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Mathematics

MATH 050 [006P]: The Predictability of Chance and Its Applications in Applied Mathematics
Quantitative Intensive (QI) [GC Math Requirement]
Roberto Camassa; Kenneth McLaughlin; Michael Minion
This course will examine the ways in which some types of behavior of random systems can not only be predicted, but also applied to practical problems. Students will learn the implications of basic probability, sampling, and prediction theory through classroom exercises that include simple games of chance as well as engaging computer demonstrations. Students will be guided to discover for themselves how some highly complex systems can be reliably predicted, and how strategies developed for gambling can help us understand more complex problems in "exact" sciences such as mathematics and physics. Students also will investigate what can be predicted about the behavior of "unpredictable" or chaotic systems.

MATH 051 [006P]: "Fish Gotta Swim, Birds Gotta Fly": The Mathematics and the Mechanics of Moving Things
Quantitative Intensive (QI) [GC Math Requirement]
Roberto Camassa; Richard McLaughlin; Michael Minion; Laura Miller
This course allows students to have hands-on exposure with a class of physical and computer experiments designed to challenge their intuition about how motion is achieved in nature. In particular, fundamental concepts like equilibrium, instability, and friction will be experienced in lab experiments and illustrated by computer simulations. Simple mathematical laws inferred from the experiments will allow students to provide interpretations of some of the surprising outcomes observed. The seminar will draw on introductory overviews, followed by several lab sessions to carry out the experimental work by teams of up to four students. The students will alternate between observation and data collection, and will prepare a report of their findings and interpretations after each lab sequence. The reports will be organized into a final paper, whose evaluation will determine the grade for the course.

MATH 052 [006P]: Fractals: The Geometry of Nature
Quantitative Intensive (QI) [GC Math Requirement]
Sue Goodman
Many natural objects have complex, infinitely detailed shapes in which we see smaller versions of the whole shape appearing throughout. Examples are the fern leaf above, or turbulent weather patterns, mountainous landscapes, clouds, galaxy systems, cell reproduction, our own capillary systems, heart rhythms and nervous systems-even the ups and downs of the stock market. Such a shape is called a fractal. Artists have used fractals to create fantastic images in galleries, on T-shirts and calendars, and imaginary landscapes in movies (the dragon curve in Jurassic Park, a moon modeled for Apollo 13, landscapes in Star Trek). Musicians have effectively modeled music from Bach to the Beatles and have created their own new music. Writers such as Tom Stoppard (Arcadia) have incorporated themes of self-similarity and fractals into their plots.We will study the basic geometric properties of fractals, learn how to design and analyze them, and study their occurrence in a variety of applications. We'll use website and supplementary material developed by Professor Bob Devaney in his Dynamical Systems and Technology Project, developed with the support of the National Science Foundation. We will read and discuss Tom Stoppard's play Arcadia, listen to and generate fractal music, and learn some computer applications for designing fractals. Group discussions and projects will play a primary role. A term paper or project gives students an opportunity to investigate more deeply some aspect or application of fractal geometry. The mathematics required for our study is basic high school geometry and algebra.

MATH 053 [006P]: Symmetry and Tilings
Quantitative Intensive (QI) [GC Math Requirement]
Sue Goodman
This seminar will approach a currently active field of research that we'll engage by studying the beautiful 'wallpaper patterns' of Dutch artist M.C. Escher, such as his depiction of tiles shaped like angels and devils filling the entire plane without gaps or overlaps. The geometrically regular patterns found in Escher's work are also seen in decorative mosaic tilings, pottery, quilt and cloth designs in many different cultures throughout history. They can be analyzed and understood mathematically using only basic high school geometric transformations (like rotations and translations). The text by David Farmer, Groups and Symmetry: A Guide to Discovering Mathematics, will form the framework of class discussions and individual and group projects. The use of two computer drawing programs Geometer's Sketchpad and Kali (easy to learn and available for use at no cost to the student) will be integrated into the course. Students will work together to discover the geometric rules governing tilings, figure out which patterns are possible and why. Projects will give students the opportunity to construct a virtual kaleidoscope, and to learn to analyze, design and create Escher-like patterns. In research projects, students could explore how mathematicians and anthropologists have used pattern analysis to examine the evolution of cultures and the development of trade. Or they could choose to study how scientists use symmetry to study the structure of crystals and some new, exceptionally strong alloys with a quasicrystalline structure, or to understand laws of growth governing various organisms. (This seminar has been developed with the aid of a Brandes Grant.)

