The Michigan Math and Science Scholars program is 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 that are still open.

ASTRONOMY

Climbing the Distance Ladder: How Astronomers Survey the Universe (New Course!) (Session 1)
Instructor: Philip Hughes

The furthest objects that astronomers can observe (the nuclei of active
galaxies) are so distant that their light set out when the Universe was only
800 million years old, and has been traveling to us for about 13 billion
years -- most of the age of the Universe. Even the Sun's neighborhood --
the local part of our Galaxy, where astronomers have successfully searched
for planets around other stars -- extends to hundreds of light years. How
do we measure the distance to such remote objects?

Certainly not in a single step! Astronomers construct the so-called
"Distance Ladder", finding the distance to nearby objects, thus enabling
those bodies to be understood and used as probes of yet more distant
regions. The first step on this ladder is to measure the distance to the
planet Venus, by timing the round-trip for radar waves bounced from its
surface. Knowing the laws of planetary motion, we can use this information
to get the distance to the Sun. Knowing the distance to the Sun, we can use
the tiny shift in the apparent position of nearby stars as the Earth orbits
during a year, to "triangulate" the distance to those stars. And so on out:
through the Galaxy, the local group of galaxies, the local supercluster,
to the furthest reaches of what is observable.

This class will explore the steps in this ladder, using lectures,
discussions, demonstrations, and computer laboratory exercises. We will make
frequent digressions to explore the discoveries made possible by knowing
the distance to objects (such as gamma-ray bursters and the acceleration
of the Universe's expansion), and get hands-on experience of using distance
as an exploratory tool, by using a small radio telescope to map the spiral
arm pattern of our own galaxy, the Milky Way.

This online course, the companion to an earlier MMSS course, Large Scale Structure of the Universe, gives the flavor of the material to be covered.

CHEMISTRY

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

Cars consume a very significant fraction of the total energy demand of the 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.

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 covered first. 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 toobserve these techniques 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.

ECOLOGY AND EVOLUTIONARY BIOLOGY

Dissecting Life: Human Anatomy and Physiology (Session 1)
Instructor: Glenn Fox

Dissecting Life will lead students through the complexities and wonder of the human body. Lecture sessions will cover human anatomy and physiology in detail. Students will gain an understanding of biology, biochemistry, histology, and use these as a foundation to study human form and function.

Laboratory sessions will consist of first-hand dissections of a variety of exemplar organisms: lamprey, sharks, cats, etc. Students may also tour the University of Michigan Medical School’s Plastination Laboratory where they can observe human dissections.

GEOLOGICAL SCIENCES

Nano-Rocks and Mega-Molecules (Session 2)
Instructor: Udo Becker

Did you ever wonder what the ingredients are that rocks and minerals are made of, what causes the color of a certain mineral, or how we know how old a certain mineral is? After learning the basics of what these ingredients are and what properties minerals have, we will look into their properties at a large scale (cm to miles), small scale (what you can see with a microscope), and submicroscopic or atomic scale. Some features that we see at the atomic scale reveal the age of a mineral, some reveal why they are magnetic, dissolve or grow. Students will be introduced to techniques that can be used to analyze microscopic properties that tell us something about reactivity in the environment, age, radioactive decay, color, magnetic properties and so on. In fact, we will not only look at rocks but use the same techniques to examine similar properties of magnetic tapes, CDs, bugs etc. Thus, we will go back and forth between the scale that you are used to (rocks as building materials, mines …) and one that is about a billion times smaller (atoms and the nanoworld).

Some examples:
Some mineral hand samples look perfectly crystalline, though we can visualize traces of radioactive decay (invisible to the naked eye), which is one method to determine the age of the rock.

You may think that a piece of rock in your hand may just sit there and not do anything. However, even at room temperature, atoms are wiggling about quite vigorously.

Crystals can have beautiful symmetry – can we find the same symmetry in our daily life?

The ability to observe the science that goes on at the “nanometer” scale will be demonstrated by an atomic force microscope (AFM) and a scanning electron microscope (SEM). For scale, the diameter of a human hair is about 50,000 nanometers!!!!  These techniques can be used to image everything from DNA to microbes to carbon nanotubes.  We will demonstrate the use of AFM to image nano-structures that are used (and taken for granted) in our everyday life, like the bumps on a compact disk or the magnetic domains on a videotape to store music and video. Using an SEM, we will take a closer look at things in the environment like the surface of a rock or the face of a beetle or the holes on the underside of a leaf that a tree uses to breathe.

What do amethyst, carnelian, tiger eye, and citrine have in common? They are all names for different types of quartz and you should not be surprised that this is but a small fraction of the variety available as quartz is made from the two most abundant elements in the Earth's crust, oxygen and silicon. Do you know what causes quartz to have so many colors? Quartz is not the only mineral that comes in to come in a mixture of colors. Sapphire and ruby belong to the same group of aluminum oxide minerals. The texture and color of a mineral can tell us about what the local environment was like from thousands to millions of years ago. The Earth has had thousands of millions of years to make all sorts of interesting rocks, minerals, mountains, and caves for us to explore. So, it is important not to take it all for granted since everything our society wants and needs comes in some form, from the Earth.

