# Courses

Crack the Code

Instructor: Caroline Turnage-Butterbaugh

Carolyn Davis, Colleen Dougherty (TA)

The word cryptography comes from the Greek words kryptos meaning secret and graphein meaning to write. Ciphers and codes are used to encrypt a message with the goal that the message can only be decrypted by the intended recipient. What types of ciphers are there, and which are the most secure? How does one develop a cipher to encrypt a secret message in a secure way? We will first explore simple ciphers and by the end of the course we will learn about the RSA cryptosystem, which is used today for secure data transmission.

If you are enthusiastic, inquisitive, and like solving puzzles, then you are ready to take this course! You will have the opportunity to work on projects with other students to investigate ciphers and the beautiful world of number theory underlying their construction

Fair Sharing

Instructor: Shan Shan

Emma Kar, Emily Kelting (TA)

How many times did you share a cake with your family and friends while envying someone who got a better piece than yours? Is there a way to cut cake fairly so that everyone is satisfied? Mathematicians and computer scientists have been intrigued by this fair division problem and proposing algorithms to solve it. However, other real life problems can be more complicated than cutting a cake. How to share apartment rent with your roommate? How to split a taxi fare among riders heading to different locations? We will look at different aspects of sharing by case study and introduce the corresponding basic math concepts and their applications.

There is no prerequisite for this course. If you are curious about how math looks in everyday life, then you are qualified!

# Lectures

• Professor (Duke)

Mathematicians helping Art Historians and Conservators

In recent years, mathematical algorithms have helped art historians and art conservators putting together the thousands of fragments into which an unfortunate WWII bombing destroyed world famous frescos by Mantegna, decide that certain paintings by masters were “roll mates” (their canvases were cut from the same bolt), virtually remove artifacts in preparation for a restoration campaign, get more insight into paintings hidden underneath a visible one. The presentation will review these applications, and give a glimpse into the mathematical aspects that make this possible.

• Professor Radmila Sazdanovic (NC State)

Catalan numbers, Chebyshev polynomials, and categorifications

What does it mean to “compute with diagrams”? We will construct a “diagrammatic algebra” based on properties of Catalan numbers and use it to recover some well-known facts about the Chebyshev polynomials. This novel approach, called categorification, provides insight into more complicated algebraic structures as well as inspiration for visual artists.

• Professor Ezra Miller (Duke)

Topology for statistical analysis of brain artery images

Statistics looks for trends in data. Topology quantifies geometric features that don’t change when shapes are squished, stretched, or bent continuously. What does one have to do with the other? When data objects are already geometric, such as magnetic resonance images of branching arteries, topology can isolate information of statistical relevance. This talk explains what we have learned about the geometry of blood vessels in aging human brains using topological methods in statistics. The main results are joint with Paul Bendich, Steve Marron, Alex Pieloch, and Sean Skwerer (at the time, a Math postdoc, Stat faculty, Math undergrad, and Operations Research grad student).

• Professor Hubert Bray (Duke)

Gravity and the Curvature of Spacetime

Einstein’s Theory of General Relativity explains gravity more accurately than any other theory by modeling the universe as a four dimensional curved spacetime manifold. We’ll discuss the mathematics behind this amazing picture of the universe.

• Professor Nancy Rodriguez-Bunn (UNC)

What calculus can tell us about life

In this talk I will discuss how we can use calculus to gain insight into complex social, ecological, and biological phenomena. We will focus on modeling urban crime and explore various important mathematical questions from the point of view of their applications.

• Professor Robert Bryant (Duke)

The Idea of Holonomy

The notion of `holonomy’ in mechanical systems has been around for more than a century and gives insight into daily operations as mundane as steering and parallel parking and in understanding the behavior of balls (or more general objects) rolling on a surface with friction. A sample question is this: What is the best way to roll a ball over a flat surface, without twisting or slipping, so that it arrives at at given point with a given orientation? In geometry and physics, holonomy has turned up in many surprising ways and continues to be explored as a fundamental property of geometric structures. In this talk, I will illustrate the fundamental ideas in the theory of holonomy using familiar physical objects and explain how it is also related to group theory and symmetries of basic geometric objects.

• Professor Colleen M Robles (Duke)

The Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem is the “crown jewel” of surface geometry. I will explain this beautiful result, what makes it so remarkable, and how one can prove/verify it using calculus.

# Group Projects

• Project 1: The dirty work problem

In real life, bigger is not always better. For example, if we are dividing unpleasant house chores, then we prefer a small portion. Can you formulate a fair division problem where each player wants the smallest fair and try to solve it? What fairness criteria do you want to use?

• Project 2: Amicable pairs

Two integers n and m are called an amicable pair if the sum of the proper divisors of n equals m and vice versa. For example, 220 and 284 form an amicable pair. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 and  284 = 1+ 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 Also, the proper divisors of 284 are 1, 2, 4, 71, and 142 and 220 = 1 + 2+ 4 + 71 + 142. What sorts of properties are known about amicable pairs? How many amicable pairs are there? Are there existing connections to cryptography? Your project should culminate in the creation or explanation of a cipher incorporating amicable numbers.

• Project 3: Negotiating to settlement in divorce

Cake-cutting algorithms are not generally applicable to real-world situations involving indivisible goods, whose value is destroyed if they are divided. People see this conflict arise very often in splitting up property in divorce. Can you formulate a fair division problem where players want to fairly divide a set of property that involve indivisible goods?

• Project 4: Creating a cipher

Suppose you and your best friend have your own alphabet that you use to exchange messages securely. Furthermore, suppose you have a long message to send to your friend that is time sensitive! In this project, you will design your own alphabet and cipher, and determine how to use a computer program to help you translate between your original message and the encrypted message written in your alphabet. Note that this is not only an artistic and computational task: you must design your cipher so that it is difficult for unintended recipients to crack. How many layers of encryption can you incorporate? How strong is your cipher? In what ways is it vulnerable to attack?

• Project 5: Exact Division

We have seen in the Cut and Choose Method that it’s better to be the chooser than the cutter. Can you come up with a fairness criteria that balance the asymmetry? Can you formulate a fair division problem that uses the criteria you proposed, and try to solve it?

• Project 6: Mersenne numbers

A Mersenne number is a positive integer of the form Mn=2n-1. When Mn is prime, we call Mn a Mersenne Prime. While we can certainly construct infinitely many Mersenne numbers (how?), does this mean there are infinitely Mersenne primes? How might one go about finding Mersenne primes? What is GIMPS? Are there existing connections to cryptography? Your project should culminate in the creation or explanation of a cipher which incorporates Mersenne primes.

• Project 7: Online division

In traditional cake-cutting setting, all players are available at the time of the division. However, in real life, people may show up late or leave early. Can you simulate a fair division problem where players arrive and depart during the process of dividing a resource?

• Project 8: Fibonacci Numbers

The famous Fibonacci series is a series of positive integers in which each number (called a Fibonacci number) is determined by summing the two preceding numbers. The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, … and so on. The Fibonacci numbers are ubiquitous in mathematics, and they are still studied in modern times. For example, a theorem of Zeckendorf (1970s) states that every positive integer can be written uniquely as the sum of distinct, non-consecutive Fibonacci numbers. For example, 10 = 2 + 5 and 17 = 13 + 3 + 1. What other properties (new and old) are known about Fibonacci numbers? Are there existing connections to cryptography? Your project should culminate in the creation or explanation of a cipher incorporating the Fibonacci numbers.