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Michigan Math
Scholars Summer 2000 General
Information
Dates: Session 1: June 25, 2000 - July 8, 2000 Session 2: July 9, 2000 - July 22, 2000 Options for Attendance: Students may elect one or both two week sessions. Students may participate in the Math Scholars Program as commuters or as residential participants. Residential students will be housed in Mary Markley Hall on Central Campus in the heart of Ann Arbor. They will be fully supervised by program staff. Both residential and commuter students will have the opportunity to participate in various activities both on and off campus. Fees: Commuter: $650/2 weeks or $1300/4 weeks Residential: $1150/2 weeks or $2300/4 weeks Financial aid is available for qualified applicants. Admission: Applications received by April 1, 2000 will receive top admissions priority. Late applications will be received on a 'space available' basis. Further Information: Upon admission, students will receive a detailed information packet including directions, contact information, health forms, and program contracts. Math Scholars personnel can arrange to meet participants at Detroit Metro Airport and accompany them to campus. This option is explained fully in the Information Packet, which can also be viewed and downloaded from our Web site. Find it under Summer 2000, General Information. We also encourage you to view this site for more information about the program, the University and the city of Ann Arbor. Welcoming Reception: On the first Sunday afternoon of each session, student and parents are invited to a Welcoming Reception in East Hall. You will be able to get to know the faculty, grad students and other staff who will be working with your students over the next two weeks. Family and friends are cordially invited to this reception. The City: The City of Ann Arbor is one of the nation’s premier university towns. It is safe, clean, green and beautiful, and abounds in cultural opportunities. The Math Scholars program offers participants many chances to partake of the City’s bounty, including the Ann Arbor Summer Festival and the Art Fair. Courses The Amazing Fibonacci (Sequence) Session II (July 9 - July 22) This course has a highly distinguished origin: It is founded on the fascination that the Fibonacci sequence holds for Professor Hochster (and for many other mathematicians) whose work on this subject won him second place in the Westinghouse (now Intel) Science Talent Search. The course will be an exploration of the mathematical ideas surrounding the sequence of Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… in which each term is the sum of the two preceding terms. The Fibonacci sequence occurs in nature in contexts that vary from the reproduction patterns of rabbits to the arrangement of the whorls on a pinecone and the petals on a sunflower. There is so much beautiful pure and applied mathematics on the Fibonacci sequence that it has its own research journal, the Fibonacci Quarterly. The astonishing properties of the Fibonacci sequence will form the basis for an exploration of mathematical ideas from number theory, combinatorics, probability and (abstract) algebra. Write f(n) for the nth Fibonacci number, so that f(1)=1, f(2)=1, f(3)=2, etc. (For example, f(8)=21.) Here are some questions that one can play with and that give a taste of the material we will cover. 1. As n gets very, very large, what can be said about the ratio f(n):f(n+1)? The answer to this question is fascinating: the ratio tends to the unique positive number T whose square is T+1. T is an irrational number, approximately equal to 1.6180339887498948482, and the ratio T:1 is called the Golden Mean. It is thought that this ratio is the rectangular shape most pleasing to the human eye and it turns out, for example, that the Parthenon–which was built many centuries before the Fibonacci sequence was discovered–has these proportions. 2. What if we want to find f(1,000,000,000)? Certainly one way to do this is to calculate f(999,999,998) and f(999,999,999) and add them, but there are certain problems with this method! Can we find a formula that will allow us to calculate f(1,000,000,000) without knowing all (or even any) of the previous f(n)'s? Remarkably, the answer to this question is "yes." In this course we will see how. 3. Note that 7 divides f(7+1)=21 while 11 divides f(11-1)=55. Note also that 7 and 11 are prime: they have no smaller positive integer factors except 1. Is something similar true for every positive prime integer p except 5? Why? 4. What is the value of f(n+1)f(n-1)-f(n)f(n)? Why? The course will focus on these and other properties of this remarkable sequence. In the meantime, can you find a surprising property of these numbers on your own? Faculty: Professor Mel Hochster Codes, Ciphers, and Secret Messages Session I (June 25 - July 8) Around 4000 years ago in Egypt, the tomb