Mathematics Students' Research Projects


In Celebration of Math Month

Three of Montana Tech’s currently enrolled math majors, Jacob Simpson, Michael Simon, and Hunter Boles did Undergraduate Research Projects this semester on interesting areas of mathematics. Jacob and Michael gave oral presentations on their work at the PiMUC (Pacific Inland Mathematics Undergraduate Conference) held at Gonzaga University in Spokane in March 2019. All three students are showcasing their works at the upcoming Techxpo on Thursday the 25th of April, 2019.

Jacob Simpson - Explorations of Two Cryptographic AlgorithmsJacob Simpson’s work is on “Explorations of Two Cryptographic Algorithms”. Whenever we send secure information electronically (say over the internet), our confidential data (from personal messages to financial information) are protected by cryptographic algorithms based on specific mathematical constrictions. The fundamental idea behind this is that of public encryption and private decryption keys, and that of a “trapdoor function”: a function which is easily computable  (within fraction of a second on today's computers) for every value in its domain, but whose inverse is practically impossible to compute in any reasonable sense using present day computing resources. For his research project, Jacob studied two such trap door functions. The first one, a number theoretic function devised by Rivest, Shamir and Adleman, forms the basis of the RSA algorithm. The basic idea of this well-known algorithm is expressed in terms of prime numbers and exploits the difficulty of factoring large integers into primes. Jacob extends the RSA algorithm to the Gaussian integers, which are a generalization of the (usual) integers. The second trapdoor function explored by Jacob is based on linear algebra, and forms the basis of the GGH algorithm of Goldreich, Goldwasser, and Halevi. Roughly stated, linear algebra is the study of the algebraic idea of linearity, and its manifold geometric applications. The GGH algorithm exploits a well-known computationally “hard” problem (the Closest Vector Problem) that arises in the theory of Euclidean lattices.

Michael Simon math researchMichael Simon’s project (“Analysis of a Root Finding Algorithm”) lies in the intersection of mathematical analysis (a deeper way of studying calculus), and (numerical) scientific computing. He explores an ancient Babylonian method for extracting square roots of positive real numbers. Using techniques from calculus, he examines why the ancient algorithm works to always produce the square root of a given positive real number to any predetermined degree of accuracy. He then extends the algorithm to the n-th root computation of positive real numbers, examines how fast the method converges, and establishes theoretical estimates of error for this algorithm.  Michael’s work showcases how every robust computational method is based on the solid footing of sound mathematical theories.

mathematics students' research projectsHunter Boles participated in an inter-disciplinary project with students and faculty in biology and mathematics, supported by the Research Assistant Mentorship Program at Montana Tech. The project, “Investigating the Responses of Weedy Plants to Climate Change,” included experiments to measure germination rates and morphological features of the plant small tumbleweed mustard (Sisymbrium loeselii) in the greenhouse. Seeds were included from four different countries: Hungary and Germany (where the plant is native) and the U.S. and India (where it is non-native). Hunter used statistical analysis to determine if the germination rates and shoot lengths were different based on country of origin. He found that the germination rate was significantly lower among seeds from Germany. He also determined that the shoot length was significantly higher for seeds from Hungary and India. Hunter is using the results of this experiment to construct a stochastic simulation model using Matlab to investigate the potential for this mustard to invade a habitat where it is non-native. The simulation begins with a ten by ten grid that is initially populated with a large number of a native species and a small number of mustard plants. The model incorporates measurements of the competitive advantage of mustard over the native species, among other parameters, to simulate the spread of both species over the grid over a specified period of time. Later, we will use this model to predict the spread of the non-native plant under a climate change scenario.

To know more details of the projects completed by these three math majors, come to Techxpo on 25th April and let them explain their work to you.