**Art and Mathematics** – *Martin Strauss (Session 3) *

With just a little historical revisionism, we can say that Art has provided inspiration for many fields within Mathematics. 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 well as folk art, as a way of studying and illustrating central mathematical concepts in familiar and pleasing material. And we’ll make our own art, by drawing, painting, folding origami papers, and more. Major topics include Projection, Symmetry, Wave Behavior, and Distortion. Projection includes the depiction of three-dimensional objects in two dimensions. What mathematical properties must be lost, and what can be preserved? How does an artwork evoke the feeling of three-dimensional space? We’ll study perspective, depictions of globes by maps, and the role of curvature. Turning to symmetry, we’ll study rotational and reflective symmetry that arise in tiling and other art and math. We’ll study more generalized symmetry like scaling and self-similarity that occurs in fractals as well as every self-portrait, and is central to mathematical concepts of dimension and un very different from the work at coarser scales—it is not self-similar. Describing light as waves and color as wavelength at once explains how mirrors, lenses, and prisms work and explains some uses of light and color in art. Finally, we ask about distorting fabrics and strings, and ask about the roles of cutting, gluing, and of stretching without cutting or gluing. Is a distorted human figure still recognizable, as long as it has the right number of organs and limbs, connected properly? 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.

**Brain and Behavior** – *Jen Cummings (Session 2) *

Ever wonder how that gelatinous blob in your head controls everything you do and think? What exactly are neurons? How do they talk to each other? And to the rest of your body? Have you ever wondered about things like: how does stress affect your body? Is exercise really that good for your brain? What happens if you miss a few nights of sleep? It makes sense that your brain affects your experiences- but can experiences actually change your brain?? We will answer these questions (and more!) in Brain and Behavior, as we explore the amazing field of behavioral neuroscience. We will begin with a section on the basic functionality of the brain and nervous system, and then will go on to investigate how the system can be affected by things like stress, learning & memory, hormones, and neuropsychiatric disorders. We will leave some time for a session on student-selected topics in behavioral neuroscience, so if there’s something else you’ve been pondering with respect to the brain, don’t worry! We’ve got you covered.

**Catalysis, Solar Energy and Green Chemical Synthesis** – *Corinna Schindler and Corey Stephenson (Sessions 1 & 2)*

“The chemist who designs and completes an original and esthetically pleasing multistep synthesis is like the composer, artist or poet who, with great individuality, fashions new forms of beauty from the interplay of mind and spirit.” -E.J. Corey, Nobel Laureate Prior to Friedrich Wöhler’s 1828 synthesis of urea, organic (carbon-containing) matter from living organisms was believed to contain a vital force that distinguished it from inorganic material. The discovery that organic molecules can be manipulated at the hands of scientists is considered by many the birth of organic chemistry: the study of the structure, properties, and reactions of carbon-containing matter. Organic matter is the foundation of life as we know it, and therefore a fundamental understanding of reactivity at a molecular level is essential to all life sciences. In this class we will survey monumental discoveries in this field over the past two centuries and the technological developments that have enabled them. Catalysis, Solar Energy, and Green Chemical Synthesis will provide a fun and intellectually stimulating hands-on experience that instills a historical appreciation for the giants whose trials and tribulations have enabled our modern understanding of chemistry and biology. Students will learn modern laboratory techniques including how to set up, monitor, and purify chemical reactions, and most importantly, how to determine what they made! Experiments include the synthesis of biomolecules using some of the most transformative reactions of the 20th century and exposure to modern synthetic techniques, such as the use of metal complexes that absorb visible light to catalyze chemical reactions, an important development in the Green Science movement. Finally, industrial applications of chemistry such as polymer synthesis and construction of photovoltaic devices will be performed. Daily experiments will be supplemented with exciting demonstrations by the graduate student instructors.

**Climbing the Distance Ladder to the Big Bang: How Astronomers Survey the Universe** – *Dragan Huterer*

**(Session 3)**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. 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? The class is recommended for students with solid high-school mathematics background, including some exposure to vectors.

