People often ask how you do research in pure mathematics. However you do it, it isn't ... easy. There is often a lot of background to learn before one even gets to a point of being able to even consider a problem. And even if the problem is solved, one can still be left wondering what just happened. One might be able to follow a proof line by line, but it is easy to get lost in the abstractness of everything and lose any intuitive understanding.
But in abstractness is also beauty. Maybe part of that beauty comes from simply being able to prove everything. In the abstractness is a certainty that whatever ``just happened'' can be made rigorous and precise. And this is exciting. Some may not experience this excitement with pure and abstract mathematics. If you need to see applications and always ask ``what is this good for?'', then maybe this project isn't the right one for you.
Mathematical rigor can be a challenge for many students of mathematics. One might have an intuitive understanding of a concept, but the formalities can get in the way of making progress.
The first part of this project will be an introduction to the topic of permutations. We will first gain a solid intuitive understanding of permutations. We will learn how to multiply permutations, find their parity, find orders of permutations, find the lengths, and other things. Most of this will be explored with examples. We will quickly learn the notation and how to do computations.
In the second (and main) part we will dive deeper and seek proper and precise definitions for the concepts involved. We will create Theorems and will attempt to prove them. We will see how choosing one definition over another will have an impact on the proofs.
This project is primarily about making proofs. Working in the context of permutations simply means that the intuitive aspects hopefully will not cause too much confusion.
Depending on how fast we move through the first two parts, we will end the project by studying more general group theory and studying how the topic of permutations fit within the abstract concept of a group. If there is interest, we may also explore how the topic of permutations is taught in modern abstract algebra courses.
Takeaways: This project gives one experience of how it is to do research in pure mathematics. You will experience having to come up with definitions, theorems, and proofs. If you like doing mathematics because of the purity and preciseness of mathematics, then this project may be for you.
Prerequisites: While this project does not require a lot of background, we do expect the students to have completed some level of Discrete Mathematics and/or Linear Algebra. It may be helpful if you have already seen things like induction proofs. You do not need to have taken group theory or abstract algebra for this project. It is okay to be scared :)
Biography: Dr. Thomas Madsen is an Associate Professor at Youngstown State University. He received his PhD in pure mathematics from the University of Oklahoma in 2014. Dr. Madsen loves studying and teaching mathematics at all levels. For students studying pure mathematics one can get the impressions that mathematics is all about taking classes. One can feel a long way from doing any level of research. Dr. Madsen believes that an early exposure to mathematical research can help generate and maintain an interest in mathematics and he hopes that this REU will energize and excite younger students into the world of mathematics.