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Art and Mathematics
(Session 2)

Instructor: Martin Strauss


With just a little historical revisionism, we can say that Art has provided inspiration for many fields within Mathematics, including color, contrast, geometry, symmetry, perspective, curvature, dimension, self-reference, and the complementary roles of local versus global properties. Conversely, Mathematics gives techniques for analyzing, appreciating, and even creating Art, as well as the basis for gallery design, digital cameras, and processing of images.

In this class we will explore the Mathematics in great works of Art, as a way of studying and illustrating central mathematical concepts in familiar and pleasing material. And we'll make our own art. As possibilities, we'll make our own tesselated drawings after Escher's lizards, as well as aperiodic tilings after the Alhambra. We'll make fractals like snowflake curves and the Mandelbrot set. We'll explore symmetries (obvious and non-obvious ones) in paintings and in Mathematics. In three dimensions, we'll make braided lanyards, origami, and build sculpture and domes after George Hart and Buckminster Fuller. Turning to engineering and signal processing, we'll manipulate images by adding or removing noise, morph one image into another, and create optical illusions that are consistent locally but inconsistent globally. We'll discuss some modern Math-based techniques for authenticating works of art. Finally, we'll discuss serious applications such as medical imaging, robotic sensing, and big data analysis of image databases.

Background in Math and interest in Art suggested. No artistic talent is necessary, though artistically talented students are encouraged to bring art supplies if they are inexpensive and easily transportable.



Climbing the Distance Ladder to the Big Bang: How Astronomers Survey the Universe
(Session 2)

Instructors: Dragan Huterer and Monica Valluri


The furthest objects that astronomers can observe 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 steps in the ladder uses radar ranging to get the distance to the nearby planets and the Sun; and 'triangulation' to get the distances to the nearest stars. Those in turn are used to obtain distances to other stars in our Milky Way Galaxy, stars in other galaxies, and out to the furthest reaches of what is observable.

This class will explore the steps in this ladder, using lectures, discussions, field trips, demonstrations, and computer laboratory exercises. Students will learn basic computer programming for a project to model the effects of gravity, and they will get hands-on experience of using a small radio telescope to map the the rotation speed of the Milky Way and measure the influence of its dark matter. We will cover concepts involving space, time, and matter that go far beyond the distance ladder, and involve some of the most fascinating mysteries in cosmology and astrophysics: What is it like inside a black hole? What is the Dark Matter? What is the Dark Energy that makes the Universe expand faster and faster? Is there other life in the Universe?



Discrete Dynamical Systems
(Session 1)

Instructor: Alejandro Uribe


In many areas of Science one is interested in modeling how a particular "something", referred to as a "system", changes with time. Such a model is called a dynamical system. Some examples are: A population of rabbits in a given environment, a billiard ball bouncing around a given billiard table, or the price of a commodity given supply and demand functions. In this course we will concentrate on models where the time is discrete, that is, the values of time are multiples of some unit. In the simplest models the evolution is given by an equation, specifying the state of the system at time n in terms of the state at time n-1. Discrete-time dynamical systems can be studied graphically, algebraically and numerically, without having to use a lot of calculus.

In this course we will study some of the many fascinating phenomena that dynamical systems can exhibit. We will start by developing a language that includes the concepts of phase space, initial condition, orbit, periodic orbit and fixed point. We will then develop the notions of stability and instability, and attractor. We will see that some systems are "integrable" or very regular, and at the opposite extreme we will try to formulate a precise definition of what a chaotic system is.

The course will include some theory (which will lead us to the idea of limit, and deep questions about the structure of the real numbers), the study of some particular models, and computer experimentation.



Explorations of a Field Biologist
(Session 1, 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 other organisms. Understanding this mass of interactions and how humans are affecting them 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.

Through our explorations of grasslands, forests, and wetlands of southeastern Michigan, you will learn many natural history facts (from identifying a turkey vulture to learning how mushrooms relate to tree health). You will also practice all the steps of doing science in the field, including making careful observations, testing a hypothesis, sampling and measuring, and analyzing and presenting results.

We will address question such as: How do field mice decide where to eat? Are aquatic insects affected by water chemistry? Does flower shape matter to bees? How do lakes turn into forests? Learning how to observe nature with patience and an inquisitive mind, and then testing your ideas about what you observe will allow you, long after this class, to discover many more things about nature, even your own back yard. Most days will be all-day field trips, including hands-on experience in restoration ecology. Towards the end of the course you will design, carry out, and present your own research project.



Fibonacci Numbers
(Session 2)

Instructor: Mel Hochster


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 a by-product of our studies, we will be able to explain how people can 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.



