The Michigan Math and Science Scholars are proud to feature courses from various departments in the College of Literature, Science and the Arts. These courses are unique in that enrollments are limited to approximately 15 students per class, thus making the ratio of students to faculty very low and allowing for an intense, fun and memorable learning experience. Go here to see the courses sorted by session.

 

BIOLOGY

Explorations of a Field Biologist (Session 2)
Instructor: Sheila Schueller

There are so many different kinds of living organisms in this world, and every organism interacts with its physical environment and with individuals belonging to both its own and to other species. Understanding this mass of interactions and how humans are affecting them in the short and long-term is a mind-boggling endeavor! We cannot do this unless we at times set aside our computers and beakers, and, instead, get out of the lab and classroom and into the field - which is what we will do in this course. You will not only learn a slew of natural history factoids, but also become well-acquainted with the how's and why's of practicing field biology, from making observations to hypothesis-testing in the field. We will address a myriad of questions from what mushrooms can tell you about trees, to how big a chipmunk territory is, to whether a flower shape matters to bees, to how lakes turn into forests, to how plants care for their young...to whatever questions arise in our explorations of the grasslands, forests, and wetlands of southeastern Michigan or in our discussions of other parts of the world. Learning this approach to addressing questions about what you see in nature will allow you, long after this class, to discover many more things about the natural world, even in your own back yard.

In many ways, the course is inspired by my reading of the autobiography of an exemplary field biologist, E.O. Wilson. Whether he is in his own back yard or in a tropical rain forest, Wilson observes nature with patience, diligence and an inquisitive mind. His career shows the value of making field observations and notes and of asking questions - and that is exactly what we practice in "Explorations of a Field Biologist." Note: This class is NOT for students who prefer to be indoors!

Genes to Genomics (Session 1)
Instructor: Santhadevi Jeyabalan

This course (formerly known as Genome Sequences and Human Health) will cover basic aspects of Mendelian and molecular genetics, and then focus on a few human disorders in detail. We will introduce human genome sequence as an aid to cloning the genes responsible for these disorders, and will demonstrate how comparative genomics allows genetic diseases to be studied using model organisms like yeast, Drosophila, nematodes and mice.

In addition to lectures, we will work with fruit flies, yeast and will do human DNA fingerprinting in a genetics laboratory. We will use computers to carry out molecular and phylogentic analysis of DNA sequences. This class should provide an understanding of Human Genome Projects and recent advances in medicine.

The Mysteries of Embryology (Session 1)
Instructors: Kathryn Tosney and Kenneth Balazovich

Embryology is the science dealing with the formation, early growth and development of living organisms. This course addresses the mysteries and processes of embryology. Questions that will be investigated include: Why do babies move in the womb? Why does the lens fit the eye? Why do all human hearts have a hole in them at birth and why isn't it lethal? Why would exposure to a particular chemical throughout the majority of pregnancy have no effect, whereas exposure during the first month of pregnancy could cause severe birth defects?

This course examines the fundamentals of embryology during lecture and through experiments with chicken embryos (before central nervous system development and the ability to feel pain) in laboratory. We will be working in groups to research how disrupting development mechanisms causes birth defects and then post our findings on the world wide web.

CHEMISTRY

Cars, Energy and Chemistry (Session 1)
Instructor: Paul Rasmussen

Cars consume a very significant fraction of the total energy demand in United States. How does the future look with respect to this energy demand and future availability? What other possibilities are there for vehicle energy? Are hybrids, fuel cells, or hydrogen going to solve the problem? How can an understanding of chemical principles help us make the necessary choices? These are questions that can be approached by studying some fundamental chemistry, such as the heat evolved in combustion, and the reactions which convert one fuel to another. This course will also show how molecules, moles and the very big numbers associated with energy consumption, can be brought together.

Surface Chemistry (Session 2)
Instructor: Zhan Chen

This course will be divided into three units: applications, properties, and techniques. The first unit will introduce students to surface science that exists within the human body, surfaces in modern science and technology, and surfaces found in everyday life. Our bodies contain many different surfaces that are vital to our well-being. Surface reactions are responsible for protein interaction with cell surfaces, hormone-receptor interactions, and lung function. Modern science has explored and designed surfaces for many applications: anti-biofouling surfaces are being researched for marine vessels; high temperature resistant surfaces are important for space shuttles; and heterogeneous catalysis, studied by surface reactions, is important in industry and environmental preservation. The usefulness of many common items is determined by surface properties: contact lenses must remain wetted; while raincoats are designed to be non-wetting; and coatings are applied to cookware for easy clean-up.