MATHEMATICS

Codes, Ciphers and Secret Messages (Session 1)
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 an 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 is 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.

Combinatorial Combat (Session 1)
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.

Fibonacci Numbers (Session 2)
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.

Fortunes Made and Lost: Financial Mathematics (Session 1)
Instructor: Michael Ludkovski

Is it possible to predict tomorrow's stock prices? Can you use mathematics to earn money through investments in a sure way? Is the financial world truly random? We will explore these and other questions by discussing the applications of probability theory to financial markets. We will discuss both the financial lingo you might encounter in the Wall Street Journal (from hedge funds to put options to futures), and the mathematics behind it. The course will end with the Black-Scholes formula that revolutionized trading stock options and earned its authors the Nobel prize. In the meantime, we will build optimal portfolios, analyze risk, discuss arbitrage and understand the benefit of insurance. We will even talk about how you can make money from hot summer weather.

The morning sessions will involve two major components. The 'business' part will focus on the terminology and setup of financial markets and associated products. The 'math' part will explore tools of probability used in modeling randomness and obtaining quantitative results. A significant portion of time will be used for thinking about computational methods and using technology to get answers (e.g. using dice to simulate a graph of a hypothetical stock). To tie the two components together, the afternoon sessions will include having each student create an imaginary electronic portfolio of $10,000. The students will be able to trade securities each day using the virtual account and will use ideas from the mornings in their investment strategies. Outcomes will be compared at the end of the session and prizes awarded to most successful traders.

Hex and the 4 C's (formerly known as Games and Geometry) (Session 2)
Instructor: Stephen DeBacker

After a very long night of homework, you finally finish your math assignment. While double-checking your work, you realize that you have done problems from page 221, not page 212 as your teacher requested. In disgust, you rip the paper out of your notebook, wad it up, and toss it back down on your notebook. Too frustrated to begin your assignment anew, your mind begins to wander. You wonder: Is there a point in the wadded up paper that lies exactly above the location from which it started?

After you pour your parent’s morning cup of joe, the coffee comes to rest while you sleepily (because of the whole homework thing) search in the fridge for the cream. After adding and stirring the cream into the cup, you watch the pretty patterns made by the swirling coffee and cream as the contents come to rest. You wonder: Is there a point in the coffee that lies at the same point both before and after the cream was stirred in?

We shall use mathematics to model and answer the above questions. Initially, the above questions will motivate our study of four fundamental concepts in mathematics, all of which begin with the letter C: continuity (what sorts of wadding/stirring are allowed), completeness (what if our paper/coffee has “gaps”), compactness, and connectedness. Interestingly, these are also the concepts one needs in order to rigorously understand why Calculus works. Our modeling will lead us to the Brouwer fixed-point theorem; a very nice topological result.

To show that the Brouwer fixed-point theorem is true, we shall also learn about the game of Hex. 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. We will explore the mathematics required to understand why every game of Hex has a winner.

Finally, we shall stitch all of the above together by showing that the fact that there are no ties in Hex implies that there is a point in your parent’s cup of joe which lies at the same point both before and after the cream was stirred in.

This course was previously called "Games and Geometry"; however, the title has been changed because there is no high school geometry in the course and but one game.

Mathematics and the Internet (Session 2)
Instructors: Jason Howald and Mark Conger

How do online computers find each other? How does email data travel over cables designed for television signals? How can gigabytes of information move over unreliable airwaves using unreliable signaling, and arrive perfectly intact? How can I have secure communication with a website run by a person I've never met? In Mathematics and the Internet, we'll answer these questions. We'll also learn to use abstract mathematical tools for studying such systems, including graph theory, probability, logic theory, and coding theory. All of these fields, heavily studied before the internet, find new practical uses every day. For fun, we will find out how to write computer code to automatically interact with and analyze the larger internet. We also try to build a calculator and a primitive cryptographic computer out of transistors and a few other parts lying around in Mark's (the Graduate Student Instructor) garage.

Mathematical Modeling in Biology (Session 2)
Instructor: Trachette Jackson and Patrick Nelson

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.

Pythagorean Triples and Number Theory (Session 1)
Instructor: Brian Conrad

In school, everyone sees several examples of so-called "Pythagorean triples" such as (3,4,5), (5,12,13), and (8,15,17): triples of positive integers that satisfy the equation in the Pythagorean theorem and have no common factor. Some triples are less well-known, such as (57, 176, 185), so we are led to ask: are there infinitely many Pythagorean triples? Moreover, can we give a simple algebraic formula that parameterizes all of them? This course will give a complete solution of not only this problem but also of some natural generalizations, culminating in an amazing synthesis of arithmetic, algebra, and geometry unlike anything you have seen before. In this way, you will gain an appreciation for the balance between experiment and theory in mathematics, acquire experience in proving seriously non-trivial theorems, and see how apparently unrelated ideas can be combined to solve a concrete problem that has played an influential role in the development of important concepts in number theory. If you enjoy thinking about mathematical ideas outside of school and like the challenge of solving hard problems, then you should be well-motivated for this course.

