Course Instructors

Gross Hall (TBA)

Orsola Capovilla-Searle

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 arenít allowed to cut and re-glue it. In that case, it is natural to ask when one knot is the same as another if Iím 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.

SWiM Counselors

Caitlyn Booms

Emily Castner

Suna Kim

Katie Taylor