Graph Theory
(Session 1)

Instructor: Doug Shaw


Ignore your previous knowledge of algebra, geometry, and even arithmetic! Start fresh with a simple concept: Take a collection of points, called vertices, and connect some of them with lines called edges. It doesn't matter where you draw the vertices or how you draw the lines - all that matters is that two vertices are either related, or not. We call that a “graph” and you’ve taken the first step on the Graph Theory road! Graphs turn up in physics, biology, computer science, communications networks, linguistics, chemistry, sociology, mathematics… you name it!

In this course we will discuss properties that graphs may or may not have, hunt for types of graphs that may or may not exist, learn about the silliest theorem in mathematics, and the most depressing theorem in mathematics, learn how to come up with good algorithms, model reality, and construct some mathematical proofs. We will go over fundamental results in the field, and also some results that were only proved in the last year or so! And, of course, we will present plenty of currently unsolved problems for you solve and publish when you get home!



Hex and the 4 Cs
(Session 1)

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, and 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.



Life, Death and Change: Landscapes and Human Impact
(Session 2)

Instructor: David Michener


You’ll never see the same world quite the same way after this course – which is important if you want to make a difference with your life. All environmental studies presume a landscape – yet what seems to be a simple landscape is often far from uniform or stable. A great deal of information critical to anyone entering the “green” sciences can be detected for analysis – including factors that may fundamentally control species diversity, habitat richness, and animal (let alone human) behavior. Resolving human from non-human agency is an engaging and important challenge, too – one scientists and “green professionals” grapple with repeatedly. This outdoor class immerses students in real-world landscapes, ranging from the nearly-pristine to the highly humanized, to solve problems which may initially appear impossible to address. You can!

Your critical thinking skills are essential rather than prior biological course and field work. Come prepared to look, measure, analyze, discus and learn how the interaction of plants, soils, climate and time (and humans) influences landscape development. Develop basic skills in plant recognition and identification – skills that you can transfer to other terrestrial communities. Address questions about the current vegetation, its stability over time and its future prospects. Gain practical insights into the realities of biological conservation as we address how we can manage species and landscapes for future generations. By the end you’ll have a conceptual skill set that helps you assess how stable and disturbed are the “natural” areas are near your home – and prepares you to put instructors on the spot in college classes to come.



Mapping the Mysteries of the Universe
(Session 1)

Instructor: Jeffrey McMahon


Astrophysical observations coupled to models based on general relativity and quantum mechanics define our modern understanding of the Universe. To reproduce observations, these models must include mysterious new physics including dark energy, dark matter, and inflation: a model for the Universe’s first moments. In this course I will introduce you to the theoretical techniques and observational methods used to map and understand our Universe and its mysteries.


Mathematics and the Internet
(Session 2)

Instructors: Mark Conger and Sunny Fawcett


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? How can a large image or sound file be transferred quickly? Why is Google so good at finding what I'm looking for? How do computers work, anyway?

The answers to all these questions involve applications of abstract mathematics. In Mathematics and the Internet, we'll develop the math on its own, but also show how it is essential to making the Internet operate as it does. Our journey will take us through logic, probability, group theory, finite fields, calculus, number theory, and any other areas of math that might come up. We'll apply our results to coding theory, cryptography, search engines, and compression. We'll also spend several days building primitive computers out of transistors, logic gates, and lots of wire. If all goes well, we'll connect them to the Internet!



Mathematics of Cryptography
(Session 1)

Instructor: Benjamin Linowitz


Ever since humans first developed the ability to write there has been an ongoing battle between codemakers and codebreakers. The armies of ancient Sparta and Rome both used ciphers to relay secret battle plans, and the ancient Mesopotamians developed encryption techniques in order to protect commercially valuable techniques for glazing pottery. From a modern perspective, the codes used by the ancients are laughably insecure. Indeed, much of what made them secure was that they were being used during a period when most people were illiterate.

Because of the advent of computers, codemakers today need to use far more sophisticated techniques in order to create secure codes. Many of these techniques are mathematical in nature. One of the cryptography systems that we will discuss in this class is called RSA and is used to ensure the security of your credit card information when you make a purchase on the internet. We'll see that at its heart, what makes the RSA system secure is that it is very hard to factor a big number. The numbers used in the RSA system are actually so big that factoring them would take you millions of years. Even if you were using a supercomputer!

This class will give an historical introduction to the mathematics of cryptography, beginning with codes used by the Roman legions and building up to the RSA cryptography system discussed above. What will really make the class unique is that there won't be any lecturing. You will discover the mathematics of cryptography by working on problems and sharing your solutions with your classmates.



