Invited Paper Session Abstracts - Open & Accessible Problems for Undergraduate Research

Knot theory is an excellent topic for undergraduate research, as the pictures are pretty, and you can involve the students in calculations right from the start to get them involved. Then they will be motivated to learn the relevant mathematics. We will talk about a variety of open problems in knot theory amenable to research by undergraduates.

Three numbers can be associated to a deformation class of closed curves on a surface S: the self-intersection number (this is the smallest number of times a representative of the curve crosses itself), the word length (that is the smallest number of letters, in a certain alphabet chosen a priori, one needs to describe the class ), and the length of the geodesic in the class (this is the length of the shortest representative of the class, where the way of computing length is chosen beforehand). The interrelations of these three numbers exhibit many patterns when explicitly determined or approximated using algorithms and a computer. We will discuss how these computations can lead to counterexamples of existing conjectures, to the discovery of new conjectures and to subsequent theorems in some cases. Many these results were discovered by or jointly with undergraduate and high school students.

To build the proverbial bridge between the mathematical sciences and the health sciences, one effective approach is to create resource-based research experience in data science. Studies show that undergraduate participation in data-science research projects has removed intra-institutional barriers encumbering access to data-resources and expertise for in-depth investigations. In particular, data dimension reduction approaches are being used to study many complex genomic and biomedical problems. This talk will present the design of a data-science research program and will share some of the resources.

In this talk we will describe two problems from elementary number theory. One concerns a theorem of Uspensky from 1929, and its possible generalizations regarding the partitioning of the natural numbers via slow Beatty sequences. The other concerns the so called very triangular numbers, their distribution among triangular numbers, and their various properties. We present a twin very triangular number theorem, a version of Bertrand's postulate for very triangular numbers, as well as some other intriguing results, all in an effort to entice undergraduate students to work on similar problems for other special subsets of the natural numbers.

One interesting enumeration problem in combinatorics asks how many trees T contain a prescribed smaller tree t. During the summers of 2010, 2011, and 2012, my teams of REU students studied this question for specific families of trees using two different definitions of what it means for one tree to contain another tree. In this talk, I’ll discuss what we know about this problem and describe other variations that are open for exploration.

Sports analytics is an exciting and accessible research area which draws student interest. Among the many mathematically inspired sports ranking systems, the Colley and Massey methods are relatively simple and can easily be introduced to undergraduate students who have taken a linear algebra course. At their most basic level, these methods are useful for sports rankings, but they are not particularly strong at predicting future outcomes of games. One way to improve these methods for ranking and predicting future outcomes is by introducing weights to these systems and using cross-validation to help determine the quality of our models. In this talk, we will share some of the undergraduate research projects done using linear algebra-based sports ranking models and share general advice on mentoring undergraduate research.