of the nobleman Khnumhotep II was nearing completion. In one particular section of the inscription, a master scribe chose to replace the usual hieroglyphic symbols with new, nonsensical ones. This act rendered that important passage of Khnumhotep's funery inscription unintelligible except to those who knew what replacement the scribe had made. The year was around 1900 BC, making this inscription the oldest surviving record of the use of secret codes. Yesterday, your neighbor ordered goods over the Internet with a credit card. When he forwarded his credit card number over the network, powerful public key encryption schemes put this information in a form that was (presumably) unreadable to the electronic eavesdropper. Although the sophistication of Internet code is vast compared to the simple substitution made by the Egyptian scribe, the security of both schemes are based on the presumed difficulty of solving certain kinds of mathematical problems. This course is about the mathematics that underlies secret codes and attacks on secret codes. We will proceed historically, beginning with substitutional codes, like the one used by our Egyptian scribe, and ending with modern public key encryption, like that used by our Internet vendors. This study of secret codes will run us into a surprisingly rich assortment of topics in discrete mathematics, statistics, information theory and number theory. For example, we'll learn a significant amount about: • permutations and random permutations • distribution of words and letters in written English • sequences of 0’s and 1’s that appear randomly generated • residues and congruences of numbers and how you can make codes from them The use of technology will be emphasized throughout. Each afternoon, students will participate in a computer lab in which they will use MAPLE, an advanced general-purpose mathematical package, to run experiments pertinent to the course material. Along the way, students will get plenty of practice creating their own codes and trying to crack those of their instructor and their classmates. Faculty: Professor Dick Canary, Professor Phil Hanlon Geometry and the Imagination Session II (July 9 - July 22) The study of geometry and topology has a rich history with problems dating back to the ancient Greeks. In this course we will touch on some topics in modern geometry and topology so as to give a flavor of what current mathematical research in these areas looks like. In geometry, we study the shapes of objects. Topology on the other hand cares little about the shape of an object–for instance, to a topologist, a doughnut, a coffee cup and a bagel all represent the same object (called a torus). A brief description of some of the topics we will cover is given below. Can a rectangle of dimension 1 by Ö2 be tiled by congruent squares? More generally, can a rectangle of dimensions L by W be tiled by congruent squares? The answer to this riddle, surprisingly enough, comes from studying electrical networks. We will also study related problems of tiling rectangles by squares and rectangular boxes by cubes. Can a sphere be turned inside out? If we are not allowed to cut it or tear it, then the answer is obviously, "No."' But if we imagine our sphere to be made of a highly elastic material that can pass through itself, then maybe it could be done. The only rule is: you are not allowed to create any sharp creases. With this rule in place, it seems impossible to turn the sphere inside out (called an eversion of the sphere). We will see sphere eversions in the award winning video, "Outside In." We will see that although a circle cannot be turned inside out, a sphere eversion is indeed possible. Consider any map of countries in an atlas, real or imagined. We want to color it so that we can distinguish one country from another. One of the most famous problems in map coloring was stated at the end of the nineteenth century: Show that any map of countries drawn on the plane can be colored with four colors. This seemingly simple problem confounded the best mathematicians for more than 100 years. This was called the Four Color Theorem and was finally proved by Appel and Haken in 1978. We will study map colorings and other problems by means of graph theory. Tying knots is part of learning to be a sailor or a Scout. This seemingly simple activity is one of the most interesting and difficult areas of study in modern topology. We will construct knots in the following fashion: take a piece of string and tie a knot in it. Now glue the two ends of the string to form a knotted loop. The result is a string that has no loose ends and is truly knotted. The simplest knot is the circle which is called the unknot. The next simplest knot is called a trefoil knot. How can we tell these knots apart? How do we know that we can’t untangle any knot to make it the unknot simply by playing with it long enough (without cutting the string, of