” he noted, which made him think that in the program it would be “impossible” for him to catch up. “I don't know how it happens, but within two years everyone is in the same level except for a couple of future medalists,” he Special thanks to Craig Desjardins says, baffled by the experience. Interviewed by Daniela Çako One of the important things to consider in applying to grad school is to look at funding. One major grant is the NSF and other grants from Department of Defense and so on. Students should be sensitive to the deadlines. In good schools, students are guaranteed funding for almost all of the years that they do research. One of the things to look forward to as a grad student is pulling fewer all nighters than as an undergraduate. “The deadlines are more extended,” he states. “Now it's thesis in 5 years,” rather than having problem sets due every day, he notes. Furthermore, graduate students are required to take fewer classes and seminars which are related to their thesis. This makes their workload somewhat easier but not really. Being a math graduate student is 26 27 Interview with Professor Stanley No, I still didn’t know. I didn’t have a good idea of what math research was. Actually, in high school, I did some research that wasn’t really original. But still, I did not see myself as being Richard Stanley received a B.S. in mathematics from Caltech in 1966 and a Ph.D. in applied able to do real research or know what it was like. I just wanted to learn as much as possible. mathematics from MIT in 1971 under the direction of Gian‐Carlo Rota. Professor Stanley’s How about being a professor? research concerns problems in algebraic combinatorics. I guess it was a long‐term goal. It seemed like something impossible to achieve. I thought I Math majors may recognize Professor Stanley from having taken 18.S34, a seminar that could never be the same as these math professors who were teaching other students, prepares students taking the Putnam exam, and 18.S66, a seminar that highlights Professor coming up with all these neat ideas just by thinking. Stanley’s focus on combinatorics. I know what you mean. I had the privilege of sitting down with him and finding out how he picked up his interest in math, how he got involved in combinatorics, why Catalan numbers are special, and his Graduate school is very good for helping you learn how to do that. passionate interest in solving chess puzzles. I read that you worked for JPL for some time. How was that related to your interest in Where did your interest in math begin? math? I would say I developed it in high school. There was a direct correlation. I was in a group there that was responsible for designing the error correcting codes that the spacecrafts were using. It was very mathematical and my Was that through competitions or was there a teacher that was influential? undergraduate advisor recommended that I apply for summer jobs there. So that fit right in Back then there were hardly any competitions. I moved to Atlanta, Georgia and there was a with my math interests. However, I never saw myself working permanently for the place. kid in my class who was doing very sophisticated math by himself and I just felt that I Which people influenced you at the time? wanted to understand more. That started me off. My undergraduate advisor was Marshall Hall. He was a very well‐known algebraist and At that point, did you know that it was research you were interested in? combinatorialist. Although back then I had no interest in going into combinatorics, he was At that point, I just wanted to learn more about mathematics. Martin Gardner had a column one of the few people who could be considered a combinatorialist. Someone else that had in Scientific American called “Mathematical Games”. There were Mobius strips and an influence on me was Donald Knuth. He got his Ph.D. at Caltech under Marshall Hall and flexigons. He had all kinds of neat problems that were very easy to state. I just wanted to he stayed there a couple years before moving to Stanford. I took the first course that he learn more. I didn’t think of myself doing research. taught and the next year I graded it. I talked to him about doing research. He tried to get me interested in CS‐type problems to work on. Did you know you would be majoring in math going to college? At that time, your interest in math was not very specific. I knew that I would. It was 90% in math, 10% in physics. In fact, my main interest was algebra. I did not think that combinatorics was a serious Did the college experience encourage you to do math research? subject. I didn’t even take the combinatorics course at Caltech. And also, at my work at JPL, 28 29 it was more of combinatorics than anything else. But even then, I didn’t really think of it as a Do you have any memorable moments studying math in college? serious research area. When I was in college, I realized how much better I was at algebra than analysis. I took a How did you get involved in combinatorics? year’s course in quantum mechanics. The first two trimesters [Caltech runs on a trimester system] were based on analysis, Schrodinger equations and all. I found that really difficult. I I went to graduate school at Harvard. When I arrived, I wanted to work in group theory. didn’t really have a good intuitive idea of what was going on. The final trimester was an There was a famous mathematician there named Richard Brauer doing group theory. But I algebraic approach based on linear algebra, matrix theory, and that all seemed so easy to didn’t really like where research in group theory was going. They were just starting