Orsola CapovillaSearle
Understanding knots
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 reglue 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.

Erika Ordog
The Seven Bridges of Konigsberg
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.
