1. Maggie Figueroa

Title: Breaking Cryptography – the Discrete Log Problem

Abstract: The goal of cryptography is to allow ciphertext to be decoded only by the intended recipient. The privacy of the message relies heavily on the key used in the encryption function. Once a key is obtained, the decryption function follows easily. Cryptosystems such as the ElGamal system and the Diffie-Hellman key exchange protocol maintain their privacy through the discrete logarithm problem. This paper expands on the methods used in order to maintain the privacy of the key used. I will also introduce Pollard’s Kangaroo method, a concept used in solving the discrete logarithm problem and how this allows us break the Diffie-Hellman key exchange protocol and the ElGamal system. (Faculty mentor: Dr. E. Stitzinger)

2. Olivia Trogdon

Title: Finding the Probability that the Population Goes Extinct

Abstract:Predicting a random event may sound like an oxymoron. However, using probability theory we can sometimes make a precise prediction about such events. Probability theory allows us to model random events, which can provide possible outcomes of what may (or may not) happen in the future. In this work, we use probability theory to determine the precise probability that a certain class of population models go extinct. Taking advantage of the notion of generating function, we record the essential information from all generation’s population size in these deterministic functions. Studying the relationship among the generating functions for different generations’ population size, we then relate the extinction probability to the fixed point of the generating function from offspring distribution and determine what is the probability that a future population will cease to exist. (Faculty mentor: M. Kang)

3. Benjamin Wendt

Title: Teaching Topology in Middle School

Abstract:For those outside the field of Mathematics, there is little understanding of what a Mathematician is or does. Thus, it is uncommon for a child to desire to be mathematician when they grow up. My research involved finding examples and research of how the topic of Topology can be used to expand student’s knowledge of how Mathematics is done and what a Mathematician does. I also created two descriptive lesson plans (with tasks backed up by evidence) that a middle school teacher can use. (Faculty Mentor: R. Sazdanovic)

4. Gautam Nagaraj

Title: Black Hole Masses for Type I AGNs from Chandra COSMOS Legacy Survey: A Classic Case of Finding the Right Fit

Abstract:Tight local relations between supermassive black hole (SMBH) masses and galaxy properties have established the fundamental connection between SMBHs and their host galaxies. However, in order to better understand the coevolution of SMBHs and their host galaxies over cosmic time, we need measurements of black hole masses, active galactic nuclei (AGN) luminosities, and galaxy stellar masses from sizable samples of AGN covering lower luminosities than the brightest quasars spanning a wide redshift range. In this study, we report masses of the SMBHs of 224 Type I AGN from the Chandra COSMOS Legacy Survey as determined by the line widths of Mg II 2798, Hb 4862, and Ha 6564 via scaling relations derived from reverberation mapping. Preliminary comparison with host galaxy luminosities and stellar masses suggests an increase in Eddington ratio with redshift, consistent with previous studies. In addition, our derived SMBH masses fall above the local AGN MBH–M* (galactic stellar mass) relation from Reines & Volonteri (2015), but it is still not clear whether this results from redshift evolution of the MBH–M* relation or from the incompleteness of the spectroscopic surveys available. (Faculty Mentor: F. Fornasini, F. Civano)