# Courses

### Strike a pose! Intro to Math Modeling When we think about mathematical problems, we usually imagine equations and proofs; however, when we think about solving real-world problems, subjects such as biology, chemistry, or engineering more frequently come to mind. Mathematical modeling is the bridge that allows a mathematician to be a biologist, chemist, or an ecologist, depending on the problem that he/she is tackling. Applying mathematical tools, real-world problems, such as the spread of infectious diseases, can be translated to equations that mimic their behavior. When a solution to the mathematical problem is obtained, the results can be interpreted back into the language of biology, chemistry, or ecology to make predictions and find new solutions, such as new treatments. With such models, a mathematician can perform “experiments” in the mathematical representations of a real-world problem, instead of undertaking experiments in a lab, which is not always possible with these issues. In this course, we will learn how to navigate through this modeling process and become versed in the modeling of various biological problems using analytical and basic programming techniques. In particular, we will focus on population dynamics, chemical reactions, cancer, epidemics, and vaccination. No calculus or programming experience is required, just curiosity and open-mindedness!

The Seven Bridges of Konigsberg Instructor: Erika Ordog

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.

# Course Materials

Strike a pose! Intro to Math Modeling

Lecture video 1 Introduction to Math Modeling

Lecture video 2 Discrete Models

Lecture video 3 Introduction to Differential Equations

Lecture video 4 Predator-Prey Models

Lecture video 5 SIR Models

Lecture video 6  Systems of Differential Equations (more advanced)

Lecture video 7  Parameter Estimation

Lecture video 8 Agent-Based Models

Description of projects

Matlab project materials (instructions are in the Powerpoint file)

The Seven Bridges of Konigsberg (Graph Theory)

Lecture handouts

You’ll need to pause the videos often to work through the problems on your own before we do them together. You should allow at least 50 or 60 minutes for each lecture.

Lecture video 1   The Bridges of Konigsberg

Lecture video 2 Vertex colorings

Lecture video 3 The chromatic polynomial

Lecture video 4 Cliques, independent sets, and graph complements

Lecture video 5 Hamiltonian paths and directed graphs

Lecture video 6 The Good Will Hunting problem

The group projects can be started after watching the second lecture video.

Project 1: Domination and the Five Queens Problem

Project 2: Vertex covers and river crossings

Project 3: Planar graphs, Euler’s formula, and Brussels sprouts

Project 4: The matrix-tree theorem

Lecture slides by Duke professor Ezra Miller

Topology for statistical analysis of brain artery images

# Group Projects

• Project 1: Who Will Stop the Pandemic?

A dangerous virus has broken out in the world, threatening to create a world epidemic. Pandemic is a board game that explores the outbreak of four diseases. Each player takes a role to help prevent the epidemic: dispatcher, medic, scientist, researcher, operations expert, contingency planner, or quarantine specialist. In this project, we will develop a model to mimic the situation presented in this board game. With simplified rules, considering only one disease and two players, and we will investigate how each player’s role affects the spread of the disease. What combination of two players will prevent an epidemic in the world?

• Project 2: 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 3: Treatment Optimization

Using a very simple model for tumor growth and treatment, we will learn how to propose novel treatment strategies and test them using virtual mice. You will learn how to use numerical tools to test several dosage options, and use mathematical tools to asses which one is the best.

• Project 4: 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 5: Love Letters

Love and relationships, so complicated and dynamic. In this project, we are going to use mathematical modeling to gain insight into love relationships. We will use the example of Romeo and Juliet’s young love and, following different assumptions about their relationship, we will create corresponding models and analyze the dynamics that emerge from each situation. In each case, what will be the long-term outcome of their love? What will be the deciding factor as to whether or not they will live happily ever after?

• Project 6: 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 7: The Game of Life

Let’s play a simple game of life. Starting when you’re 22, after graduating from college, you have to choose between going to graduate school or not. If you go to graduate school, your income will be greater, but you will be carrying debt. On the other hand, if you decide to get a job, you will not accrue debt and will start to have an income right away. However, your income will never be as high as someone with a Ph.D., MD, or similar degree. Imagine that you want to buy a house when you are 35 years old. Considering a simple economic model we will investigate how such a decision will affect the value of the house you will be able to afford.

• Project 8: 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.