Mathematics

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.

Fibonacci Numbers  –  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  –  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  –  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 Music Theory  –  Lon Mitchell
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 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 Cryptography  –  Anton Lukyanenko
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  –  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, 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.