# Available Undergraduate Research Projects

Below is a partial list of undergraduate projects that faculty have available for current students to undertake, ordered alphabetically by faculty last name and then project title.

*This is a very partial list and new content will be added as time goes on. It is currently just a template.*

**Bounded continued fractions for rationals**

*Outline: A continued fraction is a particularly elegant representation of rationals related to the Euclidean algorithm, of the form b0+1/(b1+1/(b2+1/(b3+...+1/bn))) where each bi is a positive integer. In the case of reals, the fraction continues indefinitely. A bounded continued fraction is one in which each of the bi's has some maximum value. It has been proven that any real can be represented as the sum of two bounded continued fractions, with maximum value four. Unfortunately, the method is not applicable to rationals, and we don’t know whether every rational can be represented as the sum of two finite bounded continued fractions. The aim of this project is to numerical investigate how hard it is to represent an arbitrary rational by bounded continued fractions.*

**Faculty: Stephen Lucas**

**Areas:**Number theory, computational mathematics

**Improved bounds for the kissing problem**

*Outline:*

*T*he kissing problem asks how many spheres can touch a central sphere without overlapping in n-dimensions. The solution is known for dimensions two to four, and there are bounds on the solution in higher dimensions. But these bounds are rather loose. The aim of this project is to try and improve upon these bounds using a numerical optimization algorithm that starts with a random distribution, then “juggles” them trying to fit on the central sphere. By finding examples that work and those that do, we should be able to make the bounds more strict.**Faculty: Stephen Lucas**

**Area:**Numerical analysis

**Nonlinear extrapolation for vectors**

*Outline: There are a variety of successful extrapolation techniques to accelerate the convergence (or even divergence) of iterative schemes, like Aitken’s Del-Squared method, and the epsilon algorithm. In Math 448 we discussed a vector acceleration techniques and implemented one by Irons & Tuck (1969) that failed to work! The aim of this project is to implement a few other methods and try and understand why Irons & Tuck’s method failed.*

**Faculty: Stephen Lucas**

Student: Samuel Snarr

Student

**Area:**Numerical analysis

**When:**Spring 2020 (proposed).

**Random sieves and Goldbach’s conjecture**

*Outline:*

*The sieve of Eratosthenes is an efficient way of identifying prime numbers. A sieve can also be used to randomly select numbers that have the same random distribution properties as the primes. What happens when we attempt to use these numbers to establish Goldbach’s conjecture that every even integer is the sum of two primes? What happens when we change the distribution properties so numbers are less likely to be chosen? As motivation, it appears that all even numbers after 3248 can be represented as the sum of twin primes.*

**Faculty: Stephen Lucas**

**Area:**Number theory, computational mathematics