course). We will introduce methods that will allow us to conclude that some knots are indeed more ‘knotted’ than others. We will study various types of knots, their properties and also look at connections to molecular biology and chemistry. Faculty: Professor Ravi Shankar Knots and Polyhedra, or When is a Knot not a Knot? Session I (June 25 - July 8) We all know from experience that there are many different ways to tie a knot in a piece of string: common knots include the slip knot, the trefoil knot, the granny knot, the figure eight and the square knot. But there are many, many others. If we take a piece of string, tie a (genuine) knot in it and glue the loose ends together, there is no way that we can fully untangle the string without using scissors. But there is the possibility that by playing with it long enough we can deform the original knot into a seemingly different one without the use of scissors or glue. How can we decide which knots can be transformed into each other in this way? This question is at the heart of knot theory, a relatively new and very exciting area of mathematical research. In addition to being truly profound in its own right, this question has surprising applications to biology, chemistry and physics. For example, knot theory plays a major role in the study of DNA replication and in the classification of molecules. In this course we will explore knot theory, the related constructions of polyhedra, and applications of these topics to other areas of science. Faculty: Professor Ruth Lawrence The Nature of Infinity Session I (June 25 - July 8) & Session II (July 9 - July 22) Is 0.99999... equal to 1 or just infinitely close to it? How can you do infinitely many things in a finite amount of time? What role does the infinite play in our daily lives? How can we justify the statement that one kind of infinity is bigger than another? How can a finite region have an infinite boundary? How can we, with our finite minds, even assess the quality of our reasoning on the nature of infinity? Pondering infinity has both fascinated and terrified mathematicians, philosophers and poets (and regular folk!) since time immemorial. As we shall see, "common sense" is a very poor guide when dealing with infinity. However, some of the most brilliant minds of all times have grappled with the question of infinity, and they have found some astonishing answers. Come prepared to stretch your mind to the limit! Faculty: Session I - To be announced Session II - Professor Brent Carswell Patterns in Nature: An Introduction to Dynamical Systems Session II (July 9 - July 22) Many systems in nature display pattern formation, in geology and embryology, neurology and meteorology, as well as many other disciplines. Pattern formation has been used to explain video feedback and the spotting of leopards. Here is a specific example of one kind of pattern in nature: take a layer of water that is very thin vertically, but has large surface area. Heat this layer gently and uniformly from beneath, so that the top surface attains a temperature of T1 and the bottom surface attains a temperature T2. If we heat the water enough, the water near the bottom expands, gets lighter, and rises. Meanwhile the colder, denser water at the top is sinking. In other words, the water begins to convect. However, the water cannot rise everywhere or sink everywhere. It must rise in some places, and sink in others. But the heating is horizontally uniform–every place is just like every other place. How does the water decide where to rise and where to sink? This experiment is difficult, because the heating needs to be extremely uniform, but several solutions have been observed. For instance, we can tile the horizontal plane with hexagons in such a way that the water rises in the center of each hexagon and sinks on the edges. Where before we had featureless, motionless water, now we have a hexagonal pattern. This is pattern formation. In this course we will learn why patterns are so ubiquitous in nature. We will use a branch of mathematics called group theory to learn what kinds of patterns can and cannot be formed, regardless of the physical example being studied. Faculty: Professor Tim Callahan Stupendous Statistics Session II (July 9 - July 22) Statistics is the art and science of drawing reliable conclusions on large amounts of data based on the analysis of a manageable sample. This exploratory course in introductory statistics will acquaint participants with modern statistical practices utilizing a problem-based approach to the field. Consider, for instance, one of the first examples analyzed in the course: "What's in the Bag?", which illustrates many aspects of statistical decision making processes. There are two identical bags, Bag A and Bag B. Each contains 20 vouchers, small slips of paper identical in size and appearance except for the positive and negative dollar values printed on