We will begin by studying the totality of all solutions to both the Pythagorean equation as well as variants on it (allowing coefficients on the three quadratic terms), leading to the surprising fact that such equations of "Pythagorean type" always have a parametric formula for their solutions if we can find one solution. (This also has applications in calculus!) But how can we determine if there is even a single solution for such an equation? (For example, X^2 + Y^2 = 3Z^2 has no solution in positive integers, but X^2 + Y^2 = 5Z^2 does.) This is an incredibly subtle problem with whole numbers, hopeless to solve by random guessing, and its solution by Gauss was one of his crowning achievements when he created modern number theory 200 years ago. The methods he introduced continue to play an important role today.

Following in Gauss' footsteps, we will explore (through many examples, as well as general theory) several fundamental notions in number theory, such as congruences, continued fractions, quadratic reciprocity, and quadratic forms, and learn how they all magically fit together to settle the problem of solving all equations of Pythagorean type. The daily problem sets will provide a lot of hands-on experience with the range of ideas introduced in the course. If you have some programming experience, then at the end of the course you will enjoy the challenge of converting Gauss' method into a computer program that can find all solutions to any such equation or determine (with proof) that no such solution exists!

MOLECULAR, CELLULAR AND DEVELOPMENT BIOLOGY

Genes to Genomics (Session 1 or Session 2)
Instructor: Santhadevi Jeyabalan

This course will be offered Session 1 and Session 2. This is the same course, offered in both sessions due to its popularity! Do not sign up for it twice.

This course will cover the basic aspects of Mendelian, molecular and developmental genetics, and then focus on some human disorders in detail. We will introduce the human genome sequence as an aid to cloning the genes responsible for these disorders, and we will demonstrate how comparative genomics allows genetic diseases to be studied using model organisms, including mice, zebrafish, and nematodes.

In addition to lectures, important laboratory techniques will be performed using bacteria, yeast, fruit flies, sand dollars, and chicken embryos. Computers will be used to carry out molecular and phylogenetic analysis of DNA sequences, guest lectures will be presented, and advanced genetic technology, including gene knockouts and human DNA fingerprinting, will be investigated. This workshop should provide an understanding of the many applications of genetics and the Human Genome Project, as well as recent advances in the biomedical sciences.

PHYSICS

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.

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; we shall see 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.

PROGRAM IN THE ENVIRONMENT

Modeling Daisyworld (Session 2)
Instructor: William Currie

Students will explore the interactions between the biology of plants and the physics of climate using computer models. Not only do climatic characteristics such as sunlight, thermal radiation, and temperature affect the biology of plants, but the reverse is also true: plants, by absorbing and reflecting light energy, regulate climate. Daisyworld is a hypothetical globe covered only by populations of daisies with varying colors and physical characteristics. It is used as a computer modeling tool to explore interesting and subtle dynamics in the two-way interactions between plants and climate.

Through hands-on computer simulations, we will learn how Daisyworld works to regulate its own climate. Each morning will be spent in lecture and class discussions, incorporating illustrative computer simulations shown by the professor. Afternoons will be spent in a laboratory where students will work at their own computers to explore the rich and surprising dynamics that arise in the climate and plant populations on Daisyworld. We will consider the responses of the planet's plant-climate system to chance events such as wildfires or sudden changes in cloud cover or solar luminosity. As a final project, students will work in pairs to predict which combination of daisies would be the best mixture to cultivate to give the hypothetical planet the greatest climatic stability in the face of chance events; as a class we will then test each combination against a set of unpredictable climatic events.

This course will be taught in the Dana building on the UM central campus, home of the School of Natural Resources & Environment. The 100-year old Dana building was recently renovated as a "green" building using recycled materials and incorporating no-flush restrooms and rooftop solar panels. The building was awarded a Gold rating from the US Green Building Council through their LEEDS program (Leadership in Energy and Environmental Design), making it one of the "greenest" academic buildings in Michigan.

"Why Here?" - Reading Diverse Landscapes (Session 2)
Instructor: David C. Michener

You'll never look out the window the same way after this course! Landscapes can be 'read' for a great deal of information not evident to the untrained observer. We'll be conducting class outdoors and compare different but nearby landscapes to generate compelling questions that require field observations of various types to understand and resolve. In this field-intensive class we'll explore several University-managed natural and research areas in the Ann Arbor area to learn how to orient oneself to a landscape and begin to analyze important components. From our field work, we'll address questions about the current vegetation and its stability in time; its past site history (post-settlement, pre-settlement) and future prospects. Current issues in biological conservation will raise themselves since some of the sites have native stands of rare plant species which we'll see and try to understand "why here?" We'll work with plant identification and survey skills on-site, as well as comparing photographic documentation and then better understand the limitations of our 'gut level' reading of the sites. We'll also examine areas commonly understood as "natural" that on inspection turn out to be designed and modified by humans. This may be a springboard into your future research interests here at UM!

No prior field-work or knowledge of the local climate, flora, fauna, geology, or history is expected.

STATISTICS

Political candidates drop out of elections for the U.S. Senate and New York Governorship 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 pi 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.

Michigan Math and Science Scholars
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