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Michigan Math
& Science Scholars Summer Program 1999
General Information
Travel to and from Ann Arbor
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: Session I - Dr. Ali Naddaf, Graham Denham, Session II - Dr. Phil Hanlon Error Correcting Codes Session II (July 4 - July 17) In our society, almost everything is identified by number: books have ‘ISBNs’ on the back, products in the supermarket have ‘UPC’ barcodes, and all post offices in the United States have ZIP codes, just to list a few examples . With all of these numbers running around, it is amazing that there are not more errors made during their input or transmission. Or is it? In this course, we will see that if numbers used for identification are chosen carefully using some easily defined mathematical procedures, errors made can be detected or even corrected automatically! Such procedures are examples of error correcting codes. We will discuss several examples of error correcting codes, and this will lead us to the topics of modular addition and multiplication of integers, matrices, vectors, and probability theory. We will study methods of optimizing codes for error correcting effectiveness and speed of information transfer. Faculty: Professor Darin Stephenson Foundations of Number Theory Session II (July 4 - July 17) "God made the integers. All the rest is the work of man." -Leopold Kronecker "Numbers are free creations of the human intellect. They serve as a means of grasping more easily and more sharply the diversity of things." -Richard Dedekind Which perspective do you share? This course will focus on the development of the notion of numbers. We shall define natural numbers, integers, real numbers and complex numbers, as well as arithmetical operations on them, without appealing to their intuitive meaning as lengths of segments or points in plane. We shall answer the natural question of how to create further number systems by constructing some exotic numbers such as the quaternions, the Cayley octonions and the p-adic numbers. You might wonder why we will do all of this! We will see that these number systems give solutions to some of the most ancient and difficult problems in all of mathematics, many of which first appeared in the book "Arithmetica" written by the Greek mathematician Diophantus in the third century A.D.. The recent, spectacular solution of Fermat's Last Problem by Andrew Wiles, which has been the subject of much publicity on the PBS series "Nova", as well as in Scientific American and the New York Times, is the latest achievement in this oldest area of mathematics. This course will help you to understand the ancient fascination with the integers, which drove Dr. Wiles to work for seven years on his magnificent proof. Faculty: Professor Igor Dolgachev From the Depths of the Ocean to the Edge of the Universe: Geometry, Mathematics and the World Around Us Session I (June 20 - July 3) Is there a prehistoric monster lurking in the peat-stained waters of Loch Ness? How far across is the universe? How heavy is a beam of light? What do garden snails have in common with the Parthenon in Greece? Throughout the ages, mathematics has provided incredible tools for understanding how our world works. In this course we will examine ways in which mathematics interacts with other sciences such as ecology, physiology, physics, cosmology and chemistry to provide sensible answers to seemingly impossible questions. A key concept for our work will be the notion of non-Euclidean geometry. This, the most shocking mathematical development of the nineteenth century, is the geometry in which all of Euclid’s postulates hold except for the Parallel Postulate. Non-Euclidean geometry dramatically transforms our view of the world, giving us, for example, the theory of relativity. Faculty: Professor Dale Winter Mathematical Modelling Session I (June 20 - July 3) We are often interested in analyzing complex situations to predict qualitatively and/or quantitatively the effect of some course of action. In mathematical modeling, we attempt to describe the behavior of these physical systems via mathematical devices or models. These models may use mathematical formulae or equations, graphs, physical experiments, and simulations. To construct a model, we first study the problem. Then, we determine the factors (variables) which influence the problem and select those which we think are necessary and vital for our model to produce results that are close to reality. We do not include all the possible factors since the effect of the variable may be small compared with other factors, or because it leads to a system that is too complex to handle at the current level of sophistication. The relationships among the variables, determined by empirical measurements, theory, fact or assumption, lead to the actual model. After verifying the model, we can then implement it. The success of the model is determined by the soundness of principles from which it is derived, the accuracy of the initial data, and the mathematical nature of the problem itself. In the course, we will develop some models and conduct some simulations for problems like the following:
Faculty: Professor Suzanne Weekes Mathematics and DNA Session I (June 20 - July 3) & Session II (July 4 - July 17) While most of use know that the blueprints for life are carried in very long double-stranded DNA molecules, few are aware of the interesting and challenging mathematics used now in working with DNA, and the mountains of data which biologists are accumulating about it. We will cover two important mathematical topics: "Probability and DNA," which will start with some basics about the biology of DNA, then examine some basic probability and see how it is used to evaluate information in the DNA data bank; and "The Geometry of DNA," which will examine the wonderful geometry of knots and the role they play in DNA biology. We will carry out many hands-on and Web-based computer experiments. Faculty: Session I - Professor Dan Burns Session II - Dr. Carolyn Dean The Nature of Infinity Session I (June 20 - July 3) & Session II (July 4 - July 17) 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 - Professor Jeremy Tyson Session II - Professor Andreas Blass & Professor Ken Plochinski A Day in the Life of a Michigan Math Scholar 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 & Staff Professor Andreas Blass received his Ph.D. from Harvard in 1970. He counts among his career highlights the fact that he won the Michigan Math Prize Competition and won the Putnam Fellowship for graduate study at Harvard. This fellowship is awarded annually to one of the top finishers (no winner is ever announced) of the Putnam Exam, a national competition for undergraduate mathematicians. Andreas is currently the Associate Chair for Graduate Studies in the U of M Math Department. Professor Dan Burns taught math at the Boston Latin School and took his Ph.D. at MIT. Dan directs the Math Department’s NSF Research Experiences for Undergraduates program. He volunteers with Habitat for Humanity and is an accomplished amateur vocalist. Dr. Carolyn Dean is the Director of Math Scholars. She is also very involved in the University’s statewide outreach effort. She enjoys hiking, sailing, and cooking, and can often be persuaded to put aside her work when a chance to play bridge arises. Professor Igor Dolgachev received his Ph.D. in 1970 from Moscow State (USSR). He has been on the faculty at U of M since 1978. In addition to being his profession, Mathematics is Igor's favorite hobby. He is very involved with the Honors undergraduate math program, chair’s the department’s personnel committee, and is an editor on the Michigan Mathematical Journal. Professor Phil Hanlon was born and raised in the small village of Gouverneur located near the St. Lawrence River in northern New York. He received his Ph.D. from Cal Tech, and in 1986 he joined the faculty at the U of M. Math Scholars was conceived by Phil as a means of exposing talented students to some of the most exciting work in Mathematics while they are still in high school. He is Executive Director of the program. He is married with three children ages 7, 12 and 14. He is a sports enthusiast who particularly enjoys squash, golf and running. He is also an ardent fan of the Michigan Wolverines and a member of the U of M Athletic Board. Professor Ali Naddaf was born in Iran and grew up in th> Transfer interrupted! m Courant Institute at New York University. Ali spent one year of postdoctoral training in Vancouver, Canada, before coming to U of M in 1996. His hobbies are reading classical literature, listening to soft or classical music, playing racquetball, working with computers (either programming or hacking), and electronics. Mr. Adam Parker is the Head Counselor for Math Scholars. Adam is an Honors Math major who will graduate in May and plans to begin doctoral studies in the Fall. Last year, Adam was a counselor for Math Scholars. Students were full of praise for his mathematical, organizational, and social leadership. We are very glad to announce that this year Adam will be Head Counselor as well as a Teaching Assistant for Math Scholars. Professor Ken Plochinski attended the U of M where he completed a rare triple play - earning undergraduate and graduate degrees and serving on the faculty. In 1991 he moved to Seattle and the University of Washington where he founded and directs the Math Study Center. When he's not busy doing or teaching math you can often find him out on the water racing and cruising in sailboats of all sizes. Professor Darin Stephenson received his Ph.D. in mathematics from U of M in 1994. He is an Assistant Professor at Hope College. He lives in Holland, Michigan with his wife and their 2 year old son. When he is not doing math or playing with his son, Darin enjoys singing, reading science fiction, playing tennis and golf, hiking, and statistically simulating major league baseball games. Mr. Jeremy Tyson grew up in the small college town of Blacksburg, VA. He received his bachelor’s degree from Washington University in St. Louis, and will receive his Ph.D. in Mathematics from U of M in 1999. His research is closely related to fractals, which will be a major theme of the course he teaches. Jeremy enjoys playing chess, bridge and Ultimate Frisbee. He is also an avid piano player. Professor Suzanne (Suzy) Weekes is a professor of Mathematical Sciences at Worcester Polytechnic Institute. Originally from the twin island republic of Trinidad and Tobago, she holds a Ph.D. from U of M and an undergraduate degree from Indiana University. Suzy’s research is in the area of Computational Fluid Dynamics, involving simulating the flow and movement of substances. Suzy enjoys listening to music, dancing, doing aerobics, reading and eating fine foods. Professor Dale Winter was born in New Zealand, and raised in New Zealand and Australia. He received his Ph.D. in mathematical physics in 1998 from U of M. He is currently on the faculty of Duke University. During our Summer 1997 and 1998 Math Scholars programs, Dale was the very popular Graduate Student Assistant for this course, and we are especially pleased that he is returning to Math Scholars now to teach it. Mr. Cornelius Wright is the first person to greet most U of M undergraduate math students. His friendliness and administrative expertise are keys to the success of Math Scholars. A former U of M student, Cornelius is an ardent athlete and a serious R & B aficionado. Some 1997-1998 Math Scholar Student Evaluations "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 "The counselors were great, most of the kids were great a phenomenal experience." -Kevin Cody "Now I realize how big the math world is and I want to learn more about it." -Ashley Hill "Prior to the program I was moderately to very interested in mathematics. Now I am extremely interested." -John DeWitt "Dale was awesome! Adam was really fun, cool and down to earth." -Anonymous "The program was extremely enjoyable. I am interested in participating again next year." -Matt Turk "The counselors were great." -Mike Roberson "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 "Teachers rocked!" -Laurie Rich
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