them. The contents of the bags, in terms of the face values and frequencies of the dollar amounts, are the following. Bag A: -$1000 (1), $10 (7), $20 (6), $30 (2), $40 (2), $50 (1), $60 (1). Bag B: $10 (1), $20 (1), $30 (2), $40 (2), $50 (6), $60 (7), $1000 (1). You must pick a bag to keep and you agree to receive the sum of the face values of the vouchers it contains: you will pay $560 if you pick Bag A and win $1890 if you pick Bag B. You will be shown a bag and allowed to sample some data. Based on your analysis of that data, you must decide whether to keep or exchange the bag you were shown. (You’ll only be able to examine a portion of the data, and you’ll be doing another version of this problem later with many, many vouchers. So no strategy based on seeing a large portion of the data is allowed.) How would you proceed? Although this problem touches on many sophisticated ideas, the relatively small number of data points simplifies the calculations. Throughout this course we will spend mornings discovering new concepts and implementing them on small examples. Afternoons, we will work in the Math Scholars computer labs analyzing more computationally challenging examples with state-of-the-art statistical software packages. Faculty: Professor Martha Aliaga A Day in the Life of the Michigan Math Scholar Student By 8:30am, the dorm cafeteria is filling up with Math Scholars. Those who were running computer experiments until all hours last night are a bit bleary-eyed, but the new day beckons. At 9:00am, residential and commuter participants and MMS faculty meet in our dedicated Math Scholars classrooms. For the next three hours these rooms are filled with intellectual challenges and mathematical explorations. By noon, everyone is ready for a break. After lunch there is time to hang out over ice cream or cappuccino, to browse in the original Borders’ bookstore, or to settle a bet in the dedicated Math Scholars computer labs. Around 1:00pm, participants regroup in the classrooms and labs with the advanced mathematics grad students who are helping with their projects. Mathematical research, punctuated by a break for some cold drinks and perhaps a Frisbee session, dominates the afternoon. At 4:00 pm, students head to the gym, the pool, the playing fields, or the dorm. Dinner begins around 5:00 or 5:30pm (unless there is a picnic tonight). Afterwards, there will be an outing–perhaps to the water park or the UM ice rink, and later to Top of the Park, part of the Ann Arbor Summer Festival. However, there will also be a group of students who prefer to hang out in the dorm discussing life and mathematics, and one which persuades a counselor back to the computer lab for yet another late night session. Math Scholars Faculty and Staff Dr. Martha Aliaga is an Associate Professor in the Department of Statistics at the University of Michigan in Ann Arbor. After receiving an undergraduate degree in mathematics from the University of Buenos Aires and a master’s degree in statistics, she taught statistics at various universities in Latin America before coming to Ann Arbor to begin her doctoral studies. Not only did Martha receive her Ph.D. from Michigan in 1986, but her husband and their three children each have degrees from Michigan also. Martha is coauthor of the popular and widely acclaimed college statistics text "Interactive Statistics" and was recently named a Fellow of the American Statistical Society in honor of this accomplishment. Dr. Tim Callahan received his B.A. in physics from the University of Chicago in 1989 and his Ph.D. in physics from the University of California at Berkeley in 1998, having spent five years studying particle theory before switching to dynamical systems. He is interested in the kinds of patterns that can arise when a system spontaneously breaks it symmetry. He has lived, at various times, in Kansas, New York state, St. Louis, Zagreb, Chicago and Berkeley. When not enjoying a book by Wilkie Collins or Stanislaw Lem, he can sometimes be spotted overhead, piloting a Cessna from the University of Michigan's local flying club. Dr. Dick Canary grew up in Racine, Wisconsin, graduating from Walden III Alternative High School in 1980. He received a B.A. from the New College of USF and an M.Sc. from the University of Warwick in 1985, and a Ph.D. from Princeton University in 1989. He was an Assistant Professor at Stanford University before coming to the University of Michigan, Ann Arbor in 1991, where he is now an Associate Professor in the Department of Mathematics. His research concerns the geometry and topology of 3-manifolds. His wife, Marybeth King, is a baker at the Moveable Feast. He has two children, a 12 year old daughter Becky and a 15 year old son Ian. He is an avid, but terrible, golfer. Dr. Brent Carswell lived most of his life in Northern New York State. He grew up in a small town near the Adirondack Mountains and received his