Supplemental Instruction (SI) sessions are peer-led workshops that complement classes that have a traditionally high fail rate. These professor-free workshops act as a safe place for students to freely ask questions, drive discussion, and work out problems. The goal of SI is to help students gain a deeper conceptual understanding of the concepts discussed in their classes. In this talk, I will discuss the Supplemental Instruction (SI) model, benefits for the SI student, and impact on the SI leader as well as my experience as a student and leader in the program.

I will be talking about an iterative method for finding a point in the intersection of several sets. The talk will be mostly visual, rather than symbolic, and will be accessible to students with little or no maths background. The talk may be most useful for non-maths students who are curious about what mathematics research is like and for current mathematics students who are curious about experimental mathematics. The research presented is part of a collaboration with Jonathan M. Borwein, Brailey Sims, Matthew Skerritt, and Anna Schneider.

Suppose you have 20 marbles labeled 1 to 20. You start with the marbles {1,2,3,4,5,6,7,8}. If a step consists of swapping 5 of your marbles with marbles not in your set, how many steps does it take to get to the set {3,4,5,6,7,8,9,10}?

This type of problem can be related to a family of graphs called the generalized Johnson graphs or J(v,k,i ) graphs.

The J(v,k,i ) graphs are highly symmetrical graphs that include Johnson and Kneser graphs, a notable example of which is the Peterson graph. The vertices of J(v,k,i ) are the k-element subsets of a v-element set, with two vertices being adjacent if and only if they intersect in a set of size i. Besides being awesome, these graphs have applications to coding theory and scheduling. With the help of professor John Caughman, we were able to prove formulas for distance, diameter and girth of this family of graphs. We will talk about our results, the experience of doing research for the first time, and possible directions for future research.

This expository talk will be an introduction to the new classes of topological objects called knotoids and virtual knotoids. First defined in 2011 by V. Turaev, knotoids are a generalization of classical knots obtained by restricting embeddings of S¹ in S²×I to an interval. Virtual knotoids are an analogous generalization of virtual knots. Virtual knots are representations of embeddings of S¹ in a thickened surface S_g×I first introduced in 1996 by Louis H. Kauffmann. Informally, a classical knot can be thought of as a closed knotted loop of string in 3-dimensional space. A virtual knot can be thought of as a classical knot but with the added structure of a “crossing that isn’t there” analogous to those encountered via immersions of realizations of non-planar graphs in S². In both the classical and the virtual theory, a pair of diagrams representing a knot or virtual knot are said to be equivalent if one diagram can be transformed into the other by a series of permitted maneuvers. These equivalence classes can be mapped into a category which serves as a knot invariant; a way to distinguish equivalence classes of knot diagrams. We will explore some of the more notable classical knot invariants such as the Alexander Polynomial and Kauffman bracket and advance to examples of the new collection of knotoid and virtual knotoid invariants published very recently by L. Kauffman in 2016. These objects are rich in topological and combinatorial structure and should appeal to students interested in Algebra, Graph Theory and Topology.