**Dissecting Life: Human Anatomy and Physiology** – *Mary Orczykowski (Sessions 1 & 2)*

Dissecting Life will lead students through the complexities and wonders of the human body. Lecture sessions will cover human anatomy in detail. Students will gain an understanding of physiology and histology and use these as a foundation to study human form and function. In the lab sessions, students will apply and reinforce concepts through comparative anatomy dissection, case discussions, self-experimentation, modeling, etc. In addition, students will have the opportunity to learn from osteological and dissected (plastinated and embalmed) human anatomical donors within the University of Michigan Medical School’s Gross Anatomy Laboratories.

**Forensic Physics **– *Ramon Torres-Isea (Sessions 2 & 3)*

A fiber is found at a crime scene. Can we identify what type of fiber it is and can we match it to a suspect’s fiber sample, for example from a piece of clothing? Likewise, someone claims to have valuable ancient Roman coins, a newly-found old master painting, or a Viking map of America predating Columbus’ voyage. Are they authentic or fakes? How can we determine that using some physics-based techniques? (These are real examples the Viking map proved to be a forgery). Also for example, how is a laser-based molecular-probing technique used to stop criminals from trading billions of dollars of counterfeit pharmaceuticals and endangering thousands of lives? These are a few among many examples of experimental physics methods applied to several areas of Forensics. In this session, students will be introduced to these methods and have opportunities to make measurements using molecular, atomic and nuclear forensic techniques. In addition, applications to medical imaging and diagnostics will be introduced. Students will be working at our Intermediate and Advanced Physics Laboratories with the underlying physics for each method presented in detail, followed by demonstrations and laboratory activities, which include the identification of an “unknown” sample. Various crime scenes will challenge students to select and apply one or more of the methods and use their Forensic Physics skills to conduct investigations.

**Graph Theory** – *Doug Shaw (Sessions 1, 2 & 3)*

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!

**Hex and the 4 Cs** – *Stephen DeBacker (Session 1)*

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.

**Human Identification: Forensic Anthropology Methods** – *Miranda Nicole Cosman***(Sessions 2 & 3)**

Forensic anthropology methods are used to aid in human identification with skeletal remains. Applications of forensic anthropology lie in the criminal justice system and mass disaster response. In this course, we will address questions such as: What are important differences between male and female skeletons? Utilizing skeletal remains, how would you tell the difference between a 20-year old and an 80-year old? How do you distinguish between blunt force and sharp force trauma on the skull? In this hands-on, laboratory-based course, you will be become familiar with human osteology (the study of bones] and bone biology. Through our exploration of forensic and biological anthropology methods, you will learn how to develop a biological profile [estimates of age at death, sex, ancestry and stature], assess manner of death, estimate postmortem interval, investigate skeletal trauma and pathology, and provide evidence for a positive identification from skeletal remains. Additionally, we will explore various forensic recovery techniques as they apply to an outdoor complex, including various mapping techniques. Towards the end of the course, you will work in small groups in a mock recovery of human remains and analyze the case utilizing the forensic anthropological methods learned throughout the course.

**Mathematics and the Internet** – *Mark Conger (Session 1)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!*

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 and Music Theory** – *Lon Mitchell (Session 1)*

Mathematicians can create complex and beautiful theorems from relatively basic assumptions, while Music Theorists often try to identify basic patterns and rules in complex and beautiful music. In this course, we will explore some of the recent attempts to meet in the middle, connecting mathematical patterns and structures to music from the ancient to the modern. In Mathematics, we will explore topics such as group theory, graph theory, geometry, and metric spaces, encountering some of the most important structures in the modern discipline. Fundamental results of these areas will be discussed, and students will construct and explore examples and related patterns. In Music Theory, we will take existing music by composers such as Bach and Beethoven and use mathematical structures to provide a possible explanation of what they were thinking as they composed. In addition, we will investigate the techniques of modern composers such as Arnold Schoenberg who advocated composition based on prescribed axioms. Students will be given the chance to write music using these different techniques. Although we will use the modern (Western) twelve-tone scale as a reference, our explorations will take us into discussions of tuning, temperament, and the physics of sound. We will investigate mathematical theories of what makes the best scale, how some of those scales occur in the music of other cultures, and how modern composers have engineered exotic scales to suit their aesthetics. Software allowing students to experiment with creating their own musical systems will be provided. Prospective students should have a good command of (high-school) algebra and experience with reading music in some form.