Mathematics of Decisions, Elections and Games
(Session 2)

Instructor: Michael A. Jones


You make decisions every day, including whether or not to sign up for this course. The decision you make under uncertainty says a lot about who you are and how you value risk. To analyze such decisions and provide a mathematical framework, utility theory will be introduced and applied to determine, among other things, the offer the banker makes to contestant in the television show Deal or No Deal.

Elections are instances in which more than one person's decision is combined to arrive at a collective choice. Because 1972 Nobel Prize winning economist Kenneth Arrow proved there is no best election procedure, it is useful to know what election outcomes are possible under different election procedures for the same ballots, suggesting mathematical and geometric treatments to be taught in the course. Oddly, the outcome of an election often stays more about which election procedure was used, rather than the preferences of the voters! Besides politics, this phenomenon is present any time rankings are aggregated, including Major League Baseball MVP ballotting, determining Academy Award winner, and scoring in golf tournaments.

Anytime one person’s decisions can affect another person, that situation can be modeled by game theory. That there is still a best decision to make that takes into account that others are trying to make their best decisions is, in part why John F. Nash was awarded the Nobel Prize in Economics in 1994. Besides understanding and applying Nash's results in settings as diverse as the baseball mind games between a pitcher and batter and bidding in auctions, we'll examine how optimal play in a particular game is related to a proof that there are the same number of counting numbers {1, 2, 3, …} as there are positive fractions.



Roller Coaster Physics
(Session 1)

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 (outside of the regularly scheduled weekend trip) 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.



Sampling, Monte Carlo and Problem Solving: How Analyzing Statistics Helps us Improve
(Session 1)

Instructor: Edward Rothman


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 works 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 from 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 it is just random noise? Did you know you can even calculate the number ? This way?!?!The survey and sampling techniques that allow us to draw meaningful conclusions about the whole, but based on the analysis of a small part, will be the main subject of this course.



The Physics of Magic and the Magic of Physics
(Session 2)

Instructors: Fred 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 than 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.



Discrete Dynamical Systems

Instructor: Alejandro Uribe

In many areas of Science one is interested in modeling how a particular "something", referred to as a "system", changes with time. Such a model is called a dynamical system. Some examples are: A population of rabbits in a given environment, a billiard ball bouncing around a given billiard table, or the price of a commodity given supply and demand functions. In this course we will concentrate on models where the time is discrete, that is, the values of time are multiples of some unit. In the simplest models the evolution is given by an equation, specifying the state of the system at time n in terms of the state at time n-1. Discrete-time dynamical systems can be studied graphically, algebraically and numerically, without having to use a lot of calculus.

In this course we will study some of the many fascinating phenomena that dynamical systems can exhibit. We will start by developing a language that includes the concepts of phase space, initial condition, orbit, periodic orbit and fixed point. We will then develop the notions of stability and instability, and attractor. We will see that some systems are "integrable" or very regular, and at the opposite extreme we will try to formulate a precise definition of what a chaotic system is.

The course will include some theory (which will lead us to the idea of limit, and deep questions about the structure of the real numbers), the study of some particular models, and computer experimentation.



Explorations of a Field Biologist

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 other organisms. Understanding this mass of interactions and how humans are affecting them 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.

Through our explorations of grasslands, forests, and wetlands of southeastern Michigan, you will learn many natural history facts (from identifying a turkey vulture to learning how mushrooms relate to tree health). You will also practice all the steps of doing science in the field, including making careful observations, testing a hypothesis, sampling and measuring, and analyzing and presenting results.

We will address question such as: How do field mice decide where to eat? Are aquatic insects affected by water chemistry? Does flower shape matter to bees? How do lakes turn into forests? Learning how to observe nature with patience and an inquisitive mind, and then testing your ideas about what you observe will allow you, long after this class, to discover many more things about nature, even your own back yard. Most days will be all-day field trips, including hands-on experience in restoration ecology. Towards the end of the course you will design, carry out, and present your own research project.



Graph Theory

Instructor: Doug Shaw

Ignore your previous knowledge of algebra, geometry, and even arithmetic! Start fresh with a simple concept: Take a collection of points, called vertices, and connect some of them with lines called edges. It doesn't matter where you draw the vertices or how you draw the lines - all that matters is that two vertices are either related, or not. We call that a “graph” and you’ve taken the first step on the Graph Theory road! Graphs turn up in physics, biology, computer science, communications networks, linguistics, chemistry, sociology, mathematics… you name it!