to me. I was the first to finish the final exam and I got an A+ in the class so I was really pleased classify finite simple groups and there were hundreds of pages, several hundreds of pages, by that. It made me realize that somehow I really was cut out to do algebra. I wonder how with many, many cases. But I did have some problems that came out of working at JPL, much of mathematical talent is hard‐wired, how much detail is built into you, like what area some combinatorics problems that I was curious about which I never thought of as serious of mathematics you’re going into. It seems that even that part is wired. research problems. Someone at Harvard suggested that I go to MIT to see Professor Rota. He encouraged my interest in combinatorics and he became my thesis advisor. In your opinion, how is the state of combinatorics? That’s how it all started? Right now, I think it’s a very good, extremely active subject. But I think to do really high‐ level research now, particularly in algebraic and enumerative combinatorics, which I worked That’s right. After meeting Professor Rota, I realized that one could do research in in, you have to know more than you used to. Back when I was a graduate student, you could combinatorics. It was a Mickey Mouse subject for a lot of people at the time. Some people use the simplest results from other areas like topology. Take the Euler characteristic. You still think that now, but not so much. could interpret that combinatorially and come up with all kinds of interesting results. Now, What motivated your long‐standing interest in combinatorics? the topological combinatorics gets into some of the deeper more recent aspects of topology. Combinatorial representation theory is a huge subject now, probably one of the main areas I think I just naturally liked combinatorics. I think I was more hard‐wired to like that kind of of combinatorics. At the beginning, the very simplest representation theory – groups acting mathematics, discrete type mathematics like algebra. I realized it as soon as I found some on sets – was enough to get all kinds of neat things. Now you have to be into all the latest professor who did some serious work in the area. He had this view that someone should algebras, like affine Hecke algebras, quivers, and very sophisticated, mainstream stuff that build up general principles to unify combinatorics at the time together with other branches people are working on. You have to know more now than you used to. of math. All of that was very appealing. So I’d have to say that it was due to the combination of my natural instincts and finding the right professor. In the Enumerative Combinatorics book, you list many exercises that ask different ways to prove the Catalan numbers. Where did that start, and why Catalan numbers? What do you enjoy most about being a professor? Catalan numbers just come up so many times. It was well‐known before me that they had The freedom is really good. many different combinatorial interpretations. I think there was a paper in the Monthly that had a dozen or so of these interpretations and it notes that some professor had a hundred At the time, did you know what being a professor was about? combinatorial interpretations that he came up with, unpublished. It seemed like an extension, being a student then being a professor. 30 31 When I started teaching enumerative combinatorics, of course I did the Catalan numbers. A machine could solve just by going through all possibilities, but that’s not what people are When I started doing these very basic interpretations – any enumerative course would have interested in. There are some aesthetics to that. It’s not just a question of solving the some of this – I just liked collecting more and more of them and I decided to be systematic. problem. You have to understand the themes of these problems and exactly what these Before, it was just a typed up list. When I wrote the book, I threw everything I knew in the problems are trying to show, what pieces interfere with each other in a certain way. It’s like book. Then I continued from there with a website, adding more and more problems, and trying to get a maximum amount of interesting play from these pieces. The problems are people would write to me with more interpretations. Many of them are very similar to each not always “mate in a certain number of moves”. There are “selfmates”, “helpmates”, and other, which make them really interesting. all kinds of new pieces people put in. How are Catalan numbers special? Most people do not realize that this area even exists. Not too many people are into it. If you go to my webpage and click miscellaneous and click on chess problems, you will get some They’re the most special. I think a lot of it has to do with the binary tree. Breaking structures links. up into two pieces – there are so many structures that you can do that with, even if it’s hidden. There are some other numbers you can do a lot with. (n+1)n‐1 – that’s another great Thanks for your time Professor. interesting sequence. As this is a math interview, I have to ask: Do you have a favorite number? I’d have to say my favorite number sequence is the Catalan numbers. So any number in the Catalan sequence. n‐1 Yes. The Euler numbers and (n+1) would be close seconds. Interviewed by Hyun Soo Kim Which graduate schools would you recommend for going into combinatorics? Introduction blurb taken from MIT’s math website I think you should go to the best graduate school that is compatible with your area. You should consider some combination of the whole school and the people in the area. Usually, the