# Group Projects

There are 8 group projects for SWiM. Two or three SWiM participants work on a single project of their choice. The project titles and descriptions are listed below.

1. Who will stop the epidemic?

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?

2. How random can a polynomial be?

A big property of prime numbers is that they are equi-distributed with respect to any “modulus”. Can any polynomial function behave in a similar way to the machine of prime numbers? For f(x) with degree greater than 1, is it possible that the value set of f(x) is also equi-distributed with respect to any modulus?

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.

4. Graph problems with applications in number theory

Given n vertices (or points), two vertices v_1 and v_2 are connected with a probability 1/2. What is the probability that there exists a division of V (the union of sets V_1 and V_2) such that for any u in V_1, the total number of edges between u and vertexes in V_2 is even, and vice versa? What if the probability for connecting each edge is 1/3, and so on?

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?

6. The biggest trick in number theory: from local to global

Given a polynomial with integral coefficients, how can we determine whether it is irreducible? And how do we determine its factorization into factors? An easier question is to consider the factorization of f(x) “modulo” a prime number. How does the association of these data to different prime numbers give us a hint to the original question?

7. 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.

8.  A passage leading to a class number

Consider a two variable polynomial f(x,y). Is it possible that the value set of f(x,y) is equi-distributed with respect to any modulus? How about three variable polynomial? Especially, we are interested in how many prime numbers are in the value set of f(x,y)?