1. Grant Barkley

Title: Domino Tilings and Divisibility

Abstract: A domino tiling of a region in the plane is an arrangement of 2×1 domino tiles that completely covers that region. These tilings are an example of a more general construction in graph theory called perfect matchings. We are interested in counting the number of ways to cover certain regions with dominoes. In particular, we study how the divisors of this number are affected by the geometry of the region being tiled. A new type of sub-region is considered, the presence of which is associated to powers of two in the tiling count. We apply these results to the case of rectangular regions, and additionally note some other divisibility properties that arose in our research. (Faculty mentor: Dr. R. Liu)

2. Bryan Chu

Title:  Random Particles Exploration on a Graph and Quasi-Stationary Distribution

Abstract: Random particles systems can be made into a versatile tool to investigate the geometry of a given domain. We provide a specific non-Markovian particle system of which historical empirical measures generate a dynamical system on the space of probability measures on a given graph. We prove the local stability to the equilibrium of this evolution which matches the quasi-stationary distribution for the underlying dissipative Markov chain, also coinciding with the principal eigenvector of the weighted adjacency matrix for the graph.  (Faculty Mentor: Dr. M. Kang)

3. Geneva Collins

Title: Sangaku in Spherical and Hyperbolic Geometries

Abstract: During the Edo period (1603-1867 CE) Japan was almost completely closed off from the rest of the world and developed its own mathematical tradition called wasan.  Part of this tradition was to hang tablets, known as sangaku, in the eaves of a shrine or temple as an act of devotion.  The sangaku tablets were made of solid wood and each contained a colorful illustration of one or more results in Euclidean geometry. In this presentation I will explore the generalization of one of these ancient Japanese theorems in Euclidean geometry to both spherical and hyperbolic geometry. The basics of spherical and hyperbolic geometry will be explained. This research was conducted as part of the 2018 REU program at Grand Valley State University. (Faculty mentor: Dr. W. Dickinson)

4. Thomas Lee

Title: Security of Algebraic Variations on NTRU

Abstract: The NTRU crypto-system uses the coefficients of polynomials in certain polynomial rings to conceal confidential messages. The most common attack on the NTRU crypto-system is to find the private key through the LLL lattice reduction algorithm. Hence, security of NTRU is improved if the time to termination of LLL is increased, or if LLL does not return the private key. We studied the security effects of varying the polynomial rings on which NTRU is based to determine if there exists a more secure variant compared to the original system. Security of variants was measured by time to termination of LLL, whether or not LLL returned the private key, and if a message could be decrypted with some vector returned by LLL. We conclude that there is a polynomial ring that improves the security of NTRU with respect to average time to termination of the LLL algorithm. (Faculty mentor: Dr. E. Stitzinger)