The second unit will examine the basic properties of surfaces. Lectures will focus on the concepts of hydrophobicity, friction, lubrication, adhesion, wearability, and biocompatibility. The instrumental methods used to study surfaces will be covered in the last unit. Traditional methods, such as contact angle measurements will be first covered. Then vacuum techniques will be examined. Finally, molecular level in situ techniques such as AFM and SFG will be covered, and I will arrange for students to visit them in our lab.

Multimedia powerpoint presentations will be used for all lectures. By doing this, I hope to promote high school students' interest in surface science, chemistry and science in general. A website introducing modern analytical chemistry in surface and interfacial sciences will be created.

MATHEMATICS

Combinatorial Combat (Session 2)
Instructor: Mort Brown

We will play and analyze a variety of two person competitive strategic games. All of these games will have simple rules, perfect information (i.e., you know everything that has happened at each point in the game, no luck involved), win lose or draw , and be interesting and challenging. We will study such notions as” formal strategy”, “game tree” and “natural outcome” and investigate methods of solving some of the games. We'll study Hex and a related game Y and an unrelated game “Poison Cookie” and examine John Nash's (he of the Beautiful Mind) proof that the first player has a winning strategy in each of these games even though nobody knows what the winning strategy actually is! Games such as Nim and Fibonacci Nim will allow us to see how some good math can be used to solve many games. Other games such as “Dwarfs and Giants”, “Fox and Geese”, and Dodgem will be analyzed without any mathematics but good logical skills. We'll also come across many games that are “isomorphic” (i.e., the same) even though they do not seem to be at first. If you ever played games such as Tic Tac Toe, Connect Four, Othello, Gomoku (Pente), Chess and Checkers; this course is right for you. However, no previous knowledge of any specific game is required.

Games and Geometry (Session 1)
Instructor: Stephen DeBacker

In this course, we shall look at the game of Hex, the geometry of disks, and an interesting connection between them. The game of Hex is an easy to describe board game for two players (Google “Hex game” to find a description). The game has many interesting features. For example: one of the two players must win, the first player to move should (theoretically) win, nobody knows a strategy to guarantee that the first player wins. The geometry of the disk is also interesting. For example: if we imagine that the disk is a piece of paper which we can pick up, crumple up, and then set back on itself, then at least one point in the crumpled piece of paper must lie exactly above the location where it started; this is called the Brouwer fixed point theorem. In this course, we will explore the mathematics required to understand why every game of Hex has a winner and why this is the same as the Brouwer fixed point theorem.

Codes, Ciphers and Secret Messages (Session 2)
Instructor: Carolyn Dean

Around 4000 years ago in Egypt, the tomb of 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 funerary inscription unintelligible except to those who knew what replacement the scribe had made. This inscription is the oldest surviving record of the use of secret codes.

Nowadays, you can order goods over the Internet with a credit card. When you forward your credit card number over the network, powerful public key encryption schemes put this information in a form that is 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 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 Internet vendors. This study of secret codes introduces topics in discrete mathematics, statistics, information theory and number theory. 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. Most importantly, students will get plenty of practice creating their own codes and trying to crack those of their instructor and their classmates.

Mathematical Modeling in Biology (Session 2)
Instructors: Trachette Jackson

Mathematical biology is a relatively new area of applied mathematics and is growing with phenomenal speed. For the mathematician, biology opens up new and exciting areas of study while for the biologist mathematical modeling offers another powerful research tool commensurate with a new instrumental laboratory technique. Mathematical biologists typically investigate problems in diverse and exciting areas such as the topology of DNA, cell physiology, the study of infectious diseases such as AIDS, the biology of populations, neuroscience and the study of the brain, tumor growth and treatment strategies, and organ development and embryology.

This course will be a venture into the field of mathematical modeling in the biomedical sciences. Interactive lectures, group projects, computer demonstrations, and laboratory visits will help introduce some of the fundamentals of mathematical modeling and its usefulness in biology, physiology and medicine.

