**Wednesday, April 5 | 4-4:50 p.m. | Bridges 268**

Featuring MSUM Mathematics majors Samuel Holen, Jennifer Williams, and Kelli Vosberg | The Collatz Conjecture; Parallel Lines in Hyperbolic Geometry; The Birthday Problem

**Samuel Holen will be speaking on “The Collatz Conjecture: An Analysis”**

The Collatz Conjecture, also known as the hailstone sequence, is a mathematical sequence a_n , where the starting value a 0 is a positive integer, defined in the following way:

a_{n+1} = 3a_n + 1 if a_n is odd; a_{n+1} = a_n/2 if a_n is even.

The conjecture, proposed over 80 years ago by Lothar Collatz, claims that starting from any positive integer a 0, repeated iteration will eventually yield the value 1. Simple to state and test, this conjecture starts off as a leisurely stroll in the park, but quickly becomes an Olympic marathon, leaving one drained of all enthusiasm. This talk discusses an analysis of the Collatz Conjecture. We will discuss the basics of the Conjecture along with a brief history and several examples. We will discuss number types that are shown to decrease overall as well as problematic numbers that have not been shown to decrease. We will also derive the ”worst case” numbers, numbers that increase the most consecutive times, and where they are headed. Finally, we will examine a few inductive proofs for two infinite sets of numbers that absolutely work.

**Jennifer Williams will be speaking on “Parallel Lines in Hyperbolic Geometry”**

In this presentation we will introduce a non-Euclidean geometry known as Hyperbolic geometry and one of its models, the Poincare Disk model. We will then explore how parallel lines behave in Hyperbolic geometry using some theorems and the Poincare Disk model.

**Kelli Vosberg will be speaking on “The Birthday Problem”**

What are the odds that two people share a birthday given a set number of people? We will use probability to determine the chances of this occurring for groups of people of various sizes.