two are quite correlated. The best people will be at the best schools. You should definitely discuss with an advisor. What kind of hobbies do you have? I like juggling, although I’m not very active now. Bridge is something I enjoy. I like chess problems. I don’t really like chess, but I like chess problems. It’s a serious area that is very small and extremely well‐developed into an art form. 32 33 Interview with Professor Artin I was interested in all sciences. I thought at the time that when I went to college that I would probably major in chemistry. Michael Artin received the A.B. from Princeton in 1955 and the M.A. and Ph.D. from Harvard Did you think about it in high school? in 1956 and 1960. He is currently a Professor of Mathematics at MIT. He joined the MIT mathematics faculty in 1963. Professor Artin is an algebraic geometer, now concentrating Well, I don’t think thought too much about it, but if I had thought, I probably would have on non‐commutative algebra. said chemistry. After my sophomore year, I had decided against physics. By junior year I decided against chemistry. That left biology and mathematics. Maybe I made the wrong Math majors may recognize Professor Artin from having taken the 18.701 and 18.702 series, decision – biology has been a pretty exciting field. But I have been happy doing math. which are popular among theoretical math majors. They seem to be very disjoint subjects. I had the privilege of sitting down with Professor Artin and finding out how he picked up his interest in math, how he got involved in algebra, how the brain processes math, and his That’s right. They’re two completely different things. I was just interested in completely interest in biology. independent subjects. Did your father (Emil Artin) who was an eminent mathematician influence your academic How did you end up deciding to do math? interests? I decided it would be easier to switch out of math because it was at the theoretical end. He never encouraged me particularly to go into mathematics. He spent time with me on other things too, other sciences. His father had gotten him an elaborate chemistry outfit As it turns out, you never switched out of math. when he was a kid because he was in the textile business in the early part of the 20th I didn’t. But I planned to switch out of math and move into biology when I was thirty. century and organic dyes were just taking over the textile business, so he thought that was Because everybody thought that mathematicians were washed up by thirty. I had it planned. a good career. My father did the same thing for me. He outfitted me with a fairly elaborate I was going to wander over to the biology department and start going to some of the chemistry lab set. At the time he had a student, Richard Otter, who had started out in seminars, and I actually did that. Only I realized I was too old, at thirty, and much too chemistry. He switched to mathematics and wrote a thesis on the number of trees. involved in mathematics. Mathematical trees? At the time, you were working on algebraic geometry. Yes. It was related to organic chemistry, and the mathematical part was counting Yes. Let’s see, I got my Ph.D. at age 26, and I was just starting as an Assistant Professor at hydrocarbon configurations. So that’s what my father suggested him to write his thesis on. MIT. He still had connections in the chemistry department and was able to get stuff for me. He also gave me glassware that he had made himself. That was how my interest began. Richard How did you end up choosing algebraic geometry? Otter later became a professor at Notre Dame. It was partly the personality of my teacher, Oscar Zariski. He was an algebraic geometer at Harvard when I was a graduate student. 34 35 At the time, did you consider other fields were interesting? The number of math majors has gone up dramatically in the past twenty years, but the total number going to graduate school has not gone up. Perhaps it’s that we have people win the I was thinking of topology. When I was an undergraduate, I thought that I would probably Putnam exam, and maybe people feel that if they don’t win the Putnam exam then they go into that, but that didn’t happen. I thought a lot about a famous problem, Dehn’s lemma haven’t got what it takes to be a math major. in knot theory. The famous mathematician Max Dehn published the lemma in one of his papers, but it couldn’t be proved. [Editor’s note: Dehn’s Lemma asserts that a piecewise‐ Competition seems to be a widespread concern. linear map of a disk into a 3‐manifold, with the map’s singularity set in the disc’s interior, I never could answer a single question on the Putnam exam. I never took it, but I look at the implies the existence of another piecewise‐linear map of the disc which is an embedding questions, and cannot answer a single one. That’s not a requirement for doing research in and is identical to the original on the boundary of the disc.] I worked on it without knowing mathematics. It’s a useful skill to be able to solve problems efficiently and training certainly anything. It was proved two years later, in 1957, but not by me. helps. But it is not the most important attribute for doing mathematics. So I think the Did you ever think of not going into academia? students are getting the wrong idea. It’s also true that we send twenty percent of our majors to Wall Street, although maybe not this year. Wall Street was not a very big You know, people who come to MIT often know what they want to do, they think. There employer twenty years ago. was no feeling of that type when I was at Princeton. I didn’t