Interactions between single nucleotide polymorphisms (SNPs) and complex diseases have been an important topic throughout epidemiological studies. Previous genome-wide-association studies have mostly focused on gene variables at a single locus, but did not obtain strong evidence for SNPs with complex diseases such as cancer. In our project, we performed a focused candidate gene study to test the interaction of multiple SNPs with the risk of different types of cancer. Using the R package algstat, developed by Kahle, Garcia-Puente, and Yoshida, we developed an algorithm which can test for independence between several variables and the disease. We applied our methods to the study of gene-gene interaction on cancer data obtained from the European case-control study Gen-Air. Previously pairs of SNPs were tested for independence versus certain cancers in this study, but it was not computationally feasible to test triplets of SNPs with the methods available. The algorithm developed was significantly faster so we were able to test for independence of SNP triplets and found strong evidence to reject independence of many triplet combinations of SNPs with the disease, suggesting correlation between having a certain combination of SNPs and risk of certain cancers. These results can be used for further testing of associations of these SNP combinations and certain cancers. Outside of the study of SNP-cancer association, this algorithm can be easily adjusted to perform other studies using arbitrary log-linear statistical models.

 In 1960, Graham Higman postulated that the number of conjugacy classes of the group of Unitriangular Matricies of size n over F_q is polynomial with respect to q. To better understand these groups, we explored the group of Unitriangular Matricies of size n over F_2. In particular, we sought to understand the relationship between such matricies and the set of simple graphs on n verticies. The relationship between these two sets defines a multiplication between the edge sets of simple graphs on n verticies. Hence, we awnser the following question: if G_X=([n],E_X) and G_Y=([n],E_Y) then what is G_XY=([n],E_XY)?

The Heesch number of a shape (or tile) is the maximum number of complete "layers" said shape can form around itself in a tiling. Regular hexagons can tile the two-dimensional plane in exactly one way, so tiling theorists impose "matching rules" on them to make things interesting. A matching rule on a regular hexagon dictates which of the six sides, when indexed, may touch which other sides and with what orientation. There are four possibilities: two sides cannot match, they can match with only the same orientation, they can match with only opposite orientations, or they can match in either manner. This yields approximately 4.4 trillion rule sets. The ultimate goal of this project is to use the computing cluster to find which of these rule sets give rise to finite Heesch numbers, but the existing code that exhaustively checks each rule set takes an exorbitant amount of time if the Heesch number is large or the rule set tiles the plane (has an infinite Heesch number). We are building filters that identify which rule sets tile the plane in a k-tile-transitive manner, currently as far as k=4. We hope that this will allow us to run the exhaustive Heesch number checking code on a much smaller data set, which would allow us to see results in a reasonable amount of time. 

Mathematics, in its own way, is a language, a new, and often abstract, way of viewing the world.  Through examining knot theory, we are able to contribute information to this rapidly expanding field of study, and apply our results to other research disciplines. A mathematical knot is a closed circle embedded into 3-dimensional space, where there exists crossings and regions. We examine the number of region crossing changes necessary to produce an alternating diagram, and define our new invariant as the region dealternating number. This property leads to an invariant, similar to the dealternating number, but offers new insight into the manipulation of knots. We expand our invariant under the connect sum theory to better understand the behavior of the region dealternating number. In particular, we apply the reasoning built for this invariant to prove that the (warping) span of any knot is at most 2, answering a previously unsolved question for warping span. We hope to continue to further these concepts and apply them to other families of knots and subsequent invariants. 

As sensors decrease in size, cost, and power, more and more sensors are being used to generate big data sets – a concept called the “Internet of Things.’’ These data sets must be analyzed to extract meaningful information from it. Consequently, tools or techniques for analyzing large data sets from sensor networks are important. This idea of building a data pipeline for the Internet of Things is what our project is based on. The purpose of our research project was to construct a database to store distributed sensor network data and to enable analysis of the data. Sensor data consisted of information from devices that were connected to nearby wireless networks. The data were gathered using a low-cost and low-power microcomputers, in our case, Raspberry Pis, and were collected from the sensors over several days. These were stored in the database for analysis. The large-scale data sets that result make a database necessary for analysis. The concept of gathering big data sets from sensor networks and performing data analysis can be used in optimization, prediction, control, and other applications. Data pipelines also enable analysis at different time scales from data acquisition. Designing tools for our more data-rich world is essential to understanding the mass of data that we gather each day from the sensors all around us.