For example, the cell division cycle is a sequence of regulated events which describes the passage of a single cell from birth to division. There is an elaborate cascade of molecular interactions that function as the mitotic clock and ensure that the sequential changes that take place in a dividing cell take place on schedule. What happens when the mitotic clock speeds up or simply stops ticking? These kinds of malfunctions can lead to cancer and mathematical modeling can help predict under what conditions a small population of cells with a compromised mitotic clock can result in a fully developed tumor. This course will study many interesting problems in cell biology, physiology and immunology.

Fibonacci Numbers (Session 1)
Instructor: Mel Hochster

Sample problems available here!

The Fibonacci numbers are the elements of the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... , where every term is simply the sum of the two preceding terms. This sequence, which was originally proposed as describing the reproduction of rabbits, occurs in astonishingly many contexts in nature. It will be used as a starting point for the exploration of some very substantial mathematical ideas: recursive methods, modular arithmetic and other ideas from number theory, and even the notion of a limit: the ratios of successive terms (e.g., 13/8, 21/13, 55/34, ...) approach the golden mean, already considered by the ancient Greeks, which yields what may be the most aesthetically pleasing dimensions for a rectangle. As one by-product of our studies we will be able to explain how people test certain very special but immensely large numbers for being prime. We'll also consider several games and puzzles whose analysis leads to the same circle of ideas, developing them further and reinforcing the motivations for their study.

 

Note: Due to the popularity of Financial Mathematics in the Summer of 2004, the MMSS program is offering 2 sections in the Summer of 2005!!

 

Fortunes Made and Lost: Financial Mathematics (Session 1) Section 1
Instructor: David Kausch

In the last 30 years there has been an explosion of research in the field of financial mathematics. In 1973, three economists pioneered a revolution in finance and went on to win the Nobel Prize. The revolution started with a deceptively simple formula for calculating the fair price of a stock option. Elegant, concise, and best of all practical, the Black-Scholes formula quickly became standard knowledge for academics and options traders alike. Because of the equation, trading stock options went from a guessing game to a science virtually overnight.

Financial options now make up a multi-trillion dollar industry worldwide. The field of financial mathematics has grown to a science with applications in asset management, risk assessment, sophisticated financial transactions, and insurance. The theory helps us create optimal portfolios, model expected outcomes, and protect ourselves from financial disasters through insurance and hedging.

During the morning sessions we will learn how stocks, stock options, and markets behave. We will discuss the relationship between risk and reward, between complete markets and arbitrage, and make distinctions between speculation and hedging. We will develop the Black-Scholes option pricing model and explore applying the model to more general financial transactions. The use of technology and real world financial data will be emphasized. In the afternoon sessions, each student will create an electronic portfolio starting with an imaginary balance of $10,000 at the beginning of the session. Students can trade securities each day in a make believe account. During the session students will also have modeling assignments that may be used as aid in their investment strategies. Outcomes will be compared at the end of the session in good humor and with reflection on what we have learned.

 

Fortunes Made and Lost: Financial Mathematics (Session 1) Section 2
Instructor: Mattias Jonsson

Is it possible to predict stock prices in the future or are they truly random? Can mathematics make you a millionaire by telling you how to invest in stocks or options? What are options (and futures, bonds, swaps...) anyway?

I this course we will try to answer these and other questions. There will be three interacting components. First, we will try to understand randomness using the mathematical tool of probability theory.
For example, we will be able to use (computer generated) coin tossing to produce graphs that behave much like the "Dow". Second, we will study some of the products and lingo of the financial markets and learn how to read the Wall Street Journal. Third, there will be some hands-on trading (although not with real money!) with a prize awarded to the most successful trader.

The Nature of Infinity (Session 1)
Instructor: Patrick Nelson

If only "half" of the integers are even, how come the sequence 2, 4, 6, .... seems to have just as many elements as the sequence 1, 2, 3, ...? In a similar spirit, what "fraction" of the integers are prime numbers? How can I walk one mile if I first have to walk 1/2 mile, and to do that 1/4th of a mile, and so on without end? Can I even get started? Is it possible to eat a whole cookie by taking successive bites of half of what remains? Do parallel lines intersect "at infinity", and what can that possibly mean?