think about it a lot; it just happened. What do you enjoy most about being a professor? Did you think about becoming a professor at the time? I like teaching very much, and it’s a pleasure to teach at MIT. I like research, though I’m doing less of that as I get older. Especially at that time, almost all math Ph.D.’s went to academic jobs. There were a lot of math professorships around the country. Of course, some people didn’t end up there. But What would you be doing if you did not become a professor? it’s still true for most Ph.D.’s – the first job is academic. They may go on to do other things Probably not mathematics, not directly. I think I would have been very happy doing biology. later. I’ve had about 30 Ph.D. students and about a third of them left academia at one point. That just didn’t happen. Was it common for students to know that they would become a professor? What kind of hobby do you have? I don’t know. Back then at Princeton there was no atmosphere of a math community among I guess playing music – the violin. It’s something you can do your whole life. the undergrads as there is here. In Princeton, there may have been half as many undergraduates in each class as there are at MIT now. The year I graduated from Princeton, Do you play with other faculty members? there were five math majors. However, at least three of them became mathematicians. And there were two people in the class who majored in engineering who later became Yes, I have been playing with Arthur Mattuck ever since I got here. How often we play varies. mathematicians. Math just wasn’t a big major. It’s remarkable. But the people who did a The other members have varied over the years, whoever is around. major were generally very serious. Do you hold performances? The fraction of undergraduates at MIT pursuing grad school in math is fairly small. No, we just get together to play. I refuse to play in performances. 36 37 What kind of pieces do you play? Thanks for your time Professor. It’s standard literature chamber music. String quartets, stuff like that. There are certain quartets you get tired of, then others you don’t. You certainly never tire of the late Beethoven quartets. Do a lot of math faculty members play instruments? Quite a few. It’s not so obvious why. Interviewed by Hyun Soo Kim There seems to be some kind of connection between math and music. Introduction blurb taken from MIT’s math website Actually, for a few years I tried to figure out what is algebra. Since there is an affinity between music and math, it makes you think maybe they’re in the same part of the brain. And so I tried to find where algebra is, but I didn’t succeed. That seems to be more of a biological question. It is, and it is very hard. There’s not much you can do. How do you study the brain? You can look at child development, or brain injuries – what happens when you lose a part of your brain – or you can use introspection, you think about it. They’re all helpful, but none of them are really good. I have read a fair amount about the brain injuries and what they do. I found one really interesting article by a woman in England who had a patient, an educated man who had a stroke. He lost his ability to do arithmetic, but he remembered what the rules were. She gave this one vignette which was that he was asked to add, maybe 13 and 17. And he thought, and thought, and he tried to find a way around the hole in his brain. She heard him say to himself, “Well it’s got to be an even number.” So he understood, but he couldn’t do it. That means arithmetic and algebra are different. 38 39 We include an excerpt from an article Professor Artin wrote many years ago. It talks about red. It became painfully clear that one of us was in the wrong field. After a while, he put me algebra education at the college level. out of my misery. He told me that the right answer is “fruit,” and we changed the subject. During my college days I had some summer jobs doing manual labor, and I have a sad recollection from that time which also makes me wonder in what sense abstract thinking is What is algebra, and why is it important? fragile. It is about a mentally handicapped man on one work crew who followed the By Michael Artin Brooklyn Dodgers. Though it was near the limit of this ability, this man always learned the result of the game before coming to work. He would start the morning by announcing the score several times in a loud voice. “Dodgers 5, Giants 3 yesterday.” The rest of the crew Here are three attempts at a description of algebra. usually responded gently. Then as the day went on, he would ruminate on his one piece of Algebra is information, working out its implications and reporting his conclusions to us from time to time: “Dodgers beat the Giants”, …, “Giants lost,” and so on. Each reformulation gave him the language of mathematics, the pleasure of a new insight, and I found this so remarkable that I remember it clearly working with x, today. Had it not been for his birth injury, he might have become a mathematician. So though there is clinical evidence to support the neurologists’ view, I’m not completely the study of the algebraic operations +, x and their analogues. convinced. I like the first description, the one Carole Lacampagne used in her announcement for this workshop. Algebra is one of the most abstract parts of mathematics, and I’ve always felt that language and abstraction are closely linked. The reason is that in mathematics, when the right definition is made, the concept evolves out of it. But when I mentioned this feeling to a neurologist a number of years ago, he said: “No, that’s