**Mathematics of Decisions, Elections and Games** – *Michael A. Jones (Session 3)*

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, a student’s preference for desserts and for the offer the banker makes to a contestant in the television show Deal or No Deal. Our analysis will touch on behavioral economics, including perspectives of 2017 Nobel Prize winner Richard Thaler. Elections are instances in which more than one person’s decision is combined to arrive at a collective choice. But how are votes tallied? Naturally, the best election procedures should be used. But Kenneth Arrow was awarded the Nobel Prize in Economics in 1972, in part, because he proved that there is no best election procedure. Because there is no one best election procedure, once the electorate casts its ballots, it is useful to know what election outcomes are possible under different election procedures – and this suggests 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 in other settings that we’ll consider which include: the Professional Golfers’ Association tour which determines the winner of tournaments under different scoring rules (e.g. stroke play and the modified Stableford system), the method used to determine rankings of teams in the NCAA College Football Coaches poll, and Major League Baseball MVP balloting. 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 (see the movie A Beautiful Mind, 2002). 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. We will also examine the Gale-Shapley algorithm, which is used, for example, to match physicians to residency programs and to match students to colleges (the college admissions problem). Lloyd S. Shapley and Alvin E. Roth were awarded the Nobel Prize in Economics in 2012 for their work on matching.

**Organic Chemistry 101: Orgo Boot Camp** – *Kathleen Nolta (Sessions 2 & 3)*

This course will introduce you to the techniques and concepts taught in the first term of organic chemistry at the University of Michigan. The emphasis is on lecture-based learning, small group learning, and independent presentation of problems that you have solved. While laboratory exercises will be done, they are not the main focus of the course. Topics to be covered include nomenclature and how molecules are organized structurally, including their connectivity, options for stereochemistry, and conformational manipulation. We will also explore chemical transformation by learning how to draw complete curved arrow mechanisms for some of the most fundamental reactions in organic chemistry: acid-base chemistry, nucleophilic substitutions, electrophilic additions, eliminations, and electrophilic aromatic substitutions. The emphasis will be on exploring concepts through problem solving (there will be lots of practice problems to do!), and you will have an opportunity to take examinations given to college students. Students will be able to explore the chemistry in various laboratory applications; we will also be covering the basics of infrared spectroscopy and NMR. By focusing on the concepts and trying some of the techniques, students will gain a better understanding of what organic chemistry is and how to enjoy it.

**Polynomial Party Platter** – *Ahmad Barhoumi (Session 1)*There was a time when mathematical challenges were common places! When Johann Bernoulli challenged Isaac Newton circa. 1697 to solve a certain physics problem, not only did Newton solve the problem given to him overnight, but he also responded with a little bit of shade, saying “I do not love to be dunned [pestered] and teased by foreigners about mathematical things…” What his cool attitude doesn’t tell you is that Newton actually stayed up all night to solve this problem! In fact, long before that point, challenges and contests involving solutions of polynomials were common! Almost 800 years ago, Frederick II, the Holy Roman Emperor circa. 1220, once asked Leonardo of Pisa (a.k.a Fibonacci) to solve the equation x^3 + 2x^2 + 10x = 20, which he did (up to 10 decimal places)!In this course, we will throw our hat in the ring, and compete with these mathematical giants! We will begin our discussion by trying to solve that same problem given to Fibonnaci. This will lead us to a foray into complex numbers and their properties. We will then discuss other algorithms for evaluating polynomials, properties of their solutions and approximating these solutions (or finding good candidates!). Our exploration will be primarily motivated by problems, and time spent lecturing will be kept to a minimum, with the possible exception being to discuss the drama this mathematics drummed up!