undergraduate and master’s degrees from Potsdam College. Brent received his Ph.D. from the University of Albany in 1999 specializing in the field of Complex Analysis and is now an Assistant Professor in the Department of Mathematics at the University of Michigan in Ann Arbor. Brent’s non-mathematical interests are quite diverse, but he is most serious about volleyball. In addition to being an avid player, he can often be spotted as a line judge at University of Michigan games. Dr. Carolyn Dean is the director of Math Scholars. She is also very involved in the University’s various outreach efforts. Carolyn received her Ph.D. from the University of California, San Diego. Before moving to Michigan together, Carolyn and her husband juggled her job at the University of Chicago and his job in England as creatively as possible (and with the kind assistance of their respective chairmen). Carolyn enjoys hiking, sailing and cooking, and can often be found during lunch playing bridge in the Math Commons Room with Mel Hochster. Dr. Phil Hanlon is Professor of Mathematics and Thurneau Professor in the College of Literature, Science and the Arts at the University of Michigan in Ann Arbor. He grew up in the small village of Gouverneur, located near the St. Lawrence River in northern New York State. Phil received his undergraduate degree from Dartmouth and his Ph.D. from Caltech. After postdoctoral positions and fellowships at MIT and Caltech, he joined the faculty at Michigan in 1986. Math Scholars itself was conceived by Phil as a means of exposing talented students to some of the most exciting work being done in mathematics while they are still in high school, and he is Executive Director of the program. Phil is married and has three children, ages 8, 13 and 15. He is a sports enthusiast who particularly enjoys running, squash and golf. He is also an ardent fan of the Michigan Wolverines and a member of the U of M Athletic Board. Dr. Mel Hochster is the Robert W. and Lynn H. Browne Professor of Science and Professor of Mathematics at the University of Michigan in Ann Arbor. He is also a member of the National Academy of Science. Mel attended Stuyvesant High School in New York City, during which time he won second place in the Westinghouse Science Talent Search with a project on Fibonacci numbers. After completing a B.A. at Harvard and a Ph.D. at Princeton, he taught at Minnesota and Purdue before moving to Michigan. Mel has a grown son (with a Ph.D. in statistics from Stanford), a 9 year old daughter and 4 year old triplets, which means that he and his wife rarely find themselves with nothing to do. When time permits, he enjoys relaxing by playing bridge and doing cryptic crossword puzzles. Dr. Ruth Lawrence is an Associate Professor of Mathematics at the University of Michigan in Ann Arbor. After receiving her Ph.D. from Oxford University in 1989, she was a Junior Fellow of the Harvard Society of Fellows before coming to the University of Michigan. Her research focuses on geometry, topology and physics, and she is the faculty mentor of the UM Undergraduate Mathematics Society. Dr. Krishnan (Ravi) Shankar received his Ph.D. from the University of Maryland, College Park, in 1999 under the direction of Karsten Grove. . He is now an Assistant Professor in the Department of Mathematics at the University of Michigan, Ann Arbor. His thesis studied the curvature and topology of smooth objects in higher dimensions. His passions, other than mathematics, are ballroom and swing dancing, flamenco music and movies. He also enjoys swimming, squash, fine cooking and travelling. Mr. Cornelius Wright works in the Undergraduate Math Program Office and maintains the various Web sites run by the UM Mathematics Department. He also does the administrative work for Math Scholars, and his expertise and friendliness are keys to the success of the program. Cornelius is a native of Detroit and a former U of M student. He is an excellent athlete and a serious R&B aficionado. Some Recent Math Scholar Student Evaluations "The counselors were amazing…I really enjoyed the program and I hope I will have the chance to participate next year. - Paul Johnson, MMS 1999 "I'd just like to say that this camp was probably the most enjoyable thing I've ever done. I'm really sad I have to leave." - Josie Clowney, MMS 1998, 1999 "The counselors were great, most of the kids were great–a phenomenal experience." - Kevin Cody, MMS 1998, 1999 "Teachers rocked!" - Laurie Rich, MMS 1997 "I liked math before, but this really showed how math could be used as an engineer." - Ryan Stevens, MMS 1999 "Before, math was something I was very good at, but would bore me if I did much of it. I now see it as a very interesting and deep subject that I truly enjoy." - Betsy Huebner, MMS 1997, 1998 "My interest in mathematics was large before camp. My interest after camp was immense…Infinity rules!" - Eric Jankowski, MMS 1999 |