Such paradoxical questions about infinity have been the source of much mathematical research. In this course we'll explore a number of significant ideas and results motivated by these questions. We will study the notion of cardinality of infinite sets, and we'll re-discover Cantor's theorem on the existence of infinities greater than others. We will then look at large quantities (functions), like the number of primes less than some number x. The question of the "size" of such functions will lead us to notions of asymptotic analysis. We'll take a serious look at the real numbers and ways in which irrational numbers arise from infinite processes. This will lead us to notions of dynamical systems and chaos. Finally, we'll consider geometric notions of infinity: fractal sets and non-Euclidean geometries. It will be fun, challenging, and sometimes slightly dizzying!

 

PHYSICS

The Physics of Magic and the Magic of Physics (Session 2)
Instructors: Frederick Becchetti and Georg Raithel

Rabbits that vanish; objects that float in air defying gravity; a tiger that disappears and then reappears elsewhere; mind reading, telepathy and x-ray vision; objects that penetrate solid glass; steel rings that pass through each other: these are some of the amazing tricks of magic and magicians. Yet even more amazing phenomena are found in Nature and the world of physics and physicists: matter that can vanish and reappear as energy and vice-versa; subatomic particles that can penetrate steel; realistic 3-D holographic illusions; objects that change their dimensions and clocks that speed up or slow down as they move (relativity); collapsed stars that trap their own light (black holes); x-rays and lasers; fluids that flow uphill (liquid helium); materials without electrical resistance (superconductors).

In this class students will first study the underlying physics of some classical magic tricks and learn to perform several of these (and create new ones). The "magic" of corresponding (and real) physical phenomena will then be introduced and studied with hands-on, minds-on experiments. Finally, we will visit a number of research laboratories where students can meet some of the "magicians" of physics --- physics students and faculty --- and observe experiments at the forefront of physics research.

Roller Coaster Physics (Session 2)
Instructor: David Winn

What are the underlying principles that make roller coasters run? By studying the dynamics of roller coasters and other amusement park rides in the context of Newtonian physics and human physiology, we will begin to understand these complex structures. Students first review Newtonian mechanics, a=F/m in particular, with some hands-on experiments using air-tables. This will then be followed by digital-video analysis of motion of some real-life objects (humans, cars and toy rockets) and student-devised roller coaster models. Then, a field trip to Cedar Point Amusement Park where students riding roller coasters and other rides, will provide real data for analysis back at UM. Students will carry portable data loggers and 3-axis accelerometers to provide on-site, and later off-site, analysis of the motion and especially, the "g-forces" experienced. The information collected will then be analyzed in terms of human evolution and physiology and why these limit the designs of rides.

In addition to exploring the physics of roller coasters, cutting-edge topics in physics will be introduced: high energy physics, nuclear astrophysics, condensed-matter theory, biophysics and medical physics. Students will then have a chance to see the physics in action by touring some of UM's research facilities. Students will have the opportunity to see scientists doing research at the Solid State Electronics Laboratory, the Michigan Ion Beam Laboratory and the Center for Ultrafast Optical Science.

By the end of the two-week session, students will gain insight and understanding in physics and take home the knowledge that physics is an exciting, real-life science that involves the objects and motions surrounding us on a daily basis.

STATISTICS

Sampling, Surveys, Monte Carlo and Inference (Session 1)
Instructor: Edward Rothman

Political candidates drop out of elections for the U.S. Senate and New York Governor because their poll numbers are low, while Congress fights over whether and how much statistics can be used to establish the meaning of the U.S. Census for a host of apportionment purposes influencing all our lives.

Suppose we need to learn reliably how many people have a certain disease? How safe is a new drug? How much air pollution is given off by industry in a certain area? Or even how many words someone knows?

For any of these problems we need to develop ways of figuring out how many or how much of something will be found without actually tallying results for a large portion of the population, which means we have to understand how to count this by sampling.

And Monte Carlo? Sorry, no field trips to the famous casino! But Monte Carlo refers to a way of setting up random experiments to which one can compare real data: does the data tell us anything significant, or is it just random noise? Did you know you can even calculate the number p this way?!

The survey and sampling techniques that allow us to draw meaningful conclusions about the whole based on the analysis of a small part will be the main subject of this course.

 

The Michigan Math and Science Scholars
The University of Michigan
2082 East Hall
530 Church Street
Ann Arbor, MI 48109-1043

Voice 734.647.4466
Fax 734.763.0937
mmss@umich.edu