In this course we will discuss properties that graphs may or may not have, hunt for types of graphs that may or may not exist, learn about the silliest theorem in mathematics, and the most depressing theorem in mathematics, learn how to come up with good algorithms, model reality, and construct some mathematical proofs. We will go over fundamental results in the field, and also some results that were only proved in the last year or so! And, of course, we will present plenty of currently unsolved problems for you solve and publish when you get home!



Hex and the 4 Cs

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, and 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.



Mapping the Mysteries of the Universe

Instructor: Jeffrey McMahon

Astrophysical observations coupled to models based on general relativity and quantum mechanics define our modern understanding of the Universe. To reproduce observations, these models must include mysterious new physics including dark energy, dark matter, and inflation: a model for the Universe’s first moments. In this course I will introduce you to the theoretical techniques and observational methods used to map and understand our Universe and its mysteries.


Mathematics of Cryptography

Instructor: Benjamin Linowitz

Ever since humans first developed the ability to write there has been an ongoing battle between codemakers and codebreakers. The armies of ancient Sparta and Rome both used ciphers to relay secret battle plans, and the ancient Mesopotamians developed encryption techniques in order to protect commercially valuable techniques for glazing pottery. From a modern perspective, the codes used by the ancients are laughably insecure. Indeed, much of what made them secure was that they were being used during a period when most people were illiterate.

Because of the advent of computers, codemakers today need to use far more sophisticated techniques in order to create secure codes. Many of these techniques are mathematical in nature. One of the cryptography systems that we will discuss in this class is called RSA and is used to ensure the security of your credit card information when you make a purchase on the internet. We'll see that at its heart, what makes the RSA system secure is that it is very hard to factor a big number. The numbers used in the RSA system are actually so big that factoring them would take you millions of years. Even if you were using a supercomputer!

This class will give an historical introduction to the mathematics of cryptography, beginning with codes used by the Roman legions and building up to the RSA cryptography system discussed above. What will really make the class unique is that there won't be any lecturing. You will discover the mathematics of cryptography by working on problems and sharing your solutions with your classmates.



Roller Coaster Physics

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 (outside of the regularly scheduled weekend trip) 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.



Sampling, Monte Carlo and Problem Solving: How Analyzing Statistics Helps us Improve

Instructor: Edward Rothman

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 works 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 from 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 it is just random noise? Did you know you can even calculate the number ? This way?!?!The survey and sampling techniques that allow us to draw meaningful conclusions about the whole, but based on the analysis of a small part, will be the main subject of this course.



Art and Mathematics

Instructor: Martin Strauss

With just a little historical revisionism, we can say that Art has provided inspiration for many fields within Mathematics, including color, contrast, geometry, symmetry, perspective, curvature, dimension, self-reference, and the complementary roles of local versus global properties. Conversely, Mathematics gives techniques for analyzing, appreciating, and even creating Art, as well as the basis for gallery design, digital cameras, and processing of images.

In this class we will explore the Mathematics in great works of Art, as a way of studying and illustrating central mathematical concepts in familiar and pleasing material. And we'll make our own art. As possibilities, we'll make our own tesselated drawings after Escher's lizards, as well as aperiodic tilings after the Alhambra. We'll make fractals like snowflake curves and the Mandelbrot set. We'll explore symmetries (obvious and non-obvious ones) in paintings and in Mathematics. In three dimensions, we'll make braided lanyards, origami, and build sculpture and domes after George Hart and Buckminster Fuller. Turning to engineering and signal processing, we'll manipulate images by adding or removing noise, morph one image into another, and create optical illusions that are consistent locally but inconsistent globally. We'll discuss some modern Math-based techniques for authenticating works of art. Finally, we'll discuss serious applications such as medical imaging, robotic sensing, and big data analysis of image databases.

Background in Math and interest in Art suggested. No artistic talent is necessary, though artistically talented students are encouraged to bring art supplies if they are inexpensive and easily transportable.



Climbing the Distance Ladder to the Big Bang: How Astronomers Survey the Universe

Instructors: Dragan Huterer and Monica Valluri

The furthest objects that astronomers can observe 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 steps in the ladder uses radar ranging to get the distance to the nearby planets and the Sun; and 'triangulation' to get the distances to the nearest stars. Those in turn are used to obtain distances to other stars in our Milky Way Galaxy, stars in other galaxies, and out to the furthest reaches of what is observable.