MATH 054 [006P]: The Science of Conjecture: Its Math, Philosophy and History
Quantitative Intensive (QI) [GC Math Requirement]
Jane Hawkins
How do we know what will happen next in life? How does a jury decide between guilty and not guilty? Which path will the next hurricane take? Some systems appear to be predictable while some appear to be completely random; there must be some explanation. In this course we will explore the history and present of calculating the odds of many events, from the weather to weighing evidence in a trial to card playing. We will begin by studying how evidence was weighed and odds calculated before probability and statistics existed as a mathematical subject. From there we will study the origins of probability, the science of uncertainty, and we will finish the course with a study of entropy and randomness. The course will involve reading and writing of essays, a little math, and some hands-on computer models of random and predictable events.

MATH 055 [006P]: On Volumes, Areas, and Being the Right Size
Quantitative Intensive (QI) [GC Math Requirement]
Joseph Cima
Space surrounds us in our daily lives. It seems to be almost “nothing.” But it imposes striking constraints on organic and inorganic objects. The seminar revolves around two examples of such constraints: 1. Objects in nature cannot be arbitrarily magnified, or shrunk, without raising havoc with their functionality--unless their shape or the materials they are made with are also modified. Objects have to be “the right size.” The expression comes from J.B.S. Haldane's essay “On Being the Right Size,” one of the seminar's basic readings. This is because volumes increase with the cube of the linear magnification while surface areas do so only with its square, i.e., lagging disproportionately behind—-a fact of fundamental importance in biology, physics, and engineering. A cell cannot be linearly magnified, say, 100 times preserving its shape and materials: the membrane's area will not grow enough to keep up with the cell's mass growth. The cell will starve! In a building magnified under the same conditions, beams' cross sections will have areas whose increase won't match the building's increased weight. Collapse will follow.
2. In the world of mathematics, properties of space different from the magnification principles above, surprisingly pick out the five Platonic Solids as the only possible regular polyhedra. This subtle stubbornness of space has fascinated the mathematically and philosophically minded over two millennia. (Nowadays the Platonic Solids also thrill a lot of internet aficionados!) Why are the five Platonic Solids the only three-dimensional regular polyhedra that will ever be found? Come to the seminar and be thoroughly convinced!
Expected prerequisites are good recall of high school mathematics and an interest in natural science. This seminar is most appropriate for students who have had at least some high school calculus. Students considering the course who have not had high school calculus should consult the instructor before registering.

MATH 056 [006P]: Coding and Decoding: From Thomas Jefferson to E-commerce
Quantitative Intensive (QI) [GC Math Requirement]
Karl Petersen
It is common to say that we are now living in the information age. What are the ways in which information is stored, transmitted, presented, and protected? What is information anyway? Topics for this seminar will be drawn from cryptography (secret writing throughout history, including Thomas Jefferson's cipher machine, the German Enigma machine, and security and privacy on the internet); image compression and processing (compact disks, MP3 and JPEG, transforms, error correction, noise removal); symbolic dynamics (encoding of symbol streams, like the genetic code, and associated dynamical systems and formal languages); and visualization (how can different kinds of information be vividly and usefully presented, combined, and compared?). These topics are mathematically accessible to anyone with a high-school background and offer many possibilities for experimentation and theoretical exploration. Students will undertake individual or group projects using existing software for encoding and decoding messages, enhancing and compressing images, transforming and filtering signals, measuring properties of information sources, and so on. They will report on their work in writing and orally to the seminar. Discussions will be based on readings from a course pack as well as Simon Singh's The Code Book (Doubleday, 1999), with associated theoretical investigations.

MATH 056H:  Information and Coding
Quantitative Intensive (QI)
Karl Petersen
It is common to say that we are now living in the information age. What are the ways in which information is stored, transmitted, presented, and protected? What is information anyway? Topics for this seminar will be drawn from cryptography (secret writing throughout history, including Thomas Jefferson's cipher machine, the German Enigma machine, and security and privacy on the internet); image compression and processing (compact disks, MP3 and JPEG, transforms, error correction, noise removal); symbolic dynamics (encoding of symbol streams, like the genetic code, and associated dynamical systems and formal languages); and visualization (how can different kinds of information be vividly and usefully presented, combined, and compared?) These topics are mathematically accessible to anyone with a high school background and offer many possibilities for experimentation and theoretical exploration. Students will undertake individual or group projects using existing software for encoding and decoding messages, enhancing and compressing images, transforming and filtering signals, measuring properties of information sources, and so on. They will report on their work in writing and orally to the seminar. Discussions will be based on readings from a course pack as well as Simon Singh's The Code Book (Doubleday, 1999), with investigations in probability, number theory, combinatorics, and information theory to provide theoretical foundations.