**Relativity: A Journey through Warped Space and Time **– *Daniel Mayerson (Session 3)*

Einstein forever altered our understanding of the nature of space and time with his theories of relativity. These theories tell us that the speed of light is a universal constant, declare that the fabric of space and time is warped by matter, and demand that matter moves through spacetime by following its curvature. Introduced 100 years ago, these concepts clash mightily with our everyday physical intuition, but are nevertheless cornerstones of modern-day physics. In this course, we will explore the exciting world of relativity (both the special and general theories). After briefly reviewing classical mechanics (Newton’s laws), we will use thought experiments to understand the ideas behind relativity and see how they are actually ultimately simpler and more natural than classical mechanics. Along the way we will encounter strange paradoxes that push the limits of our understanding and learn powerful mathematics that will allow us to quantify our relativistic understanding of the universe. Using our knowledge, we will delve into black holes, learn how GPS systems work, and debate the possibility of time machines and wormholes.

**basic concepts in geometry (e.g. coordinates, distance formulae) and physics (e.g. position, velocity, acceleration). A working knowledge of elementary calculus is recommended (e.g. what a derivative is and how to take one). We will introduce a little bit of multivariable calculus (e.g. partial differentiation) and integration techniques, so prior knowledge of those is a bonus. An open, curious and interested mind is absolutely necessary; you must be willing to think deeply about physics and the nature of our universe!**

*Prerequisites*:**Social Choice and Mathematics of Voting **– *Iian Smythe (Session 2)*

When a group of people have to make a choice from a slate of options, whether that be ice cream flavors, award nominees, or political representatives, how should they go about it? At first, there seems to be an obvious answer: Have all of the people vote for their favorite and simply choose the one with the most votes! But when there are more than two options, this can get tricky. What if none are favored by a clear majority? And how do we weigh people’s relative preferences among the different options?

In this course, we will consider a variety of different ways (called “voting methods” or “social choice functions”) for making such choices, including first-past-the-post, ranked-choice or instant run-off voting, and point-based schemes. Real world examples will be studied from throughout history up to today, including Ann Arbor’s very recently adopted new voting system for municipal elections. We will then consider a number of potentially desirable properties that these methods may or may not have, each capturing a different notion of “fairness”. The course will culminate in Arrow’s Impossibility Theorem, which says that, however desirable they may be, there is no voting method which satisfies all of these fairness properties in all circumstances. The necessary mathematical background is the ability to count, think carefully, and be creative. A familiarity with US civics may be helpful for providing additional context.

**Surface Chemistry** – *Zhan Chen (Session 2)*

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 ling 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, studies 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 deigned to be non-wetting; and coatings are applied to cookware for easy cleanup. 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 students will be able to observe these techniques in the lab. Multimedia PowerPoint presentations will be used for all lectures. By doing this, it’s hoped 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.

**Sustainable Polymers** – *Anne McNeil (Session 1)*From grocery bags and food packaging to contact lenses and therapeutics, there is no doubt that polymers have had a positive impact in our lives. Most of these polymers are made from petroleum-based feedstocks, which are dwindling in supply. And although some plastics are recycled, most of them end up contaminating our lands and oceans. Through hands-on lab work and interactive lessons, this class will introduce the future of polymer science – that is: polymers made from sustainable materials that ultimately biodegrade! Students will conduct research experiments to make, analyze, and degrade renewable plastics. We will also examine commercial biodegradable materials and plastics used for energy and environmental remediation, and practice science communication through a creative stop-motion animation project.

**The Physics of Magic and the Magic of Physics – **Georg Raithel** (Session 1)**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.