wrong. The power of abstraction is much more shallowly rooted in the brain than language. People with a brain injury often lose the ability to think abstractly, though the language capability remains.” Compiled by Hyun Soo Kim This was interesting, and so I asked him: “How do you test abstract thinking?” “Oh, we have standard tests. For instance, why is an apple like an orange?” The thing is, I flunked the test. I thought: “Well, they’re both sort of round.” But the apple has those dents, so I rejected that answer. Then an orange is orange and an apple might be 40 41 This leads up to the Guts round, held in the auditoriums as described above. Students are The Harvard‐MIT Math Tournament given sets of three problems at a time. When the team thinks they have solved the three problems (or decides to skip them), the runner on their team hands it in and grabs the next packet of On February 21, 2009, seven hundred high school students from around the world will be problems. It is widely considered the most exciting round of the contest. crammed into 10‐250 and 26‐100. Some will be huddled over their lap desks furiously scribbling out The Guts Round wraps up the day‐long tournament. calculations, others working with other members of their team of eight in urgent whispers, and one person from each team on the edge of their seat, ready to grab an answer from a teammate and The awards sprint to the nearest station of graders. A gigantic overhead projector in each room will display the cumulative scores of the teams in real time, as the graders and proctors work to continually update The awards also make HMMT one‐of‐a‐kind among math competitions. Frisbees bearing the system and keep everything in line. the HMMT logo are awarded to the top teams, and top individuals have received painted Klein This is just one round, the “Guts Round”, of the annual Harvard‐MIT Mathematics bottles, Abaci with the sides engraved, and decks of playing cards as prizes. Tournament (HMMT), a math contest for high school students run entirely by MIT and Harvard undergraduates. The lighthearted awards ceremony, combined with the unconventional nature of the The first HMMT was held in 1998 at Harvard University1. It started out as a local tournament contest and the sheer difficulty and beauty of the problems, makes HMMT a truly exciting and with a few hundred participants. As the word spread, top schools within driving distance began to worthwhile experience. participate, such as Phillips Exeter Academy. Last year a team from Florida, teams from all up and down the East coast, and international participants from Canada, China, Thailand, and Turkey joined How you can get involved the crowd. This year, five teams from China are already signed up for the 2009 tournament. Write problems! Sitting down and writing an interesting math problem is not as difficult as it sounds. What makes our contest so appealing that students fly in from the other side of the world to Email any problems you write up (with answers and solutions) to hmmt‐problem‐firstname.lastname@example.org. If participate? your problem is used on the contest, you will be recognized on our program brochure. The tests Sign up for our mailing list! The mailing list hmmt‐email@example.com is our list for new members. You can email hmmt‐firstname.lastname@example.org to be added to hmmt‐list. This is the first step to becoming The format of HMMT is unique among high school math competitions. It kicks off in the involved with the organization of the contest. morning with two individual tests that each last one hour. Every contestant chooses either to take the “General Tests”, which consist of a mixed bag of problems of various difficulty, or two of the Come to the meetings! Once you are on the mailing list, you will receive announcements about our “Subject Tests”. There are four Subject Tests: Algebra, Calculus, Combinatorics, and Geometry. Each upcoming meetings, such as our bi‐weekly problem‐writing sessions and test collating parties. individual test consists of ten questions, increasing in difficulty and point value. For instance, There is also free food at the problem writing sessions, to fuel your brain for writing awesome problem number 3 from last year's Combinatorics test asked: problems. Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so Volunteer on the day of the contest! Again, you should email hmmt‐email@example.com to sign up to that every pair consists of animals of different species? (Assume that all animals are volunteer on February 21, 2009. Save the date! distinguishable from each other.) Next comes the Team Round. The teams of eight go to various classrooms and are given a set of questions that usually revolve around one or two topics. The questions have multiple parts, and many of the parts require the students to write mathematical proofs. 1 The Team Round takes a substantial amount of time to grade, so we then give the students HMMT started out as a joint competition with tournaments held at Rice University and Washington a lunch break and hold “mini‐events”. Some of our mini‐events include Set, juggling, Rubix cubes, University in St. Louis. In 1999, it was run jointly with the Stanford Math Tournament, and starting in 2000 hypercubes, and any other interesting topic that a volunteer signs up to teach. HMMT became an independent contest. m 3 We like you. Make suggestions. Provide feedback. Contribute. Please let us know at uma‐firstname.lastname@example.org. Our website: http://web.mit.edu/uma/www/index.html.
Pages to are hidden for
"math majors magazine"Please download to view full document