Courses

Understanding Knots

Caitlyn Booms, Emily Castner (TA)

You can picture a knot by taking a piece of string, tangling it and gluing the ends together. You can twist and pull on the knot but you aren’t allowed to cut and re-glue it. In that case, it is natural to ask when one knot is the same as another if I’m allowed to move it around without cutting it. This is a surprisingly hard question to answer. One strategy is to associate to every knot a number or a polynomial so that if two knots are the same then they are assigned the same value. Then when two knots have different values we know for sure that they are distinct. This is called a knot invariant. Examples of knot invariants are the unknotting number, tricolorability, and the Jones polynomial. In this course we will learn about different knot invariants, how they were developed and what they can tell us about knots.

The Seven Bridges of Konigsberg

Instructor: Erika Ordog

Suna Kim, Katie Taylor (TA)

The city of Konigsberg consisted of two sides of the Pregel River and two large islands, all connected to each other by seven bridges. It was a famous and deceptively simple problem to determine whether one could walk across each of the bridges once and only once. In proving that this was an impossible task, Leonhard Euler developed a new field of mathematics called graph theory. A graph, or a network, is a set of vertices and edges and has applications in the study of molecular structures, computer science, computational linguistics, and social networks. In this course we will study network concepts such as vertex colorings, spanning trees, and Hamiltonian cycles.

There is no prerequisite for this course other than an interest in studying an area of math that is not usually encountered in high school. In this course, students will work on projects in both the pure and applied aspects of networks.

Lectures

- - 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.

- - Professor Richard T. Durrett (Duke)
    
    Truth is stranger than fiction: A look at some improbabilities
    
    Probability is full of surprises and paradoxes, most of which result from doing the calculation incorrectly. We will illustrate this using some familiar old stories and new ones: the Monty Hall problem, cognitive dissonance in Monkeys, the birthday problem, lottery coincidences, the sad story of Sally Clark, the 2016 election, and gerrymandering in North Carolina. Some of these topics can be found in my blog or in previous versions of this talk, which can be reached by links on my web page.

- - Professor Irina Kogan (NCSU)
    
    A story of two postulates
    
    “I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallel alone”, wrote a Hungarian mathematician Farkas Bolyai to his son János, horrified at the thought that his son is attracted by the problem of parallels. János was not deterred, however, and discovered, simultaneously with Lobachevski, a consistent geometry in which the Euclidean parallel postulate does not hold. We will zoom through the path of mathematical thought that took over 2000 years, noticing which results from a high school geometry text-book do not rely on the parallel postulate, and which are altered completely in hyperbolic geometry.

- - Professor Jayce R. Getz (Duke)
    
    An invitation to modern number theory via elliptic curves
    
    We will define elliptic curves and use them as a touchstone to introduce a key idea in modern number theory relating analysis and algebra.

- - Professor Katie Newhall (UNC)
    
    Applied Stochastic Dynamics
    
    As an applied mathematician, I look to mathematics to help understand the physical world around us and how it behaves. A ping-pong ball dropped follows a mathematical function for the distance traveled. But imagine a strong wind randomly swirling around, knocking the ping-pong ball back and forth as it falls. Where will it end up? What’s the most likely place it will end up? A combination of probability and calculus can answer these questions! I will discuss not a ping-pong ball, but a magnetic model of spins and “active matter” as a simplified collection of micro-organisms. These are all example systems that exhibit Stochastic dynamics.

- - Professor Alexander A. Kiselev (Duke)
    
    How fluids move
    
    Attempts to understand the nature of fluid motion have occupied minds of researchers for many centuries. Fluids are all around us, and we can witness complexity and subtleness of their behavior in every day life, in ubiquitous technology, and in dramatic phenomena such as tornado or hurricane. I will discuss derivation of the fundamental equations describing fluid motion, and talk about some of the modern challenges in mathematical fluid mechanics.

- - Professor Thomas P. Witelski (Duke)
    
    Complex systems from simple dynamics
    
    Complicated and dramatic system-wide behaviors can be sparked by simple individual events. Examples include avalanches on mountains and messages “going viral” in social networks. Intricate behaviors like pattern formation, synchronization, and chaos will be shown to occur from problems as simple as the dynamics of a bouncing ball.

Group Projects

Project 1: Domination and The Five Queens Problem

What is the smallest number of queens that we can place on a chessboard so that every space on the board is either occupied by a queen or being attacked by at least one queen? How can you use graph theory to find a suitable arrangement of queens on an nxn chessboard? To find a solution to this problem, you will learn about the dominating set of a graph.

Project 2: Unknotting knots

A knot invariant is an algorithm for assigning a number, polynomial, etc. to a knot which will be the same if two knots are the same. If the two knots are different the knot invariant will hopefully be different. In this project you will work toward defining an invariant which involves looking at a diagram of a knot and seeing how many crossing you have to change to get the unknot (a circle). Further directions could include finding classes of knots that your invariant helps distinguish. We will also briefly explore whether this invariant is useful in studying tangled DNA.

Project 3: Vertex Covers and River Crossings

Suppose a farmer wants to transport his wolf, goat, and cabbage across a river, but he only has one extra seat in his boat. Since the wolf wants to eat the goat and the goat wants to eat the cabbage, he can’t leave the wolf and the goat or the goat and the cabbage alone. What is the best way to transport all three across the river? How can we understand this problem in terms of the vertex cover of a graph? If the problem becomes more complicated, how many seats does the farmer need in his boat for the problem to have a solution?

Project 4: Braid invariants

Any two knots which are the same can be related by a series of what are called Reidemeister moves. Every knot has what is called a braid representation. For this project, you will work on finding moves analogous to the Reidemeister moves, but which relate braid representations of the same knots.

Project 5: Planar Graphs, Euler’s Formula, and Brussels Sprouts

A graph is called planar if it can be drawn without any of its edges crossing. For this project, you will learn about Euler’s formula for planar graphs, which relates the number of vertices, edges, and faces of a graph in one equation. You will also learn about a paper-and-pen game called “Brussels sprouts.” Can you use Euler’s formula to ensure that you win the game every time?

Project 6: Satellite knots

Given two knots, one can make what is called a satellite knot. For this project, your goal is to determine whether or not there is a unique way to express a knot as a satellite. In other words, can you find two sets of knots which are different, but give you the same knot when you satellite them? If not, why not?

Project 7: Kirchoff’s Matrix-Tree Theorem

How can we associate a matrix to a graph? How many possible spanning trees does any given graph have? You will learn about the Laplacian matrix of a graph and its eigenvalues to see how to calculate the total number of spanning trees of a graph.

Project 8: The Jones polynomial and alternating knots

A knot can be drawn in the plane with a knot diagram. One simple knot invariant is the minimum number of crossings required in the diagram of knot. The Jones polynomial is a very interesting (and complicated) invariant of knots which assigns a (Laurent) polynomial to a knot. However, it turns out that there is a nice relationship between the crossing number and the Jones polynomial for a class of knots called alternating knots. The goal of this project will be to determine what this relationship is.