This class will explore the steps in this ladder, using lectures, discussions, field trips, demonstrations, and computer laboratory exercises. Students will learn basic computer programming for a project to model the effects of gravity, and they will get hands-on experience of using a small radio telescope to map the the rotation speed of the Milky Way and measure the influence of its dark matter. We will cover concepts involving space, time, and matter that go far beyond the distance ladder, and involve some of the most fascinating mysteries in cosmology and astrophysics: What is it like inside a black hole? What is the Dark Matter? What is the Dark Energy that makes the Universe expand faster and faster? Is there other life in the Universe?



Explorations of a Field Biologist

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 other organisms. Understanding this mass of interactions and how humans are affecting them 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.

Through our explorations of grasslands, forests, and wetlands of southeastern Michigan, you will learn many natural history facts (from identifying a turkey vulture to learning how mushrooms relate to tree health). You will also practice all the steps of doing science in the field, including making careful observations, testing a hypothesis, sampling and measuring, and analyzing and presenting results.

We will address question such as: How do field mice decide where to eat? Are aquatic insects affected by water chemistry? Does flower shape matter to bees? How do lakes turn into forests? Learning how to observe nature with patience and an inquisitive mind, and then testing your ideas about what you observe will allow you, long after this class, to discover many more things about nature, even your own back yard. Most days will be all-day field trips, including hands-on experience in restoration ecology. Towards the end of the course you will design, carry out, and present your own research project.



Fibonacci Numbers

Instructor: Mel Hochster

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 a by-product of our studies, we will be able to explain how people can 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.



Life, Death and Change: Landscapes and Human Impact

Instructor: David Michener

You’ll never see the same world quite the same way after this course – which is important if you want to make a difference with your life. All environmental studies presume a landscape – yet what seems to be a simple landscape is often far from uniform or stable. A great deal of information critical to anyone entering the “green” sciences can be detected for analysis – including factors that may fundamentally control species diversity, habitat richness, and animal (let alone human) behavior. Resolving human from non-human agency is an engaging and important challenge, too – one scientists and “green professionals” grapple with repeatedly. This outdoor class immerses students in real-world landscapes, ranging from the nearly-pristine to the highly humanized, to solve problems which may initially appear impossible to address. You can!

Your critical thinking skills are essential rather than prior biological course and field work. Come prepared to look, measure, analyze, discus and learn how the interaction of plants, soils, climate and time (and humans) influences landscape development. Develop basic skills in plant recognition and identification – skills that you can transfer to other terrestrial communities. Address questions about the current vegetation, its stability over time and its future prospects. Gain practical insights into the realities of biological conservation as we address how we can manage species and landscapes for future generations. By the end you’ll have a conceptual skill set that helps you assess how stable and disturbed are the “natural” areas are near your home – and prepares you to put instructors on the spot in college classes to come.



Mathematics and the Internet

Instructors: Mark Conger and Sunny Fawcett

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? How can a large image or sound file be transferred quickly? Why is Google so good at finding what I'm looking for? How do computers work, anyway?

The answers to all these questions involve applications of abstract mathematics. In Mathematics and the Internet, we'll develop the math on its own, but also show how it is essential to making the Internet operate as it does. Our journey will take us through logic, probability, group theory, finite fields, calculus, number theory, and any other areas of math that might come up. We'll apply our results to coding theory, cryptography, search engines, and compression. We'll also spend several days building primitive computers out of transistors, logic gates, and lots of wire. If all goes well, we'll connect them to the Internet!



Mathematics of Decisions, Elections and Games

Instructor: Michael A. Jones

You make decisions every day, including whether or not to sign up for this course. The decision you make under uncertainty says a lot about who you are and how you value risk. To analyze such decisions and provide a mathematical framework, utility theory will be introduced and applied to determine, among other things, the offer the banker makes to contestant in the television show Deal or No Deal.

Elections are instances in which more than one person's decision is combined to arrive at a collective choice. Because 1972 Nobel Prize winning economist Kenneth Arrow proved there is no best election procedure, it is useful to know what election outcomes are possible under different election procedures for the same ballots, suggesting mathematical and geometric treatments to be taught in the course. Oddly, the outcome of an election often stays more about which election procedure was used, rather than the preferences of the voters! Besides politics, this phenomenon is present any time rankings are aggregated, including Major League Baseball MVP ballotting, determining Academy Award winner, and scoring in golf tournaments.

Anytime one person’s decisions can affect another person, that situation can be modeled by game theory. That there is still a best decision to make that takes into account that others are trying to make their best decisions is, in part why John F. Nash was awarded the Nobel Prize in Economics in 1994. Besides understanding and applying Nash's results in settings as diverse as the baseball mind games between a pitcher and batter and bidding in auctions, we'll examine how optimal play in a particular game is related to a proof that there are the same number of counting numbers {1, 2, 3, …} as there are positive fractions.



