By: Leah Widlarz & Sydney Vitalbo
Around the end of our Fall 2024 semester, we were both presented with the opportunity to participate in a math research project with Dr. Laubacher, our Calculus 3 professor. He presented it as a continuation of his previous research in homological algebra. Even though neither of us had been involved in math research before, we decided to take him up on his offer and try it out!
We worked on this research for the entire spring semester, about 16 weeks. Once a week, we met with Dr. Laubacher, where he would introduce new concepts through definitions, examples, and proofs– which are logical arguments that demonstrate the truth of a mathematical statement. In some cases, we would then additionally work on another proof relating to the particular concept we were going over that day. Some of the concepts we went over were rings, fields, modules, chain complexes, and k-algebras– basic structures in the field of Homological Algebra. On our own, we completed proofs provided by Dr. Laubacher that aided in working up to the main point of the research, such as The 3 x 3 Lemma and The 5 Lemma. Our main task was to prove that a presimplicial homotopy defines an equivalence relation, meaning that we needed to prove that a presimplicial homotopy is reflexive, symmetric, and transitive. To put it in simpler terms, an example of an equivalence relation is the equals sign. The equals sign is reflexive, which means that A = A. The equals sign is also symmetric, which means that if A = B then B = A. Being transitive means that if A = B and B = C, then A = C. An example of something that is not an equivalence relation is the less than symbol. The less than symbol is not reflexive, as A < A is false. The less than symbol is not symmetric or transitive either. Equivalence relations are incredibly important to mathematics as a whole because they let us know how we can use certain operators, like the equals sign and the less than symbol. We were able to prove that presimplicial homotopies define equivalence relations, which was reflected in Lemma 3.4 of the paper on our research.
Toward the end of our project, we focused on creating a poster to present at St. Norbert’s Undergraduate Research Forum. Using LaTeX, we made the poster which included a variety of definitions that we had used over the course of the project. More specifically, these definitions gave an overview on fields, rings, modules, and k-algebras. Two of the main sections were on morphisms and homotopies, ultimately leading up to the main result: that a presimiplicial homotopy defines an equivalence relation. The purpose of the main result was to show that even though objects might not look and/or seem equivalent, they can be. For example, just as the fractions 1/2 and 2/4 look different, they actually mean the same thing and can be used interchangeably in calculations. We added visualizations on our poster to help highlight this idea. A challenging part of presenting our research was being creative in how we explained it to our audience, taking into account that not many would have a background in theoretical math and therefore might not fully understand the topics within our research.
We did a whole lot of math, but it was a ritual at our weekly meetings to taste-test Dr. L’s banana oatmeal chocolate chip cookies! While we were working on this project, he was in the process of perfecting his recipe, and every week we snacked on the latest batch while talking about our project. After presenting our poster at the Research Forum, we earned an Honorable Mention award! Our paper was also accepted to be published in the journal Categories and General Algebraic Structures with Applications. It will be coming out in 2025! Even if we don’t end up pursuing careers in theoretical mathematics, we’re so grateful for this opportunity to try something new and be a part of the math research process! We encourage anyone interested in research to seek out opportunities like this because it truly was a fun time. If an opportunity happens to fall into your lap, as this did to us, we encourage you to take it enthusiastically and with an open mind.
If you are interested in reading our paper, click on this link: https://arxiv.org/abs/2404.06368