MATH 057 [006P]: The Fourth Dimension
Quantitative Intensive (QI) [GC Math Requirement]
James Stasheff
The idea of a fourth dimension has a rich and varied history--in mathematics but also in the physics of relativity, in surrealistic art, and in philosophy. Modern computer graphics permit visualization and increased interaction of mathematics and the other disciplines. The seminar will explore the concept of the fourth (and higher) dimensions both mathematically and more widely in human thought, both technical and popular.

MATH 058 [006P]: Math and Art: Symmetry without Fear
Quantitative Intensive (QI) [GC Math Requirement]
Thomas Brylawski
I will (with student input) mathematically classify (using only high school geometry) rosette patterns (such as an asterisk or a hubcap), the eight frieze patterns (such as a zigzag or a zipper), and the seventeen wallpaper patterns (such as a checkerboard or a honeycomb). Then, the students will take over exhibiting patterns from various cultures and local instances (e.g., brick patterns on the UNC campus). I will also teach how to create Escher-like patterns and students can create such patterns (using, e.g., The Geometer's Sketchpad program available for extra credit). The serious student by the end of the course will understand and appreciate how mathematicians classify things: which they consider the same and which different. Also he or she will see how visual beauty gives rise to mathematical beauty and vice-versa.

MATH 058: Math, Art and the Human Experience:  We All do Math
Quantitative Intensive (QI)
Mark A. McCombs
This course is designed to engage students in an exploration of the relevance of mathematical ideas to fields typically perceived as "non-mathematical" (e.g. art, music, film, literature). Equally important will be an exploration of how these "non-mathematical" fields, in turn, influence mathematical thought. In each case, course activities and assignments have been designed to illuminate the fact that even the most complex mathematical concepts grow out of real people's attempts to understand better their world. By the end of the course, students should be able to

• Identify and assess how mathematical ideas influence and are influenced by ideas expressed through art, music, literature, religion, etc
• Compare and contrast different philosophies concerning the nature of mathematics
• Articulate their own well-reasoned ideas concerning the nature of mathematics
• Discuss the evolution of fundamental mathematical concepts in a historical as well as a cultural context
• Discuss the work and lives of important mathematicians in relation to the “non-mathematical” work of their contemporaries
• Identify and assess how their own understanding of mathematical ideas influences the way they interact with the world

Course assignments and activities will include weekly readings and short homework writing assignments (2–3 paragraphs), one longer paper (8­–10 pages), and a portfolio of mathematical art (e.g., painting, origami, poetry, music).

MATH 059 [006P]: The Mystery and Majesty of Ordinary Numbers
Quantitative Intensive (QI) [GC Math Requirement]
William Smith
Problems arising from the arithmetic of ordinary counting numbers have for centuries fascinated both mathematicians and non-mathematicians. This seminar will consider some of these problems (both solved and unsolved). There will be some investigations into areas of current applications of the theory of numbers. Some elementary questions will be discussed that arise in current research projects including those of the instructor. The seminar is intended for the general student. No mathematics background beyond the very basic arithmetical skills in algebra is required. There will be some use of the computer to deal with the arithmetic of large numbers; however, no previous computer skills are required.

MATH 060 [006P]: Simulated Life
Quantitative Intensive (QI) [GC Math Requirement]
Tim Elston
Increasingly scientists are using computational models to help understand the complex behavior of biological systems. For example, computational models have been used to explain how thousands of fireflies can synchronize their flashing, as well as to understand how individual cells respond to a changing environment. This seminar will introduce students to the thought process that goes into developing computational models of biological systems. The class will also expose students to techniques for simulating and analyzing these models and build intuition into the dynamic behavior produced by the models.

MATH 061 [006P]: The Language of Mathematics: Making the Invisible Visible
Quantitative Intensive (QI) [GC Math Requirement]
Ladnor Geissinger
The development of current technology, from MP3 players, DVDs, computers, internet security, and smart devices, to satellites and GPS, space stations and interplanetary navigation, as well as much of modern science, would have been impossible without mathematics. Mathematics is everywhere, and is essential for understanding our complex physical environment and our data-drenched society, but often math goes unnoticed by the casual user of technology. The content of this course has been condensed by Keith Devlin (the mathguy who reports on NPR) into two soundbites - memes - cultural viruses that he's aggressively disseminating: mathematics is the science of patterns, and mathematics makes the invisible visible. In this seminar we will discuss various kinds of patterns that give rise to different branches of mathematics, and see how these math tools help us to understand complex invisible structures. The core text will be The Language of Mathematics: Making the Invisible Visible by Keith Devlin. In it he follows 8 general themes covering patterns of counting, of reasoning and communicating, of motion and change, of shape, of symmetry and regularity, of position, of chance, and the fundamental patterns of the universe. This should give a good idea what math is, what math does, and why it appears everywhere. Devlin also wrote the book for the PBS TV series Life by the Numbers, and we will watch several of these videos and discuss them and the book.

