# Session 3 (July 21 – August 2)

An Introduction to Cryptography: From the Caesar Cipher to the One Time Pad and BeyondPat Boland (FULL)
How do we transmit private information in a secure, yet feasible way? This question has challenged humans for thousands of years and has become increasingly more important with the technological advances of the 20th and 21st centuries. This course will study a number of cryptographic techniques and the mathematics used to implement and analyze each. We will attempt to pay homage to the work of former University of Michigan undergraduate student Claude Shannon in his development of modern cryptographic theory. For example, we will ponder: What technique should we use if the “enemy” knows the system? Mathematically we will introduce and use elements of combinatorics, probability and statistics, modular arithmetic, elementary number theory (including factorization as a means to study the RSA algorithm), and the concept of random number generation. This course will be interactive with a focus on group work and scholar presentations. We will also use the University computer labs to help implement and analyze ciphers.

Climbing the Distance Ladder to the Big Bang: How Astronomers Survey the Universe  – Dragan Huterer (FULL)
The furthest objects that astronomers can observe are so distant that their light set out when the Universe was only 800 million years old; the light from these objects has been traveling to us for about 13 billion years. 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, and demonstrations. Students will learn basic computer programming, culminating in a project to model the motion of massive bodies interacting gravitationally. We will go to a nearby “mountain” near Ann Arbor to do night-time observing, guided by members of a local amateur astronomers’ club. 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.

From Physics through Biology to Medicine Ari Gafni (FULL)
From its humble beginning in the early 19th century in explaining the mechanics of steam  engines, the branch of physics called thermodynamics evolved to provide a foundation on  which the scientific discipline called biophysics was built. Current biophysicists use a variety of  concepts and tools from physics chemistry and biology to address important problems in basic,  applied, and medical sciences. In this course we will discover how biophysicists approach  scientific problems, what tools they use in their research, and highlight several interesting  areas of current research. The lectures will begin by reviewing the rules of thermodynamics in  a clear and intuitive way, including demonstrations and lab experiments. We will then move to  discuss the intriguing and complicated question of how a protein molecule, initially produced as  a long linear chain of amino acids devoid of biological activity, undergoes metamorphosis into  a precisely folded structure that is perfectly designed to fulfill its specific function. This  question, called the protein folding problem, has been studied by both theoretical and  experimental approaches and therefore serves as an excellent introduction into biophysics.  Using hemoglobin as our protein example, we will explore its biological function in transporting  oxygen from the lungs to tissues and discuss how it performs this task with great efficiency.  We will learn how hemoglobin’s structure was solved and how this knowledge has been used to explain in detail its mechanism of function. Finally, we will see how using purely biophysical  approaches led to the discovery of the molecular origin of the devastating disease sickle cell  disease, a disease that involves an aberrantly folded hemoglobin molecule. This discovery led  to the development of a therapeutic approach to this disease. We will end by discussing  several other protein folding diseases where research to explain their molecular origin is still at  the forefront of biophysics.

Graph Theory –  Doug Shaw (FULL)
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 to solve and publish!

Human Identification: Forensic Anthropology Methods –  Isabel Hermsmeyer (FULL)
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 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.

Introduction to Quantum Computing – Vanessa Sih (FULL)
The development of quantum physics at the beginning of the 20th century made possible current technology, including computer chips, solar cells, and flat screen displays.  We are now at an exciting time when quantum computers are being developed that could more efficiently solve some problems than existing “classical” computers.  However, quantum physics is mysterious and predicts behavior that is not intuitive.  What does it mean for a particle to tunnel through a barrier?  How can objects exist in a superposition like the Schrodinger’s cat, which is both dead and alive?  How is a quantum computer different from a “classical” computer?  This course will introduce students to quantum theory and its applications in modern technology and quantum computing and incorporate a mix of group problem solving and hands-on activities, including demonstrations, laboratory activities, and simulations.

Mathematics and Music Theory –  Lon Mitchell (FULL)
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 (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.

Science of Happiness –  Dina Gohar (FULL)
This course will introduce you to the exciting field of positive psychology–the scientific study of positive experiences, traits, relationships, and the institutions and practices that facilitate their development. Although psychological science has traditionally concentrated on “fixing what is wrong” (e.g., treating depression, anxiety, and other disorders), positive psychology focuses on “cultivating what is right” (e.g., promoting happiness and flourishing) and what makes life worth living. What truly makes us happy? How can YOU feel happier and more satisfied in your life? As you will learn, core research findings suggest that happiness is inextricably linked to: 1) using your strengths and contributing to something bigger than yourself, 2) staying grateful and optimistic, and 3) cultivating strong social connections. You will not only learn about but also practice some research-based strategies to improve both your learning and your own happiness and life satisfaction this summer. Through lively lectures, seminar-style discussions, activities, and interactive technology (e.g., documentaries, TED talks, etc.), we will examine the major topics of concern in positive psychology–pleasure, engagement, and meaning in life, and a critical source of these experiences: interpersonal relationships–and explore its applications to your everyday life as teenagers. We may also have some time to cover student-selected topics related to happiness in our last week.

What Really Happens at Night in a Museum?: Multidisciplinary Approaches for Exploring Ecology and Evolution Randy Singer (FULL)
Everything we know about our planet has been learned from the collection of biological research specimens. Anytime a new species is discovered, a new behavior is witnessed, or an ecosystem is explored all the data are documented and stored in a museum. Not the type of museum you might be thinking about, but rather a research museum collection. When you visit a natural history museum you are typically only seeing about 1% of what the museum actually has stored away in cabinets, jars and on sheets of paper. These specimens and data are stored for use by researchers to ask and answer almost any type of question. When all the data from all the specimens in every collection across the world are combined we form an irreplaceable network of data that can only be compared to a time machine. Want to go back to the age of dinosaurs and see what they are eating? Want to see how primates communicate with one another? Want to explore the farthest reaches of the ocean, but you don’t have a submarine? Museums can do this and more! Come on an adventure between the shelves, on the pages and in the digital realm of natural history collections and learn about how we can explore our planet and protect its future through the use of museum specimens!