The Physics of Magic and the Magic of Physics

Instructors: Fred 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 than 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.



Climbing the Distance Ladder to the Big Bang: How Astronomers Survey the Universe
(Session 2)

Instructors: Dragan Huterer and Monica Valluri


The furthest objects that astronomers can observe 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 steps in the ladder uses radar ranging to get the distance to the nearby planets and the Sun; and 'triangulation' to get the distances to the nearest stars. Those in turn are used to obtain distances to other stars in our Milky Way Galaxy, stars in other galaxies, and out to the furthest reaches of what is observable.

This class will explore the steps in this ladder, using lectures, discussions, field trips, demonstrations, and computer laboratory exercises. Students will learn basic computer programming for a project to model the effects of gravity, and they will get hands-on experience of using a small radio telescope to map the the rotation speed of the Milky Way and measure the influence of its dark matter. We will cover concepts involving space, time, and matter that go far beyond the distance ladder, and involve some of the most fascinating mysteries in cosmology and astrophysics: What is it like inside a black hole? What is the Dark Matter? What is the Dark Energy that makes the Universe expand faster and faster? Is there other life in the Universe?



Mapping the Mysteries of the Universe
(Session 1)

Instructor: Jeffrey McMahon


Astrophysical observations coupled to models based on general relativity and quantum mechanics define our modern understanding of the Universe. To reproduce observations, these models must include mysterious new physics including dark energy, dark matter, and inflation: a model for the Universe’s first moments. In this course I will introduce you to the theoretical techniques and observational methods used to map and understand our Universe and its mysteries.


Explorations of a Field Biologist
(Session 1, 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 other organisms. Understanding this mass of interactions and how humans are affecting them 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.

Through our explorations of grasslands, forests, and wetlands of southeastern Michigan, you will learn many natural history facts (from identifying a turkey vulture to learning how mushrooms relate to tree health). You will also practice all the steps of doing science in the field, including making careful observations, testing a hypothesis, sampling and measuring, and analyzing and presenting results.

We will address question such as: How do field mice decide where to eat? Are aquatic insects affected by water chemistry? Does flower shape matter to bees? How do lakes turn into forests? Learning how to observe nature with patience and an inquisitive mind, and then testing your ideas about what you observe will allow you, long after this class, to discover many more things about nature, even your own back yard. Most days will be all-day field trips, including hands-on experience in restoration ecology. Towards the end of the course you will design, carry out, and present your own research project.



Life, Death and Change: Landscapes and Human Impact
(Session 2)

Instructor: David Michener


You’ll never see the same world quite the same way after this course – which is important if you want to make a difference with your life. All environmental studies presume a landscape – yet what seems to be a simple landscape is often far from uniform or stable. A great deal of information critical to anyone entering the “green” sciences can be detected for analysis – including factors that may fundamentally control species diversity, habitat richness, and animal (let alone human) behavior. Resolving human from non-human agency is an engaging and important challenge, too – one scientists and “green professionals” grapple with repeatedly. This outdoor class immerses students in real-world landscapes, ranging from the nearly-pristine to the highly humanized, to solve problems which may initially appear impossible to address. You can!

Your critical thinking skills are essential rather than prior biological course and field work. Come prepared to look, measure, analyze, discus and learn how the interaction of plants, soils, climate and time (and humans) influences landscape development. Develop basic skills in plant recognition and identification – skills that you can transfer to other terrestrial communities. Address questions about the current vegetation, its stability over time and its future prospects. Gain practical insights into the realities of biological conservation as we address how we can manage species and landscapes for future generations. By the end you’ll have a conceptual skill set that helps you assess how stable and disturbed are the “natural” areas are near your home – and prepares you to put instructors on the spot in college classes to come.



Discrete Dynamical Systems
(Session 1)

Instructor: Alejandro Uribe


In many areas of Science one is interested in modeling how a particular "something", referred to as a "system", changes with time. Such a model is called a dynamical system. Some examples are: A population of rabbits in a given environment, a billiard ball bouncing around a given billiard table, or the price of a commodity given supply and demand functions. In this course we will concentrate on models where the time is discrete, that is, the values of time are multiples of some unit. In the simplest models the evolution is given by an equation, specifying the state of the system at time n in terms of the state at time n-1. Discrete-time dynamical systems can be studied graphically, algebraically and numerically, without having to use a lot of calculus.