MATH 062 [006P]: Combinatorics
Quantitative Intensive (QI) [GC Math Requirement]
Ivan Cherednik
A leading expert in Modern Combinatorics wants to share his vision of the subject with students. The seminar is a perfect background for future specialists in computer science, mathematics, biology, economics, physics, for those who are curious about cryptography and how the stock market works, as well as for everyone who likes mathematics. High school Algebra II is the only prerequisite.
Combinatorics is "the theory of combinations and the science of counting". Practical and theoretical problems of all sorts - mundane matters like designing a railroad time table, or calculating how many phone numbers a community needs, foundations of computer science, genomics, architecture, nuclear physics, military tactics, and the top priority directions like cryptography and stock market - have all been improved by the discipline of Combinatorics.
In this class, we will learn about Combinatorics by puzzling through (among other things) dominoes, magic square, roulette, and the stock market. Our aim is to understand the ingenious and fascinating methods that have been developed to solve (what mathematicians call) "problems involving finite sets of objects". Future card sharks, stock brokers, physicists, biologists, engineers, administrators, and anyone else who has to make sense of large numbers of things: this class is for you.

MATH 062H:  Combinatorics
Quantitative Intensive (QI)
Ivan Cherednik
Combinatorics is a branch of mathematics concerned with the study of finite and discrete objects.
This seminar will introduce students to the fundamentals and history of combinatorics, and sharpen their methodological skills. The seminar will be organized around the following topics:

1) Puzzles: covering by dominos, magic squares, 36 officers
2) Combinations: from coin tossing to lotto, dice and poker
3) Fibonacci numbers: rabbits, recurrences, population growth
4) Arithmetic: designs, cyphers, prime numbers, finite fields
5) Catalan numbers: from playing roulette to stock market

Students will be involved in individual research projects, group projects, and written and oral presentations. The seminar provides an excellent background for students who are interested in mathematics, physics, computer science, biology, economics, statistical physics, cryptography, or the stock market, but no prerequisite is assumed beyond high school algebra.

MATH/MASC 063 [006P]: From the Sound of Music to the Perfect Storm
Quantitative Intensive (QI) [GC Natural Science - no lab]
Roberto Camassa; Alberto Scotti
We are constantly surrounded by phenomena that are wave-like in nature. We communicate over short distances with sound waves, while we use electromagnetic waves over long distances. We see waves when we stand at beach, and the weather we experience is controlled very often by wave-like features of the jet stream. In this seminar, we will develop the conceptual framework necessary to understand waves, starting from laboratory observations. The main goal is to expose the common traits of waves, and how they can be used to enhance our understanding and predict the outcome of a broad range of important physical phenomena.

MATH 065 [006P]: Colliding Balls & Springs: The Microstructure of How Materials Behave
Quantitative Intensive (QI) [GC Math Requirement]
Sorin Mitran
Leucippus and Democritus of Miletus introduced the idea of "atoms" (that which cannot be split) in the 5th Century BCE. Ever since, natural philosophers and scientists have been fascinated by the idea of relating phenomena we directly observe to overall behavior of collections of particles. We follow this intellectual journey combining the history of the times, the applications to science, and the mathematics developed to study atoms. The noted 20th Century American physicist Richard Feynmann said: "If I had to keep one concept to restart science after some disaster it would be the atomic hypothesis" - we will learn why.

MATH 066 [006P]: Seeing the World with Non-Euclidean Eyes
Quantitative Intensive (QI) [GC Math Requirement]
Patrick Eberlein
For more than 2000 years mathematicians tried to prove that the mathematical universe was flat, despite contradictory physical evidence. The attempt to prove Euclid's parallel postulate (described below) proved to be as satisfying and fruitful as pushing on a door marked "pull". When mathematicians finally pulled on the door in the 19th century they opened it to a new nonEuclidean world far stranger and richer than anything they had imagined. This course will investigate nonEuclidean geometry (hyperbolic and spherical) from historical, mathematical and practical perspectives. The mathematical approach will be largely algebraic, in contrast to the traditional axiomatic method. I intend to emphasize that algebra and geometry are not distinct warring subjects, but are closely related in many ways. All algebraic methods will be developed from scratch. The historical part of the course would center on attempts to prove the parallel postulate from the axioms of Euclid, why these attempts ultimately failed and the models of nonEuclidean geometry that make this failure clear. The parallel postulate states that given any line L and a point P not on L there exists a unique line L' containing p that is parallel to L. Many false proofs of the parallel postulate have been given over the past 2000 years, some by famous mathematicians. Analysis of these false proofs will underscore the importance of critical thinking and the need to verify facts that seem "obvious" at first glance.