Paul N's Senior Project Blog
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Project Title: Unifying Mathematical Perspectives: Connecting Seemingly Unrelated Areas of Math BASIS Advisor: Lyle Koonce Internship Location: University of Texas San Antonio Onsite Mentor: Dr. Sandy Norman |
Project Abstract
As someone with a strong background in calculus but lacking experience in other fields of mathematics, the goal of this internship is to learn from professors in the field to expand my base of knowledge in mathematics into other fields such as real and complex analysis, number theory, and topology. Most importantly, I hope to be able to make connections between these fields and learn techniques of mathematical reasoning in the process in a course that models what a real university-level mathematics course would be like. The critical thinking and connection skills I learn here are exactly what college classes in theoretical mathematics seek to teach, and also exactly what is needed to have a successful career in mathematics. This project is taking place at the University of Texas at San Antonio, since I have taken various summer courses at UTSA and have gotten well-acquainted with the campus and staff. I am engaging in various 1:1 discussions with Dr. Sandy Norman and learning material that he would teach to his actual classes. To help simulate the college learning environment, I am also taking notes, doing textbook readings, and working out examples with him. At the end of this project, I should see dramatic improvement in both the disciplines that I am new to and those which I already have experience with, as well as countless other fields of science, since topics such as Fourier analysis have direct applications in the natural sciences and physics especially.
#10 – Metrics and Non-Euclidean Derivatives
Hi, and welcome to the final week of my Senior Project. My original goal for this week was to find a way to use pseudoconics to analyze Non-Euclidean geometries via metrics, and while this did not end up happening, I ended up finding an arguably much more interesting application for pseudoconics within Non-Euclidean geometry. Firstly,... Read More
#9 – Using Pseudoconics to Analyze Non-Euclidean Geometries
Hi, and welcome to week 9 of my Senior Project! This week was cut short by a visit to Michigan, but I still got a lot more done than I expected to. The first thing that I did was that I worked on creating a graph to visualize the transformation between a Euclidean and Non-Euclidean... Read More
#8 – Finding Non-Euclidean Surfaces From a Graph
Hi, and welcome to week 8 of my Senior Project! This week, I focused on finishing my task from last week to discover a method of finding a Non-Euclidean surface upon which the geodesic projection of a curve onto the surface would produce the parametric projection of another curve. Turns out, this task is a... Read More
#7 – Exploring Pseudoconics with Non-Euclidean Geometry
Hello, and welcome to week 7 of my Senior Project! This week, I worked with my advisor to learn the basics of analyzing Non-Euclidean geometry through 3D surfaces, and then I applied this knowledge to find a way to connect pseudoconics to Non-Euclidean geometry through geodesics. For those who are unfamiliar with Non-Euclidean geometry, Euclidean... Read More
#6 – Pseudoconics and Conic Sections
Hello, and welcome to week 6 of my Senior Project! If you recall, last week, I mentioned that I began learning about Non-Euclidean geometry to see if there was some way to connect it to pseudoconics. So far, I have had two ideas on how to explore this connection: my first idea was that it... Read More
#5 – Applications of Pseudoconics and Non-Euclidean Geometry
Hello, and welcome to week 5 of my Senior Project! This week, the three main topics that I focused on exploring were 3-dimensional linear pseudoconics, applying pseudoconics to answer a particular question, and learning the basics of Non-Euclidean geometry. In the last 2 posts, apart from wrapping pseudoconics around spheres, all of my discussions about... Read More
#4 – 2D and 3D Pseudoconics with Eccentricity
Welcome to week 4 of my Senior Project! In last week's post, I mentioned that I had begun working on pseudoellipses and pseudohyperbolas in 2 dimensions. This week, me and my advisor were able to figure out these generalized pseudoconics in 2 dimensions as well as in 3 dimensions, and the results are astonishing. First,... Read More
#3 – More Pseudoconics
Welcome to week 3 of my Senior Project! At this point, we have begun to completely deviate from the original plan for our Senior Project and instead have been putting tons of effort into studying pseudoconic curves and learning more about them. There is so much new mathematics to be discovered with these curves! First... Read More
#2 – Pseudoconic Curves
Welcome to week 2 of my Senior Project! This week's topic was proofs and derivations, and while we did do some deep dives into number theoretical proof techniques such as infinite ascent and descent, there was something much more interesting that we stumbled upon this week that I would like to share. Many people who... Read More
#1 – Introduction and Deriving the Fibonacci Sequence – 3/9/25
Welcome to my Senior Project! My name is Paul, and in this project, I am exploring the interconnectedness of various disciplines of mathematics by learning about these disciplines and studying the ways in which they can be related to each other. Specifically, I will be studying the fields of sequences, proofs, number theory, geometry, algebra,... Read More