3D Visualization of Knots and Braids– Janice Pappas
In everyday life, we may encounter an object that appears to be partially occluded, not by other objects, but by parts of the object itself. What does this mean? Visually, one kind of partially occluded object is a knot. Another such object is a braid.Mathematically, a knot is a non-intersecting closed curve, and a braid is the intertwining of a number of “strings” with ends that never turn back on themselves. In this course, we will identify and explore mathematical knots and braids and their properties. We will make physical knots and braids and identify their properties as well. We will determine when unknotting or unbraiding is possible and look at the similarities and differences between mathematical and physical knots and braids. Some of the questions we will address are: Why do certain materials knot more easily than others? What is an effective knot? What is the relation between the number of knot crossings and strength? We are interested in exploring the utility of knots with regard to ropes, wires, paper, and other materials and applied to, for example, boat mooring lines, shoelaces, cables, hangers, surgery sutures, and more. Knot folding techniques will also be studied. Braids will be explored in a similar way. Our goal is to address questions involving mathematical and physical knots and braids. We will also study knots and braids as they are found in a biological or chemical setting. As we delve into concepts concerning knots and braids, we will also learn how to view knot and braid models in virtual reality. We will learn how to use different software packages to devise knots and braids. We will have hands-on time with various materials to create knots and braids, study their properties, and conduct experiments. Students will conduct collaborative research and use software packages to create projects involving knots and/or braids. Projects will be visualized using software packages and at Groundworks in the Duderstadt Center. Using the available resources at Groundworks, students will be able to visualize in 3D the knot projects they create and conduct analyses. A student symposium will be convened in which student projects will be presented.
Prerequisites: having had a science, math or computer science course is helpful, but not necessary. Just bring your enthusiasm for learning about knots, braids and 3D visualization.
Art and Mathematics – Martin Strauss
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.
Catalysis, Solar Energy and Green Chemical Synthesis – Corinna Schindler and Corey Stephenson
“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.
Dissecting Life: Human Anatomy and Physiology – Glenn Fox FULL
Dissecting Life will lead students through the complexities and wonder of the human body. Lecture sessions will cover human anatomy and physiology in detail. Students will gain an understanding of biology, biochemistry, histology, and use these as a foundation to study human form and function. Laboratory sessions will consist of first-hand dissections of a variety of exemplar organisms: lamprey, sharks, cats, etc. Students may also tour the University of Michigan Medical School’s Plastination and Gross Anatomy Laboratories where they may observe human dissections
Forensic Physics – Ramon Torres-Isea
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
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!
Human Identification: Forensic Anthropology Methods – Maire Malone
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
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 – Michael A. Jones 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 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.
Sustainable Polymers – Anne McNeil
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.