My research lies in the area of algebraic number theory. More specifically, I study the solution sets of systems of equations over finite fields. I also have considerable experience working on machine learning based projects, having completed two summer internships in the cyber security division at Oak Ridge National Laboratory. I am also interested in math education.
Math words I think about: Finite Fields, Homogeneous Forms, Diagonal Forms, Polynomials in Many Variables, A-Systems, Chevalley-Warning Theorems, Machine Learning, Cyber Security, Adversarial Machine Learning, Growth Mindset, Mindfulness
(Submitted: joint with David Leep)
Abstract: This paper corrects an error in the proof of Theorem 1.4 (3) of our earlier paper, Further Improvements to the Chevalley-Warning Theorems. The error originally appeared in Heath-Brown's paper, On Chevalley-Warning Theorems, which invalidates the proof of Theorem 2 (iii) in that paper. In this paper, we use a new method to give a correct proof of Theorem 1.4 (3). The correction in this paper also xes the proof of Theorem 2 (iii) in Heath-Brown's paper. The proof in this paper provides slightly stronger estimates for some of the inequalities that were used in Further Improvements to the Chevalley-Warning Theorems.
(Contributed: Differential Equations Chapter)
Abstract: The 2023 Joint Mathematics Meetings was held in Boston, Massachusetts. A Professional Enrichment Program (PEP) titled “Introductory Python Jupyter Notebooks for College Math Teachers” brought together members of the authorship team of this classroom resource project.
(joint with David Leep)
Link: arxiv.org/abs/2207.04114
Abstract: We study lower bound estimates for the number of solutions of systems of equations over finite fields. Heath-Brown improved the lower bounds given by the classical Chevalley-Warning Theorems by excluding systems of equations whose solutions forms an affine space. We improved each of Heath-Brown's results and demonstrate sharpness in several cases.
(joint with Julie Vega and Andrés Vindas Meléndez)
Link: scholarship.claremont.edu/jhm/vol12/iss1/31/
Abstract: As mathematicians working in higher education we reflect on meritocracy and growth mindset with a focus on the relationship between the two. We also note the subtle differences between growth mindset and grit. Our reflection ends with suggestions for how to move forward in the math classroom and throughout the collegiate level.
(joint with Berat Arik and Jared M. Smith)
Link: https://www.osti.gov/servlets/purl/1486945
Abstract: In this work, we take a fundamentally different approach to the problem of analyzing a device for compromises via malware; our approach is OS and instruction architecture independent and relies only on having the raw binary data extracted from the memory dump of a device. Our system leverages a multi-hundred TB dataset of both compromised host memory dumps extracted from the MalRec dataset and the first known dataset of benign host memory dumps running normal, non-compromised software. After an average of 30 to 45 seconds of pre-processing on a single memory dump, our system leverages both traditional machine learning and deep learning algorithms to achieve an average of 98% accuracy of detecting a compromised host.
US 11,620384 B2, Independent Malware Detection Architechture, Filed Sept. 2018, Approved April 2023