In this course we will study some of the many fascinating phenomena that dynamical systems can exhibit. We will start by developing a language that includes the concepts of phase space, initial condition, orbit, periodic orbit and fixed point. We will then develop the notions of stability and instability, and attractor. We will see that some systems are "integrable" or very regular, and at the opposite extreme we will try to formulate a precise definition of what a chaotic system is.

The course will include some theory (which will lead us to the idea of limit, and deep questions about the structure of the real numbers), the study of some particular models, and computer experimentation.



Mathematics and the Internet
(Session 2)

Instructors: Mark Conger and Sunny Fawcett


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? How can a large image or sound file be transferred quickly? Why is Google so good at finding what I'm looking for? How do computers work, anyway?

The answers to all these questions involve applications of abstract mathematics. In Mathematics and the Internet, we'll develop the math on its own, but also show how it is essential to making the Internet operate as it does. Our journey will take us through logic, probability, group theory, finite fields, calculus, number theory, and any other areas of math that might come up. We'll apply our results to coding theory, cryptography, search engines, and compression. We'll also spend several days building primitive computers out of transistors, logic gates, and lots of wire. If all goes well, we'll connect them to the Internet!



Art and Mathematics
(Session 2)

Instructor: Martin Strauss


With just a little historical revisionism, we can say that Art has provided inspiration for many fields within Mathematics, including color, contrast, geometry, symmetry, perspective, curvature, dimension, self-reference, and the complementary roles of local versus global properties. Conversely, Mathematics gives techniques for analyzing, appreciating, and even creating Art, as well as the basis for gallery design, digital cameras, and processing of images.

In this class we will explore the Mathematics in great works of Art, as a way of studying and illustrating central mathematical concepts in familiar and pleasing material. And we'll make our own art. As possibilities, we'll make our own tesselated drawings after Escher's lizards, as well as aperiodic tilings after the Alhambra. We'll make fractals like snowflake curves and the Mandelbrot set. We'll explore symmetries (obvious and non-obvious ones) in paintings and in Mathematics. In three dimensions, we'll make braided lanyards, origami, and build sculpture and domes after George Hart and Buckminster Fuller. Turning to engineering and signal processing, we'll manipulate images by adding or removing noise, morph one image into another, and create optical illusions that are consistent locally but inconsistent globally. We'll discuss some modern Math-based techniques for authenticating works of art. Finally, we'll discuss serious applications such as medical imaging, robotic sensing, and big data analysis of image databases.

Background in Math and interest in Art suggested. No artistic talent is necessary, though artistically talented students are encouraged to bring art supplies if they are inexpensive and easily transportable.



Discrete Dynamical Systems
(Session 1)

Instructor: Alejandro Uribe


In many areas of Science one is interested in modeling how a particular "something", referred to as a "system", changes with time. Such a model is called a dynamical system. Some examples are: A population of rabbits in a given environment, a billiard ball bouncing around a given billiard table, or the price of a commodity given supply and demand functions. In this course we will concentrate on models where the time is discrete, that is, the values of time are multiples of some unit. In the simplest models the evolution is given by an equation, specifying the state of the system at time n in terms of the state at time n-1. Discrete-time dynamical systems can be studied graphically, algebraically and numerically, without having to use a lot of calculus.

In this course we will study some of the many fascinating phenomena that dynamical systems can exhibit. We will start by developing a language that includes the concepts of phase space, initial condition, orbit, periodic orbit and fixed point. We will then develop the notions of stability and instability, and attractor. We will see that some systems are "integrable" or very regular, and at the opposite extreme we will try to formulate a precise definition of what a chaotic system is.

The course will include some theory (which will lead us to the idea of limit, and deep questions about the structure of the real numbers), the study of some particular models, and computer experimentation.



Fibonacci Numbers
(Session 2)

Instructor: Mel Hochster


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 a by-product of our studies, we will be able to explain how people can 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.



Graph Theory
(Session 1)

Instructor: Doug Shaw


Ignore your previous knowledge of algebra, geometry, and even arithmetic! Start fresh with a simple concept: Take a collection of points, called vertices, and connect some of them with lines called edges. It doesn't matter where you draw the vertices or how you draw the lines - all that matters is that two vertices are either related, or not. We call that a “graph” and you’ve taken the first step on the Graph Theory road! Graphs turn up in physics, biology, computer science, communications networks, linguistics, chemistry, sociology, mathematics… you name it!

In this course we will discuss properties that graphs may or may not have, hunt for types of graphs that may or may not exist, learn about the silliest theorem in mathematics, and the most depressing theorem in mathematics, learn how to come up with good algorithms, model reality, and construct some mathematical proofs. We will go over fundamental results in the field, and also some results that were only proved in the last year or so! And, of course, we will present plenty of currently unsolved problems for you solve and publish when you get home!



