**Art and Mathematics** – *Martin Strauss
(Session 2) *

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- FULL) *

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- OPEN & 2- FULL)*

“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 2)*

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.

**Data, Distributions, and Decisions: The Science of Statistics** – *Adriene Beltz
(Session 2- FULL)
*Will that medication work? Did your SAT scores significantly improve since you last took the test? Does “fake news” affect the way people vote? Statistics are needed to answer these questions! They are the backbone of empirical investigations in psychological, social, neuro, political, and medical science. They help determine if the patterns we see in data reflect true group differences and real-world relationships, or not. This course serves as a general introduction to the field of statistics, as utilized by empirical scientists. In it, we will discuss the collection, analysis, and interpretation of data with an eye toward answering pressing questions. Specifically, we will cover: (1) probability and distributions, (2) basic group comparisons, (3) correlation and regression, and (4) the general linear model (e.g., how different statistical tests are all related and can provide the same result). There will be some lecture with slides, online videos and visualizations, and group work, but the majority of the course will focus on out-of-class data collection and hands-on coding and analysis in R.

**Data Science of Happiness** – *Dina Gohar
*

**(Session 2)***This course is an introduction to*

**positive psychology-**the study of positive experiences, positive traits, positive relationships, and the institutions and practices that facilitate their development–a rapidly expanding area of study that is of great interest and benefit to students. The course will use an active and experiential approach that encourages reflective learning in combination with lively lectures, seminar style discussions, interactive technology and collaborative activities to creatively and critically examine the major topics of concern in positive psychology, such as pleasure, engagement, and meaning in life, as well as a critical source of these experiences: positive interpersonal relationships, and areas of controversy (e.g., what is happiness and how do you measure it?). To get first-hand experience, students will complete several empirically supported wellness-enhancing exercises and reflect on their experiences in class, such as planning an ideal day and having it, doing a secret good deed, and writing a letter of gratitude to someone who hasn’t been properly thanked. As a capstone project, students may also design their own hands-on activities to teach younger students about positive psychology and possibly demonstrate them to students at a nearby community center or summer camp.

**Forensic Physics** – *Ramon Torres-Isea
(Sessions 1 & 2)*

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)*

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!

**Greatest Hits in Vertebrate Evolution** – *Carlos Peredo
(Session 1)
*This course will serve as an introductory course designed to introduce students to the concepts of evolution by studying the fossil record. The course will cover some of the major transitions in the evolution of vertebrates, such as the emergence of fish onto land, the origins of flight in dinosaurs, and the transition to bipedality in early hominids. We will pay extra attention to my area of expertise: major evolutionary transitions in marine mammals, including their return to the sea from land, the origin of echolocation, and the transition to filter feeding. This course will aim to teach students about the broader biological mechanisms that drive natural history and will involve both traditional and hands on learning opportunities.

**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** – *Maire Malone
(Sessions 1 & 2)*

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.

**Mathematical Modeling in Biology** – *Trachette Jackson and Patrick Nelson
(Session 1- FULL)*

Mathematical biology is an exciting interdisciplinary field that combines applied mathematics, scientific computing, biology, ecology, physiology and medicine. This branch of mathematics is growing with phenomenal speed! For the mathematician, biology opens up new and exciting areas of study, while for the biologist, mathematical and computational modeling offers another powerful research tool that can provide insight into the complexity of a biological system. Mathematical biologists typically investigate problems in diverse and exciting areas such as the topology of DNA, cell physiology, the study and spread of infectious diseases, population ecology, neuroscience, tumor growth and treatment strategies, and organ development and embryology. This course will be a venture into the field of mathematical modeling in biology and the biomedical sciences using techniques from calculus, dynamical systems and scientific computing. Interactive lectures, group projects, computer demonstrations, and guest speakers will help introduce some of the fundamentals of mathematical modeling and its usefulness in biology, physiology and medicine. For example, the cell division cycle is a sequence of regulated events which describes the passage of a single cell from birth to division. There is an elaborate cascade of molecular interactions that function as the mitotic clock and ensures that the sequential changes that take place in a dividing cell take place on schedule. What happens when the mitotic clock speeds up or simply stops ticking? These kinds of malfunctions can lead to cancer and mathematical modeling can help predict under what conditions a small population of cells with a compromised mitotic clock can result in a fully developed tumor. Students who can speak the languages of mathematics and computation along with biology and medicine will be able to solve some of the most challenging problems of the 21

^{st}century. Wouldn’t it be amazing if mathematics could guide future experiments that lead to a cure AIDS or Cancer?

**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 1- FULL)*

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
(Session 2- FULL)*

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.

**Relativity: A Journey through Warped Space and Time** – *Daniel Mayerson
(Session 2- FULL)*

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 new knowledge, we will delve into black holes, learn how GPS systems work, and debate the possibility of time machines and wormholes. Prerequisites: basic concepts in geometry (e.g. coordinates, distance formulae) and working knowledge of elementary calculus (e.g. what a derivative is and how to take one). We will introduce some 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!

**Survey in Modern Physics** – *Jun Nian
(Session 2- FULL)
*How can we describe curved spacetime? What is the difference between black hole and worm hole? Is a time machine ever possible? What is Schrödinger’s cat? What is quantum entanglement? Are there parallel universes? What are four elementary interactions in nature? What are anti-matter, dark matter and dark energy? What is Big Bang Theory? What is String Theory? We may have seen these words in many movies and science fictions, but what do they really mean? To answer these questions and to understand the concepts mentioned above, we need to first learn two fundamental pillars of modern physics, relativity and quantum mechanics, both of which take years of physics courses. This mini course is aimed at providing a crash course in these subjects. We will begin with elementary physics taught in high school, and then step by step survey in relativity and quantum mechanics, or more generally in modern physics. The key concepts will be demystified with a lot of examples, demonstrations and discussions. To help understand physics problems and methods, some tutorials and labs will be provided, accompanying the lectures everyday. Basic knowledge in calculus will be helpful, but not necessary. Good knowledge in mathematics and physics at high school level should be enough. Some advanced mathematics will be introduced during the course. The only prerequisite is the passion in science and modern physics