Hex and the 4 Cs
(Session 1)

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, and 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.



Mathematics and the Internet
(Session 2)

Instructors: Mark Conger and Sunny Fawcett


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? How can a large image or sound file be transferred quickly? Why is Google so good at finding what I'm looking for? How do computers work, anyway?

The answers to all these questions involve applications of abstract mathematics. In Mathematics and the Internet, we'll develop the math on its own, but also show how it is essential to making the Internet operate as it does. Our journey will take us through logic, probability, group theory, finite fields, calculus, number theory, and any other areas of math that might come up. We'll apply our results to coding theory, cryptography, search engines, and compression. We'll also spend several days building primitive computers out of transistors, logic gates, and lots of wire. If all goes well, we'll connect them to the Internet!



Mathematics of Cryptography
(Session 1)

Instructor: Benjamin Linowitz


Ever since humans first developed the ability to write there has been an ongoing battle between codemakers and codebreakers. The armies of ancient Sparta and Rome both used ciphers to relay secret battle plans, and the ancient Mesopotamians developed encryption techniques in order to protect commercially valuable techniques for glazing pottery. From a modern perspective, the codes used by the ancients are laughably insecure. Indeed, much of what made them secure was that they were being used during a period when most people were illiterate.

Because of the advent of computers, codemakers today need to use far more sophisticated techniques in order to create secure codes. Many of these techniques are mathematical in nature. One of the cryptography systems that we will discuss in this class is called RSA and is used to ensure the security of your credit card information when you make a purchase on the internet. We'll see that at its heart, what makes the RSA system secure is that it is very hard to factor a big number. The numbers used in the RSA system are actually so big that factoring them would take you millions of years. Even if you were using a supercomputer!

This class will give an historical introduction to the mathematics of cryptography, beginning with codes used by the Roman legions and building up to the RSA cryptography system discussed above. What will really make the class unique is that there won't be any lecturing. You will discover the mathematics of cryptography by working on problems and sharing your solutions with your classmates.



Mathematics of Decisions, Elections and Games
(Session 2)

Instructor: Michael A. Jones


You make decisions every day, including whether or not to sign up for this course. The decision you make under uncertainty says a lot about who you are and how you value risk. To analyze such decisions and provide a mathematical framework, utility theory will be introduced and applied to determine, among other things, the offer the banker makes to contestant in the television show Deal or No Deal.

Elections are instances in which more than one person's decision is combined to arrive at a collective choice. Because 1972 Nobel Prize winning economist Kenneth Arrow proved there is no best election procedure, it is useful to know what election outcomes are possible under different election procedures for the same ballots, suggesting mathematical and geometric treatments to be taught in the course. Oddly, the outcome of an election often stays more about which election procedure was used, rather than the preferences of the voters! Besides politics, this phenomenon is present any time rankings are aggregated, including Major League Baseball MVP ballotting, determining Academy Award winner, and scoring in golf tournaments.

Anytime one person’s decisions can affect another person, that situation can be modeled by game theory. That there is still a best decision to make that takes into account that others are trying to make their best decisions is, in part why John F. Nash was awarded the Nobel Prize in Economics in 1994. Besides understanding and applying Nash's results in settings as diverse as the baseball mind games between a pitcher and batter and bidding in auctions, we'll examine how optimal play in a particular game is related to a proof that there are the same number of counting numbers {1, 2, 3, …} as there are positive fractions.



Mapping the Mysteries of the Universe
(Session 1)

Instructor: Jeffrey McMahon


Astrophysical observations coupled to models based on general relativity and quantum mechanics define our modern understanding of the Universe. To reproduce observations, these models must include mysterious new physics including dark energy, dark matter, and inflation: a model for the Universe’s first moments. In this course I will introduce you to the theoretical techniques and observational methods used to map and understand our Universe and its mysteries.


Roller Coaster Physics
(Session 1)

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 (outside of the regularly scheduled weekend trip) 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.



The Physics of Magic and the Magic of Physics
(Session 2)

Instructors: Fred 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 than 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.



Sampling, Monte Carlo and Problem Solving: How Analyzing Statistics Helps us Improve
(Session 1)

Instructor: Edward Rothman


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 works 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 from 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 it is just random noise? Did you know you can even calculate the number ? This way?!?!The survey and sampling techniques that allow us to draw meaningful conclusions about the whole, but based on the analysis of a small part, will be the main subject of this course.